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+H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf filter=lfs diff=lfs merge=lfs -text +X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf filter=lfs diff=lfs merge=lfs -text +_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/0NE0T4oBgHgl3EQf_QLA/content/tmp_files/2301.02825v1.pdf.txt b/0NE0T4oBgHgl3EQf_QLA/content/tmp_files/2301.02825v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bda268947b6990b9296f8fa12f07c233f79505e2 --- /dev/null +++ b/0NE0T4oBgHgl3EQf_QLA/content/tmp_files/2301.02825v1.pdf.txt @@ -0,0 +1,5981 @@ +Discrete holography in dual-unitary circuits +Llu´ıs Masanes∗ +London Centre for Nanotechnology, University College London, UK +Department of Computer Science, University College London, UK +January 10, 2023 +Abstract +We introduce a family of dual-unitary circuits in 1+1 +dimensions which constitute a discrete analog of confor- +mal field theories. +These circuits are quantum cellular +automata which are invariant under the joint action of +Lorentz and scale transformations. +Dual unitaries are +four-legged tensors which satisfy the unitarity condition +across the time as well as the space direction, a property +that makes the model mathematically tractable. +Using +dual unitaries too, we construct tensor-network states for +our 1+1 model, which are interpreted as spatial slices of +curved 2+1 discrete geometries, where the metric distance +is defined by the entanglement structure of the state, fol- +lowing Ryu-Takayanagi’s prescription. The dynamics of +the circuit induces a natural dynamics on these geome- +tries, which we study for flat and anti-de Sitter spaces, +and in the presence or absence of matter. +We observe +that the dynamics of spaces with two or more particles +differs from that of zero or one, suggesting the presence +of black holes. But this contrasts with the fact that the +family of models appears to be non-chaotic. +1 +Introduction +In 1997 Juan Maldacena proposed a duality between (i) +certain theories of quantum gravity with negative cosmo- +logical constant in d + 1 spacetime dimensions, and (ii) +conformally-symmetric quantum field theories (CFT) in d +spacetime dimensions [1]. This duality is also known as +the AdS/CFT correspondence, because when the cosmo- +logical constant is negative, the ground state of classical +gravity is anti-de Sitter space (AdS). Important aspects +of the correspondence were soon elaborated by other au- +thors [2, 3], and today Maldacena’s paper gathers 22889 +citations. +A central feature of this duality is that the spacetime +curvature on the gravity side corresponds to the entangle- +ment structure and its dynamics on the CFT side [4–13]. +In [13] this is expressed as holography being a hydrody- +namic description of the boundary entanglement with en- +tropies as its macroscopic phase space. This opens the pos- +∗l.masanes@ucl.ac.uk +sibility of describing the entanglement dynamics of other +many-body systems (different than CFT) in terms of ge- +ometry with an extra dimension. One direction where this +exploration has been fruitful is that of discrete hologra- +phy, where one considers quantum systems with discrete +degrees of freedom (e.g. spin chains) that are dual to dis- +crete geometries [14–26]. Some advantages of the discrete +approach are: mathematical tractability and rigour, more +straightforward simulation on classical and quantum com- +puters, and more accessible experiments. +In this work we present a family of quantum circuits +which are invariant under the joint action of Lorentz +and scale transformations. (Recall that invariance under +Lorentz transformations alone is impossible on the lattice.) +Our circuits are constructed with any given two-site dual- +unitary gate +and its complex conjugated +following +the pattern +T = · · · +· · · +(1) +so that causality is strictly respected. This type of dynam- +ics is called quantum cellular automata (QCA) [27–30], +and it is the discrete-spacetime version of quantum field +theories. (Recall that in “lattice field theory” time is con- +tinuous, which implies the loss of either causality or unitar- +ity, both respected by QCAs.) In this work we introduce +conformal QCAs in 1+1 dimensions, and leave the general- +isation to higher dimensions for the future. To our knowl- +edge, these are the first QCAs whose evolution operator T +has a form of Lorentz and scale invariance. An essential in- +gredient of these QCAs are dual unitaries: four-legged ten- +sors which satisfy the unitarity condition across the space +as well as the time direction (7-8), implying the unitarity +of circuit (1). Dual-unitary circuits have recently attracted +a lot of attention because they provide mathematically- +tractable models of quantum chaos [31–34]. +After a detailed presentation of conformal QCAs (Sec- +tion 2) we explore the holographic properties of these mod- +els. +In Section 3 we analyse how tensor-network states +for 1+1-dimensional QCAs define 2+1 discrete geometries +with metric distance defined by the entanglement struc- +ture of the state, following Ryu–Takayanagi’s prescrip- +1 +arXiv:2301.02825v1 [hep-th] 7 Jan 2023 + +tion [4–7]. General Relativity in 2+1 dimensions does not +have propagating gravitational degrees of freedom, but it +can have non-trivial dynamics for the manifold boundary, +which is what we analyse in this work. +The fact that +our tensor-network states are constructed with the same +dual-unitary tensor as the evolution operator T makes the +dynamics of these discrete geometries very natural (see +Figure 1), avoiding the need of any duality map. We con- +struct tensor networks states representing AdS space, AdS +double-sided black hole and thermal AdS. The discreteness +of time in QCAs implies the absence of ground and ther- +mal states, which opens the possibility of describing other +spaces, like the finite piece of flat space with boundary +shown in Figure 1. +In Section 4 we analyse the dynamics of these spaces in +the presence of matter, and observe two very different be- +haviours. When there is only one particle, this oscillates +from one point of the boundary to its antipodal counter- +part, undergoing an orbit with a short period. That is, the +total state evolves similarly in the cases of zero and one +particle. On the other hand, when there are two particles +or more, the dynamics appears to be of scrambling type, +exploring a large space of states which do not close an or- +bit. The complexity of this dynamics justifies describing +these states by their time average, leading to a mixed state. +Also, it suggests the possibility that these states contain +some probability amplitude of having a black hole. How- +ever, simulations indicate that conformal QCAs do not +have quantum-chaotic dynamics [32]. That is, even if we +construct a circuit (1) with a random dual unitary +(which displays Wigner-Dyson level statistics), the par- +ticular structure of the evolution operator (1) produces +Poissonian level statistics. This is a very anomalous phe- +nomenon that deserves to be explored with more detail in +future work. +2 +Conformal QCAs +In this section we introduce conformal QCAs and discuss +some of their properties, like scale and Lorentz invariance. +2.1 +Quantum circuits in 1+1 dimensions +Consider a chain of n sites labelled by integers x ∈ Zn +modulo n, where n is multiple of four. In each site there is +a quantum systems with Hilbert space Cq, so the Hilbert +space of the chain is Hn = (Cq)⊗n. +We denote by A +the algebra of matrices acting on Cq, and by Ax,y,... the +algebra of matrices acting on sites x, y, . . . ∈ Zn. +The +dynamics of the system is given by the time-translation +operator T defined through the circuit +T = +�� +x oddvx +��� +x evenux +� +(2) += · · · u +v +u +v +u +v +u +v +u +v +x=−1 +0 +1 +2 +x=−1 +0 +1 +2 +· · · +t +(3) +where the two-site operators ux, vx ∈ Ax,x+1 are unitary. +The action of T on a local operator at the origin a ∈ A0 +is +TaT † = +u† +u +a +v +v +v† +v† +∈ A−1,0,1,2 , +(4) +reflecting the causality of T. +The (two-site) space- +translation operator +S = · · · +· · · +(5) +allows for writing the translation-invariance of the dynam- +ics as +ST = TS . +(6) +2.2 +Dual unitaries +If we represent the two-site unitary u ∈ A0,1 as a four- +legged tensor u = +then we can write its transpose as +uT = +and its conjugation by the swap operator s = +as sus† = +. Also, we denote complex conjugation with +a darker shade u∗ = +, so that the Hermitian conjugate +is u† = +. +With this notation we can say that +is +unitary if it satisfies the two equivalent conditions += +and += +. +(7) +Also, we say that +is a dual unitary if, in addition to +unitarity, it satisfies the two equivalent conditions += +and += +, +(8) +which can be phrased as “unitarity in the space direction”. +Dual unitarity implies that, for any traceless local op- +erator a ∈ A0, the partial trace of uau† ∈ A0,1 on site 1 +vanishes, +tr1(u0a0u† +0) = a += a = 10 tr(a) = 0 . +(9) +In other words, the operator uau† ∈ A0,1 is a linear com- +bination of terms of the form 10 ⊗ c1 and b0 ⊗ c1 with +traceless b, c, but not of the form b0 ⊗ 11. That is, all +terms in uau† act non-trivially on site 1. +Returning to our model, if u, v are dual unitaries then a +local operator a ∈ Ax on an even site x evolving as T taT −t +grows towards the right at maximal speed, which in lat- +tice units is c = 2. The operator T taT −t may also grow +towards the left and develop a highly non-local and com- +plex structure, but that is not necessary. Alternatively, +if the initial operator a ∈ Ax is located on an odd site x +then T taT −t grows towards the left at maximal speed −2. +2 + +t = 0 +0 +t = 0.5 +0 +t = 1 +0 +t = 1.5 +0 +t = 2 +0 +t = 2.5 +0 +t = 3 +0 +t = 3.5 +0 +t = 4 +0 +t = 4.5 +0 +t = 5 +0 +1 +Figure 1: Dynamics of flat space with a boundary of n = 20 sites. Each rectangle represents a tensor-network +state of the QCA, where black dots in a blue line are the free legs that correspond to Cq systems, and site x = 0 is +marked. At each time step t we depict the state T t|Ψfl⟩ ∈ H20, and at semi-integer times t we depict TevenT ⌊t⌋|Ψfl⟩, +where ⌊t⌋ is the largest integer less than or equal to t. +The sequence produces an orbit of period ∆t = 5. +If +we approximate one of these rectangles by an infinite spatial strip, then this dynamics resembles that of General +Relativity, where the width of the spatial strip decreases until it collapses and bounces back (see [35,36]). +Hence, we see that the even/odd location x ∈ Zn plays +the role of a momentum ±2 quantum number. In sum- +mary, every perturbation in a dual-unitary circuit grows +at maximal speed towards the right, the left, or both, as +in CFT. +2.3 +Free particles and quantum chaos +In order to simplify the discussion of this subsection (only) +we restrict ourselves to circuits (2) with v = u. The first +thing to do when we are given a dual unitary u = +is to +obtain the spectral decomposition of the maps Ω+ : A0 → +A1 and Ω− : A1 → A0, defined as +Ω+(a0) = 1 +q tr0(u0a0u† +0) = 1 +q a +, +(10) +Ω−(a1) = 1 +q tr1(u0a1u† +0) = 1 +q +a . +(11) +The eigenvectors of Ω+ +with unimodular eigenvalue +Ω+(e) = eime satisfy +TexT † = eimex+2 , +(12) +for all even x. Analogously, the eigenvectors of Ω− with +unimodular eigenvalue Ω−(e) = eime satisfy +TexT † = eimex−2 , +(13) +for all odd x. +In the dual-unitary and QCA literature, +these operators are respectively called right/left-moving +solitons [34] and gliders [28]. +When acting on a state, +these operators can create a free particle with velocity ±2 +and quasi-mass m. +The eigenvectors Ω±(e) = λe with eigenvalue modulus +less than one |λ| < 1 grow under T in a scrambled fashion +which fills up all the lightcone. This dynamics displays +many signatures of quantum chaos, including the profile +of the spectral form factor [31–33]. +2.4 +Definition of conformal QCA +Let us define the family of circuits introduced and analysed +in this work. For any given dual unitary u = +we define +the following time-translation operator +T = · · · +0 +0 +· · · +(14) +where site x = 0 is marked. Note that, for any four local +unitaries a, b, c, d, the new dual unitary +u′ = a0b1u c0d1 = +a b +c d +(15) +defines a new circuit T ′ via (14) which is equal to T up to +a local change of basis, +T ′ = (· · · a0b1d2c3a4 · · · )T(· · · a0b1d2c3a4 · · · )† . +This reminds the structure of a gauge theory, but it is not +the same. +3 + +Another property of the structure of T is that it pro- +duces a cancellation of the phases of travelling solitons, +forcing them all to be massless. +That is, if e0 satisfies +u0e0u0 = eime1 then TexT † = ex+4 for all even x (and +analogously for odd x). Recall that CFTs can only have +massless particles - however this is not sufficient to be con- +formal. In the next section we show that T is invariant +under scale transformations. +2.5 +Scale invariance +The discussion in this section requires the size of the chain +n to be a multiple of 8. We start by defining the contrac- +tion isometry C : Hn → H n +2 as +C = q− n +8 +� +· · · +0 +· · · +� +(16) +which maps an n-site chain to an n +2 -site chain. The par- +ticular structure of operators C and T together with the +dual unitarity of their building block +allows to easily +calculate the product CT in a manner that is independent +of the choice of +: +CT = · · · +· · · += · · · +· · · += · · · +· · · += · · · +· · · +(17) +Continuing in a similar fashion we obtain +CT = · · · +· · · += · · · +· · · +(18) +The above can be synthesised as +C2nT 2 +2n = TnC2n , +(19) +where we have added a subindex to the operators to indi- +cate the size of the chain where they act on. The above +equation tells us that, the action of Cn produces a rescal- +ing of space and time by a factor 1 +2. +By using the dual unitary constraints (7) and (8) we +can calculate the action of C on a local operator a ∈ Ax, +CaxC† = +� 0 +if x = 0, 1, 2, 3 mod 8 +a x +2 +if x = 4, 5, 6, 7 mod 8 +. +(20) +This action depends on the position x in a very non- +smooth way. However, the action of C is smooth on oper- +ators having a good field-theory limit +Φn(x) = +� +y∈Zn +ϕ(y − x)ay , +(21) +where ϕ(y) is a smearing function (e.g. a gaussian) centred +around the origin and spreaded over a large number of +lattice units. In this case we have +CΦn(x)C† ≈ 1 +2Φ n +2 ( x +2) , +(22) +where the subindex of Φn stresses that the field operators +on the left- and right-hand sides act on chains of different +sizes. +It is possible to construct contraction isometries with +scale factor different than +1 +2. +This can be achieved by +separating the vectors +in (16) by a length different +than 4. +2.6 +Lorentz transformations +In this subsection we give a brief summary of Lorentz +transformations on conformal QCAs, and refer to Sec- +tion 5 for the complete presentation. +Lorentz transfor- +mations are more clearly discussed in an infinite chain, so +here and in Section 5 we assume n = ∞. +In Section 5 we define the isometry Rl : H∞ → H∞ +which jointly implements a contraction and a spacetime +transformation which resembles a Lorentz boost towards +the right, with velocity v parametrised by the positive +integer l as +v = +2 +√ +4 − 2l + l2 . +(23) +These transformations commute with the translations in +the diagonal direction (x, t) = (1, 1), +Rl(ST) = (ST)Rl . +(24) +The action of Rl on a local operator a(x, t) := T taxT −t +can be informally described as +Rla(x, t)Rl = +� a(x′, t′) +for most (x, t) +complicated +for a few (x, t) +(25) +4 + +where, in the regime |x| ≫ l, the transformed coordinates +(x′, t′) can be written as +x′ += +� +1 − 1 +l +� +x + 2 +l t +t′ += +� +1 − 1 +l +� +t + 1 +2lx +� +. +(26) +The label “complicated” in (25) stands for a transforma- +tion that is not purely spacetime, as in the first case. Next, +note that transformation (26) preserves Minkowski’s met- +ric up to a scale factor +(ct′)2 − x′2 = +� +1 − 2 +l +� � +(ct)2 − x2� +. +(27) +(Recall that the speed of light is c = 2.) Now, we can +remove the scale transformation from (26) by dividing the +new coordinates (x′, t′) by the scale factor +� +1 − 2/l, so +obtaining the pure Lorentz transformation. Once this is +done, we simplify the first equation by imposing x = 0, +obtaining +x′ +� +1 − 2/l += +2/l +� +1 − 2/l +t = +v +√ +1 − v2 t , +(28) +where the second equality follows from the standard form +of a Lorentz transformation with velocity v and x = 0. +This second equality can be use to isolate v as a function +of l and confirm the relation (23). +In Section 5 we also define the isometry Ll : H∞ → H∞, +which jointly implements a contraction and a spacetime +transformation which resembles a Lorentz boost towards +the left, with velocity v parametrised by the positive inte- +ger l as +v = +−2 +√ +4 − 2l + l2 . +(29) +The action of Ll on a local operator a(x, t) := T taxT −t is +Lla(x, t)Ll = +� a(x′, t′) +for most (x, t) +complicated +for a few (x, t) +(30) +where, in the regime |x| ≫ l, the transformed coordinates +(x′, t′) can be written as +x′ += +� +1 − 1 +l +� +x − 2 +l t +t′ += +� +1 − 1 +l +� +t − 1 +2lx +� +. +(31) +Note that this transformation also preserves Minkowski’s +metric up to the same scale factor +� +1 − 2/l. +Like with the scale transformations described in the pre- +vious subsection, the action of Bl and Ll becomes smooth +on operators of the form Φ(x, t) = T tΦ(x)T −t, where the +smeared operator Φ(x) is defined in (21). In particular we +have +RlΦ(x, t)R† +l ≈ +� +1 − 1 +l +� +Φ(x′, t′) +(32) +for all (x, t), not just “most”, avoiding the “complicated” +cases of (25) and (30). Naturally, the coordinates (x′, t′) +in (32) satisfy (26). +Interestingly, the conjugated operators R† +l and L† +l imple- +ment a Lorentz boost together with a dilation (instead of a +contraction). And in this case, the boost directions are re- +versed: R† +l is a left boost and L† +l a right boost. Therefore, +the composition R† +l Ll produces a Lorentz transformation +without a scale transformation, but it has a non-trivial +kernel, and hence, it is not an isometry. +2.7 +Two-layer conformal circuits +To simplify notation we redefine u as the coarse-grained +(dual) unitary +u = += +(33) +where the double-arrow notation encapsulates the new +symmetry uT = sus†. +We also redefine q so that the +Hilbert space of a coarse-grained site has dimension q. +Now we can write the evolution operator (14) as a two- +layer circuit +T = · · · +· · · +(34) +In the rest of this work, our starting point is a dual unitary +u = +with the symmetry uT = sus†, and the time- +translation operator (34). Note that this dynamics is more +general than the coarse-grained four-layer circuit. +3 +Discrete holography +In this section we construct tensor-network states for 1+1 +QCAs, and show that they can be interpreted as spacial +slices of 2+1 discrete geometries with metric distance de- +fined by their entanglement structure. +3.1 +Tensor-network states and dynamics +By using the four-legged tensor +, its complex conju- +gated and their rotated versions, we can construct tensor- +network states for the chain Hn. +One example of such +states for a chain of n = 20 sites is +|Ψfl⟩ = 1 +q5 +−1 +0 +1 +2 +, +(35) +where each black dot in the blue line represents a free leg +of the tensor network, and hence, a Cq system of the chain. +The label “fl” stands for “flat”. Naturally, the black dots +represent the sites of the chain Z20 in the same order, like +the marked sites x = −1, 0, 1, 2. +5 + +When the state evolves via the quantum circuit |Ψfl⟩ → +T |Ψfl⟩, the corresponding tensor network also evolves. To +simplify the calculation of this evolution, it is convenient +to separate each of the two layers of the time-translation +operator as T = ToddTeven. With this notation we can +write the evolution of |Ψfl⟩ as +Teven |Ψfl⟩ = += +, +(36) +and +ToddTeven |Ψfl⟩ = += +, +(37) +where here and in the rest of the paper we ignore the +normalisation of states. Note that the action of T on the +tensor network is not fully determined until we know the +position of one site, like for example x = 0. +The complete evolution of |Ψfl⟩ is depicted in Figure 1. +Remarkably, this evolution is cyclic and has a very small +period (∆t = 5) when compared with the approximate +recurrence time of a typical state in H20, which would +be doubly exponential in the size (∼ exp q20). This (flat- +space) tensor network can be generalised to any boundary +size n multiple of four, with corresponding period ∆t = n +4 . +Of course, there are many other tensor-network states, and +we analyse some of them below. +Consider a state |Φ⟩ and a positive integer ∆t satisfying +T ∆t |Φ⟩ = |Φ⟩. If ∆t is the smallest such integer, then +they generate the orbit T t |Φ⟩ for t = 0, . . . , ∆t − 1. (An +example of orbit is depicted in Figure 1.) Given any such +orbit we can construct some eigenstates of T as +|Φω⟩ = +∆t−1 +� +t=0 +ei 1 +∆t ωt T t |Φ⟩ , +(38) +T |Φω⟩ = ei 1 +∆t ω |Φω⟩ +(39) +for ω = 0, . . . , ∆t − 1. Hence, it is meaningful to associate +to this subspace a dynamical mode of (quasi) energy +E = 1 +∆t . +(40) +3.2 +Entanglement geometry +The above tensor-network states can be interpreted as 2D +spatial geometries, where the metric distance is fixed the +a) +h = 5 log q +b) +h = 2 log q +c) +2 +4 +2 +1 +1 +2 +4 +4 +2 +3 +5 +1 +1 +3 +5 +3 +3 +3 +5 +2 +2 +6 +6 +1 +Figure 2: Entanglement and geodesics in flat space +(tensor network from Figure 1 at t = .5). Figures a) b): +the entanglement entropy h between the region consisting +of red dots and the rest of the chain is equal to the small- +est number of black lines (times log q) that are crossed +by a curve enclosing the red dots. In very light grey, the +position of the gates. Figure c): distance between each +location and that with the pink dot. Each white rhom- +bus represents a location in the bulk, which in this case +has flat curvature. Locations in the boundary of the bulk +correspond to links in the chain Z20. +entanglement structure of the state, via Ryu-Takayanagi’s +prescription [4–7]. This identifies the von Neumann en- +tropy of a set of consecutive sites in the chain with the +length of the shortest curve beginning and ending at the +boundary of the set (see Figure 2). Note that we can apply +this prescription to spaces other than AdS. +Let us obtain the metric distance of the simplest tensor- +network state +|Ψfl⟩ = 1 +q +0 +1 +2 +3 +. +(41) +The unitary condition (7) implies that the reduced density +matrix of subsystem {0, 1} is proportional to the identity +matrix, which implies that |Ψfl⟩ is maximally entangled +with respect to the bipartition 01|23. Similarly, the dual- +unitary condition (8) implies that |Ψfl⟩ is maximally en- +tangled with respect to the bipartition 03|12. This in turn +implies that each site (0,1,2 or 3) is maximally entangled +with the rest. This precise entanglement structure is con- +tained in the geometry +, where each of the 4 triangles +represents a location in the surface, and the distance be- +tween two locations is given by the number of black lines +that are crossed when travelling from one location to the +other. +In the previous example, the emergent geometry does +not contain interior points. Figure 2 depicts the geome- +try of a more complex tensor-network state, which has a +non-trivial interior. We have checked the equality between +entropy and distance for all bipartitions, but we have not +proven that this geometry uniquely captures the entangle- +ment structure of the state. But in any case, this geom- +etry is special, because it has the same structure as the +underlying tensor network. Now, we can use the distance +defined in Figure 2 to calculate the length of any curve +6 + +in the bulk, not necessarily the shortest one connecting a +pair boundary points. This reveals that the geometry in +question is a piece flat space with boundary. +In the continuum setup (AdS/CFT) there is an exten- +sive literature [8–13] addressing the problem of how to ob- +tain the bulk geometry given the entanglement structure +of the CFT state. In the following subsections we discuss +the geometry of relevant states in Hn, which generate dis- +crete versions of AdS space with and without a black hole, +and the double-sided AdS black hole. +3.3 +Anti-de Sitter state +The dilation isometry Dn : Hn → H2n maps a chain of +length n (multiple of 4) to a chain of length 2n. Its par- +ticular form is +Dn = q−n/8� +· · · +−2 −1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 +0 +1 +2 +3 +4 +5 +· · · +� +, +(42) +where we have included the site labels x of the input and +output chains. Note that the dilation D is the Hermitian +conjugate of the contraction C defined in (16), except that +C is defined for the four-layer circuit (14) and here we +define D for the two-layer circuit (34). +The toy AdS state is recursively defined as +|Ψtoy +4 ⟩ = 1 +q +0 +1 +2 +3 +, +(43) +|Ψtoy +2n ⟩ = Dn |Ψtoy +n ⟩ , +(44) +which results in +|Ψtoy +n ⟩ = D n +2 · · · D16D8D4 +� +�� +� +ϱads +|Ψtoy +4 ⟩ . +(45) +Note that the size of the chain is n = 2ϱads4, where the pos- +itive integer ϱads denotes the number of recursions. This +tensor network is depicted in Figure 3, where we can see +that each recursion corresponds to a value of the radial +coordinate ϱ = 0, 1, . . . , ϱads. Also, the intermediate state +|Ψtoy +2ϱ4⟩ in the recursion (45) represents a spatial slice of +AdS with the radial coordinate restricted to the interval +[0, ϱ]. +By proceeding as in (36) and (37) we can check that the +4-site toy AdS state (43) is an eigenstate of the evolution +operator +T4 |Ψtoy +4 ⟩ = |Ψtoy +4 ⟩ . +(46) +Also, by proceeding as in (18) we obtain the equality +T 4 +2nDn = DnT 2 +n , +(47) +which implies that, when the radial coordinate ϱ decreases +by one unit, time slows down by a factor of two: +T 4 +2n |Ψtoy +2n ⟩ = DnT 2 +n |Ψtoy +n ⟩ . +(48) +In Section 4 we show that the relationship (48) between +time at different radial locations applies to local clocks +made of matter. +The state |Ψtoy +n ⟩ is not an eigenstate of Tn, it evolves +in time through the orbit shown in Figure 4. The corre- +sponding period can be calculated by using equations (46) +and (47), obtaining +(Tn)2ϱads+1 |Ψtoy +n ⟩ = |Ψtoy +n ⟩ . +(49) +That is, the period of this orbit is proportional to the +length of the chain ∆t = 2ϱads+1 = n +2 . Again, note that +this is much shorter than the recurrence time of a typical +state in Hn, which would be doubly exponential in the +size (∼ exp qn). Using (40) we associate to empty AdS an +energy +Eads = 2 +n . +(50) +The geometry of the tensor network of |Ψtoy +n ⟩ is not reg- +ular along the radial direction. For instance, the geodesic +distance from a boundary point x to the centre (ϱ = 0) +depends on x, as shown in the red lines of Figure 3. In- +terestingly, the T-eigenstate +|Ψads +n ⟩ = +n +2 −1 +� +t=0 +T t |Ψtoy +n ⟩ , +(51) +has a more regular geometry in the large-q limit. Specifi- +cally, it produces the metric distance +∆s2 +Ψads = log2q +� +−22ϱ∆τ 2 + ∆ϱ2 + 22ϱ ∆θ2 +π2 +� +, +(52) +to leading order in q. (The proof of this fact will be pre- +sented elsewhere.) Recall that this distance characterises +a discrete geometry, hence, the increments ∆τ, ∆ϱ, ∆θ are +discrete. +The radial coordinate ϱ ∈ [0, ϱads] ⊂ Z has +already been introduced. The time coordinate τ at the +boundary is related to the QCA time via t = log q 2ϱadsτ. +Note that the proper time defined by (52) reflects the ra- +dial dependence implied by (48). The angular coordinate +θ ∈ [0, 2π] is discretised in 2ϱ4 steps of size ∆θ = 2π2−ϱ−2. +Hence, each step has one unit of proper distance and the +proper distance of a complete circle is log q 2ϱ4, the log- +arithm of the dimension of the chain in the correspond- +ing intermediate recurrence step. Finally, note that the +distance (52) strongly resembles AdS’s distance in global +coordinates +ds2 +AdS = α2 � +− cosh2ϱ dτ 2 + dϱ2 + sinh2ϱ dθ2� +. +(53) +3.4 +Thermofield double state +In the previous subsection we saw that eigenstates produce +smoother geometries. However, eigenstates require super- +positions of tensor networks, and hence, are less easy to +7 + +s = 0 +s = 3 log q +D16 +ϱads = 3 +D8 +ϱ = 2 +D4 +ϱ = 1 +|Ψtoy +4 〉 +ϱ = 0 +31 +0 +8 +13 +18 +24 +T 2 +4 +1 +Figure 3: Toy AdS state. The left figure shows (continuum) AdS tiled with equal-size triangles. The right figure +displays the the tensor network of the toy AdS state for a chain of size n = 32 with periodic boundary conditions. +Each level of the four recursions is identified by the corresponding value of the radial coordinate ϱ. In red, the minimal +length paths from boundary locations x = 13, 18 to the centre, and their length s. Which shows that this “toy” version +of AdS is very non-smooth. +t = 0 +0 +t = 0.5 +0 +t = 1 +0 +t = 1.5 +0 +t = 2 +0 +t = 2.5 +0 +t = 3 +0 +t = 3.5 +0 +t = 4 +0 +t = 4.5 +0 +t = 5 +0 +t = 5.5 +0 +t = 6 +0 +t = 6.5 +0 +t = 7 +0 +t = 7.5 +0 +t = 8 +0 +1 +Figure 4: Dynamics of the toy AdS state |Ψtoy +32 ⟩ represented in Figure 3. The position of the chain location x = 0 +is represented at each time step. +8 + +D16 +D8 +D4 +|Ψads +4 +〉 +0 +T 2 +4 +1 +Figure 5: Thermofield double state. This tensor net- +work represents a toy version of a two sided black hole in +AdS with boundary radius ϱads = 3 and horizon radius +ϱh = 0, which implies boundary size n = 32 and horizon +size a = 4. Periodic boundary conditions are understood +in the two chains and the throat (the piece connecting the +two symmetric sides). The throat has been growing for +2 local time steps, which implies that the QCA has been +evolving for t = 2 × 23 = 16 time steps. +visualise. For this reason, in this subsection, we continue +using non-eigenstate tensor-network states. +The thermofield double (TFD) is a joint state of two +identical chains Hn⊗Hn evolving in time via the dynamics +T ⊗T ⊺, where T ⊺ is the transpose of T (see Figure 5). This +state is characterised by the largest and smallest values of +the radial coordinate outside the throat ϱ ∈ [ϱh, ϱads] ⊆ Z. +These parameters fix the size of each chain to n = 2ϱads4, +and the area (length) of the horizon to a log q, where we +define a = 2ϱh4. The TFD with n = a is the maximally- +entangled state between the two chains +|Ψtfd +a,a⟩ = +� +x∈Za +|ψ⟩x , +(54) +|ψ⟩x = +1 +√q +q +� +k=1 +|k⟩x ⊗ |k⟩x , +(55) +where |ψ⟩x is the maximally-entangled state between site +x of one chain and site x of the other chain. When n > a +the TFD can be recursively generated via +|Ψtfd +2n,a⟩ = Dn ⊗ D∗ +n |Ψtfd +n,a⟩ , +(56) +where D∗ +n is the complex conjugate of Dn, defined in (42). +The TFD can be explicitly written as +|Ψtfd +n,s⟩ = (D n +2 · · · D2aDa +� +�� +� +ϱads−ϱh +) ⊗ (D n +2 · · · D2aDa +� +�� +� +ϱads−ϱh +)∗ |Ψtfd +a,a⟩ . +(57) +Next, let us analyse the dynamics of the TFD. Proceed- +ing as in (18) we obtain the identity +(T ⊺ +2n)4D∗ +n = D∗ +n(T ⊺ +n)2 , +(58) +which is not equivalent to (47), although here it produces +a similar result: when the radial coordinate ϱ is decreased +D16 +D8 +D4 +|Ψads +4 +〉 +0 +T 2 +4 +1 +Figure 6: Thermal AdS state. This tensor network rep- +resents the toy version of a simple black hole in AdS, for a +chain of size n = 32 and horizon area a = 4. The two blue +lines correspond to the input and output of the density +matrix. Periodic boundary conditions are understood in +the chain and the horizon. +by one, time slows down by a factor two +(T2n ⊗ T ⊺ +2n)4 |Ψtfd +2n,a⟩ = (Dn ⊗ D∗ +n)(Tn ⊗ T ⊺ +n)2 |Ψtfd +n,a⟩ . +Now we recover the standard fact that the throat worm- +hole grows linearly in time, since the action of T ⊗ T ⊺ +cannot be simplified +Ta ⊗ T ⊺ +a |Ψtfd +a,a⟩ = T 2 +a ⊗ 1 |Ψtfd +a,a⟩ , +(59) +as also illustrated in Figure 5. +Note however, that the +propper time in the throat is slower than that on the +boundary by an exponential factor 2(ϱads−ϱh). +3.5 +Thermal anti-de Sitter state +If we perform the partial trace on the right chain in the +TFD then we obtain the mixed state +ρn,a = trright|Ψtfd +n,a⟩⟨Ψtfd +n,a| += (D n +2 · · · D2aDa +� +�� +� +ϱads−ϱh +)1a(D n +2 · · · D2aDa +� +�� +� +ϱads−ϱh +)† , +(60) +where 1a is the identity acting on Ha (see Figure 6). The +fact that Dn are isometries implies that the entropy of ρn,a +is a log q. When n = a the state ρads +a,a is time-independent +and maximally mixed, which corresponds to infinite tem- +perature. +Like in the previous variants of AdS, here we also have +the following property. +When n > a, when the radial +coordinate ϱ decreases by one unit, time slows down by a +factor two +(T2n)4ρ2n,a(T † +2n)4 = Dn (Tn)2ρn,a(T † +n)2D† +n . +(61) +This implies that the evolution of the thermal AdS state +undergoes a cycle of period ∆t = 2ϱads−ϱh+1 = 2n +a . Hence, +we associate to it an energy +E +Eads += a +4, +(62) +9 + +where we have substituted the energy of pure AdS Eads +obtained in (50). +At this stage, it is not clear how to +interpret this identity. +4 +Spaces with matter +In the previous section we considered tensor-network +states (45) constructed with the building blocks +and +, and interpreted them as curved empty spaces for the +bulk. In this section we consider bulk spaces with matter. +We say that a state contains matter (when interpreted as a +bulk state) when it cannot be written as a tensor network, +or a super-position thereof, constructed with the building +blocks +and +. The states with matter that we analyse +are empty spaces with the addition of some (non-building +blocks) operators on a small number of links. When only +one link is affected we interpret it as the position of a +particle. +4.1 +Ambiguity in the position of particles +Let us see that, if the dual unitary +is chaotic (has no +solitons), then there are particle states with a well-defined +position for the particle. Let us start by considering the +(empty) flat space state +|Ψfl⟩ = +0 +, +(63) +and apply an arbitrary operator += a0 ∈ A0 at the chain +site x = 0 +a0 |Ψfl⟩ = +. +(64) +We interpret this state as the empty space (63) with one +particle at its boundary. However, the same sate can also +be written as +a0 |Ψfl⟩ = +, +(65) +where the two-site operator +is defined by += +, +(66) +which then satisfies += +, +(67) +and implies the equality between (64) and (65). Analo- +gously, the operator += +, +(68) +allows to write the same state (64) as +a0 |Ψfl⟩ = +. +(69) +And, if we use the operator += +, +(70) +then we can write the same state as +a0 |Ψfl⟩ = +. +(71) +Clearly, there is a large number of ways of writing the +same state. +If a is a soliton of +, or a linear combination thereof, +then the operators +, +and +defined in (66), (68) +and (70) act non-trivially on a single site only. Contrary, +if +is generic then, in addition of having no solitons, +we expect that all alternative ways of writing the one- +particle state (64) involve operators with terms acting non- +trivially in more than one site, like for example += +b⊗c+· · · This fact eliminates the ambiguity of the particle +position. Therefore we use the following prescription: If +a state can be written as a tensor network of the building +blocks, and one extra tensor on a single link, then we +interpret this state as a space with one particle at the +position of mentioned link. +This prescription allows us to put a particle at any lo- +cation in the bulk (not necessarily the boundary), like for +example +. +(72) +However, In the following subsection we see that this pre- +scription cannot be applied to all single-particle states. +4.2 +Dynamics of spaces with one particle +We have already seen the dynamics of the empty-space +state |Ψfl⟩ in Figure 1. Now, let us explore what happens +when we add one particle a0 |Ψfl⟩ at the boundary site +x = 0. Figure 7 shows that the geometry of T ta0 |Ψfl⟩ +evolves like that of T t |Ψfl⟩, if we represent the particle +10 + +t = 0 += a +t = 0.5 += +t = 1 += +t = 1.5 += +T +t = 2 += +t = 2.5 += +t = 3 +t = 3.5 +t = 4 += T +t = 4.5 +t = 5 +1 +Figure 7: Dynamics of flat space with one particle - sequence of states at different times t. At t = 0, initial +state of geometry (in grey) and particle (red dot represents arbitrary operator a). At each half time step t, state of +geometry and particle, and below, definition of the operator representing the particle in terms of operators defined in +previous time steps. Each green operator is the transpose of the red operator with the same form. At t = 2 and t = 2.5 +the operators shrink - this requires the dual unitary and the operator a to be real, because then operator equalities +at t = 2 and t = 2.5 are equivalent to those at t = 1 and t = 0.5 respectively. At t = 5, the geometry returns to its +original state but the particle is in the antipodal position. Hence the period is ∆t = 10. +with certain inserted operators, also defined in Figure 7. +This evolution has a very short period (∆t = 10) when +the dual unitary and the operator are real += +, +a = a∗. +(73) +(The same holds if, instead of real, the operator is symmet- +ric a = a⊺.) This implies that the subsequent operators += += +(74) +are real too; and it makes the operator equalities at t = +2 and t = 2.5 the transposition of equalities at t = 1 +and t = .5 respectively, rendering them equivalent. This +implies the shrinking of operators at t = 2 and 2.5, and a +return to the original a. Note that the recurrence time of +a0 |Ψfl⟩ ∈ Hn is t = n +2 , while that of a typical state in Hn +is t ∼ exp qn. +Figure 7 also shows that some intermediate states can- +not be written with a single-site tensor representing the +particle. Hence, as mentioned above, the position of the +particle in these states is not well defined. +When the particle ax is located at other points of the +boundary of flat space, the corresponding state ax |Ψfl⟩ ex- +periences an evolution similar to that with x = 0. Specif- +ically, for any x, the particle reaches the antipodal point +at t = n +4 , and the period of the evolution is ∆t = n +2 . Fig- +ure 12 shows the evolution of the same flat space when the +initial location of the particle is a corner (x = −2). +Figure 13 shows the dynamics of toy AdS with a particle +at the boundary location x = 0, that is a0 |Ψtoy +32 ⟩. In this +case we also observe that the particle reaches the antipo- +dal point at t = n +4 and the recurrence happens at t = n +2 . +However, contrary to flat space, certain initial positions +of the particle (e.g. a9 |Ψtoy +32 ⟩) generate a state with recur- +rence time much longer than ∆t = +n +2 . In the following +section we observe that two-particle states also give rise +to this second type of dynamics, with longer recurrence +times. +4.3 +One vs two-particle dynamics +In this subsection we compare the dynamics of states with +different numbers of particles. We do so by numerically +obtaining the dimension of the effective subspace explored +by the evolution of each such state. +This is done with +the dynamics of the four-layer circuit (14) generated by a +randomly sampled dual unitary +with real entries and +local dimension q = 3. We use the four-layer circuit to get +rid of the constraint += +and, in this way, enlarge the +size of the set of dual unitaries. This fact increases the +similarity between the properties of different instances of +the random dual unitary, due to concentration of measure. +For the sake of simplicity we perform the above- +described comparison with the small empty-space state +|Ψ⟩ = +0 +1 +−1 +∈ H8 . +(75) +We calculate its dynamics by proceeding as in (36) and +(37) but with the four-layer circuit T = T[4]T[3]T[2]T[1] +11 + +described in (14), obtaining the cycle +T[1] |Ψ⟩ = +0 +, +T[2]T[1] |Ψ⟩ = +0 +, +T[3]T[2]T[1] |Ψ⟩ = +0 +, +T |Ψ⟩ = +0 +. +(76) +We observe that the period is ∆t = 1, and so, |Ψ⟩ is an +eigenstate of T. +If we add a particle at the boundary +location x = 0 by applying an operator += a ∈ A then we +obtain the state +a0 |Ψ⟩ = +0 +1 +−1 +, +(77) +which has the similar evolution +T[1]a0 |Ψ⟩ = +0 +, +T[2]T[1]a0 |Ψ⟩ = +0 +, +T[3]T[2]T[1]a0 |Ψ⟩ = +0 +, +Ta0 |Ψ⟩ = +0 +, +(78) +where += a⊺ ∈ A is the transpose. That is, after one time +step, the particle is in the antipodal location (x = 4), +so the period of the state a0 |Ψ⟩ is ∆t = 2. +We have +checked that the same dynamical behaviour happens for +any initial location x ∈ Z8 of the particle. In summary, +all single-particle states generate a closed orbit of period +∆t = 2, which allows us to construct exact eigenstates via +(38). Because of the existence of simple eigenstates, we +consider the zero and one-particle subspace “integrable”. +Next, let us compare the above results with the dynam- +ics of the following two and three-particle states +, +, +, +(79) +generated with a random matrix += a ∈ A with real +entries. For each such pure state |Ψ⟩⟨Ψ| we numerically +calculate its time average after t time steps +ρ(t) = +1 +t + 1 +t +� +r=0 +T r|Ψ⟩⟨Ψ|T −r . +(80) +And for each such mixed state we calculate the effective +dimension of its support +deff(t) = +� +trρ2(t) +�−1 , +(81) +which takes into account the different weights of the eigen- +values, as in the second-order Renyi entropy via log2 deff = +h2(ρ) = − log2 ρ2. For example, if a states ρ is propor- +tional to a projector ρ2 ∝ ρ then deff is equal to the di- +mension of the projector. The effective dimension of the +above two and three-particle states is plotted in Figure 8. +We observe that the corresponding curves attain a much +larger value than those of the zero and one-particle states, +20 +40 +60 +80 +100 +120 +140 +5 +10 +15 +20 +25 +30 +35 +1 +11 +1 +1 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +Figure 8: Effective dimension of 1,2,3 particles. Each +curve shows the dimension deff(t) of the effective subspace +as a function of time t, generated by the evolution of the +tensor-network state next to it. The top curve is gener- +ated by a random state with real entries. The dynamics is +generated by the four-layer circuit (14) with a real random +dual unitary. We observe a qualitative distinction between +the single-particle state and the rest. The differences be- +tween the other four states are not essential, because they +vary according to the instance of random dual unitary. +which have deff(∞) = 1, 2; and take a longer time to equi- +librate. In addition, the lack of convergence to an integer +value suggests that the corresponding evolutions do not +generate a closed orbit, and hence, do not have simple +eigenstates associated to them. Figure 8 also shows that +the behaviour of the states (79) is similar to that of a +random state in H8. This establishes a sharp distinction +between one particle or less on one side, and two particles +or more on the other. These results open the possibility +that the dynamics in the two or more particles subspace +is quantum-chaotic. However, the results of the following +subsection suggest that the contrary is true. +4.4 +Absence of quantum chaos +First of all, let us see that our method for sampling ran- +dom dual unitaries generates an ensemble qualitatively +similar to that of Haar-random unitaries (with no dual- +ity constraint). Actually, since we generate random dual +unitaries with real entries, we need to compare these with +Haar-random orthogonal matrices. In order to do so, we +generate a set U of 85 random dual unitaries u ∈ U with +real entries, and we calculate the spectral form factor +KU(t) = 1 +|U| +� +u∈U +��tr(ut) +��2 . +(82) +The result are the red points in Figure 9, which are con- +trasted with the form factor of the orthogonal group +KSO(d)(t) = +� 2t − t log2(1 − 2t/d) +if t < d +2d − t log2 +� +2t+d +2t−d +� +if t ≥ d +, +(83) +12 + +5 +10 +15 +2 +4 +6 +8 +10 +12 +A then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +where += a| 2 A is the transpose. That is, after one time +step, the particle is in the antipodal location (x = 4), so +the period of the state a0 | i is �t = 2. We have checked +that the same dynamical behaviour happens for any initial +location x 2 Z8 of the particle. In summary, all single- +particle states generate a closed orbit of period �t = 2, +which allows us to construct exact eigenstates via (38). +Because of the existence of simple zero and one-particle +eigenstates, we consider these cases somehow “integrable”. +Next, let us compare the above results with the dynam- +ics of the following two and three-particle states +, +, +, +(79) +generated with a random matrix += a 2 A with real +entries. For each such pure state | ih | we numerically +calculate its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r| ih |T �r . +(80) +And for each such mixed state we calculate the e↵ective +dimension of its support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values as in the second-order Renyi entropy via log2 de↵ = +h2(⇢) = � log2(⇢2). For states ⇢ proportional to a pro- +jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the +dimension of the projector. +The e↵ective dimension of +the above two and three-particle states is plotted in Fig- +ure 8. The corresponding curves attain a much larger value +than those of the zero and one-particle states, which have +de↵(1) = 1, 2; and take a longer time to equilibrate. In +addition, the lack of convergence to an integer value sug- +gests that the corresponding evolutions do not generate a +closed orbit, and hence, do not have simple eigenstates as- +sociated to them. Figure 8 also shows that this behaviour +is very similar to that of a random state | i 2 H8. This +establishes a sharp distinction between one particle or less +on one side, and two particles or more on the other. These +results open the possibility that the dynamics of two or +more particles is quantum-chaotic. However, the results +of the following subsection suggest that the contrary is +true. +1 +11 +1 +1 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +Figure 8: +E↵ective +dimension +of +1,2,3-particle +states. Each curve shows the dimension of the e↵ective +subspace de↵(t) as a function of time t, generated by the +evolution of the tensor-network state next to it. The top +curve is generated by a random state with real entries. +The dynamics is generated by the four-layer circuit (14) +with a random real dual unitary. We observe a qualita- +tive distinction between the single-particle state and the +rest. The di↵erences between the other four states are not +essential, because they vary according to the instance of +random dual unitary. +Figure 9: Spectral form factor of dual unitary. +4.4 +Absence of quantum chaos +First of all, let us see that our method for sampling real +random dual unitaries generates an ensemble qualitatively +similar to that of Haar-random real unitaries (i.e. orthog- +onal matrices) without the duality constraint. In order to +do so, we generate 85 random dual unitaries u 2 U with +real entries, and calculate the spectral form factor +K(t) = 1 +85 +X +u2U +��tr(ut) +��2 . +(82) +In Figure +12 +A then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +where += a| 2 A is the transpose. That is, after one time +step, the particle is in the antipodal location (x = 4), so +the period of the state a0 | i is �t = 2. We have checked +that the same dynamical behaviour happens for any initial +location x 2 Z8 of the particle. In summary, all single- +particle states generate a closed orbit of period �t = 2, +which allows us to construct exact eigenstates via (38). +Because of the existence of simple zero and one-particle +eigenstates, we consider these cases somehow “integrable”. +Next, let us compare the above results with the dynam- +ics of the following two and three-particle states +, +, +, +(79) +generated with a random matrix += a 2 A with real +entries. For each such pure state | ih | we numerically +calculate its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r| ih |T �r . +(80) +And for each such mixed state we calculate the e↵ective +dimension of its support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values as in the second-order Renyi entropy via log2 de↵ = +h2(⇢) = � log2(⇢2). For states ⇢ proportional to a pro- +jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the +dimension of the projector. +The e↵ective dimension of +the above two and three-particle states is plotted in Fig- +ure 8. The corresponding curves attain a much larger value +than those of the zero and one-particle states, which have +de↵(1) = 1, 2; and take a longer time to equilibrate. In +addition, the lack of convergence to an integer value sug- +gests that the corresponding evolutions do not generate a +closed orbit, and hence, do not have simple eigenstates as- +sociated to them. Figure 8 also shows that this behaviour +is very similar to that of a random state | i 2 H8. This +establishes a sharp distinction between one particle or less +on one side, and two particles or more on the other. These +results open the possibility that the dynamics of two or +more particles is quantum-chaotic. However, the results +of the following subsection suggest that the contrary is +true. +1 +11 +1 +1 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +Figure 8: +E↵ective +dimension +of +1,2,3-particle +states. Each curve shows the dimension of the e↵ective +subspace de↵(t) as a function of time t, generated by the +evolution of the tensor-network state next to it. The top +curve is generated by a random state with real entries. +The dynamics is generated by the four-layer circuit (14) +with a random real dual unitary. We observe a qualita- +tive distinction between the single-particle state and the +rest. The di↵erences between the other four states are not +essential, because they vary according to the instance of +random dual unitary. +Figure 9: Spectral form factor of dual unitary. +4.4 +Absence of quantum chaos +First of all, let us see that our method for sampling real +random dual unitaries generates an ensemble qualitatively +similar to that of Haar-random real unitaries (i.e. orthog- +onal matrices) without the duality constraint. In order to +do so, we generate 85 random dual unitaries u 2 U with +real entries, and calculate the spectral form factor +K(t) = 1 +85 +X +u2U +��tr(ut) +��2 . +(82) +In Figure +12 +Figure 9: Spectral form factor of dual unitary with +real entries and local dimension q = 3, averaged over 85 +random instances (red dots). Spectral form factor of the +orthogonal group SO(d) of dimension d = q2 = 9 (blue +line). Both plots display the initial “dip” and are qualita- +tively similar. This signals the presence of quantum chaos +in random dual unitaries. +with d = q2 = 9. We observe in Figure 9 that the two form +factors are qualitatively similar. In particular, both dis- +play the so called “dip” for t < 9. This behaviour signals +the presence of quantum chaos in random dual unitaries. +With the above mentioned 85 instances u ∈ U, we con- +struct 85 instances of the four-layer evolution operator T +for the chains with n = 4 and n = 8. +Using formula +(82) but replacing u by T, we calculate the correspond- +ing form factors and plot them in Figure 10. +We ob- +serve that both form factors are essentially flat, with no +“dip”, a behaviour characteristic of Poisson level statistics, +which appears in “integrable” systems. This is very sur- +prising, because both evolution operators are constructed +with unitaries u which, as discussed above, display quan- +tum chaos. Hence, we conclude that the special structure +of the conformal circuit (14) cancels the chaos present in +the building block +. +It is proven in [37] that the asymptotic value of the form +factor is +K(∞) = +� +E +g2 +E , +(84) +where E are the eigenvalues of the evolution operator and +gE the corresponding degeneracy. In the absence of degen- +eracies (gE = 1) we have K(∞) = qn, which is the stan- +dard value in generic models including random unitaries. +However, the two approximately constant plots in Fig- +ure 10 show values for K(t) much larger than the Hilbert- +space dimension qn. This signals that the spectrum of T +is highly structured in a way that leads to large degener- +ation. This structure will be studied in future work. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +50 +100 +150 +500 +1000 +1500 +2000 +100 +200 +300 +400 +500 +600 +700 +500000 +1.0×106 +1.5×106 +2.0×106 +2.5×106 +3.0×106 +3.5×106 +A then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +where += a| 2 A is the transpose. That is, after one time +step, the particle is in the antipodal location (x = 4), so +the period of the state a0 | i is �t = 2. We have checked +that the same dynamical behaviour happens for any initial +location x 2 Z8 of the particle. In summary, all single- +particle states generate a closed orbit of period �t = 2, +which allows us to construct exact eigenstates via (38). +Because of the existence of simple zero and one-particle +eigenstates, we consider these cases somehow “integrable”. +Next, let us compare the above results with the dynam- +ics of the following two and three-particle states +, +, +, +(79) +generated with a random matrix += a 2 A with real +entries. For each such pure state | ih | we numerically +calculate its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r| ih |T �r . +(80) +And for each such mixed state we calculate the e↵ective +dimension of its support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values as in the second-order Renyi entropy via log2 de↵ = +h2(⇢) = � log2(⇢2). For states ⇢ proportional to a pro- +jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the +dimension of the projector. +The e↵ective dimension of +the above two and three-particle states is plotted in Fig- +ure 8. The corresponding curves attain a much larger value +than those of the zero and one-particle states, which have +de↵(1) = 1, 2; and take a longer time to equilibrate. In +addition, the lack of convergence to an integer value sug- +gests that the corresponding evolutions do not generate a +closed orbit, and hence, do not have simple eigenstates as- +sociated to them. Figure 8 also shows that this behaviour +is very similar to that of a random state | i 2 H8. This +establishes a sharp distinction between one particle or less +on one side, and two particles or more on the other. These +results open the possibility that the dynamics of two or +more particles is quantum-chaotic. However, the results +of the following subsection suggest that the contrary is +true. +1 +11 +1 +1 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +Figure 8: +E↵ective +dimension +of +1,2,3-particle +states. Each curve shows the dimension of the e↵ective +subspace de↵(t) as a function of time t, generated by the +evolution of the tensor-network state next to it. The top +curve is generated by a random state with real entries. +The dynamics is generated by the four-layer circuit (14) +with a random real dual unitary. We observe a qualita- +tive distinction between the single-particle state and the +rest. The di↵erences between the other four states are not +essential, because they vary according to the instance of +random dual unitary. +4.4 +Absence of quantum chaos +First of all, let us see that our method for sampling real +random dual unitaries generates an ensemble qualitatively +similar to that of Haar-random real unitaries (i.e. orthog- +onal matrices) without the duality constraint. In order to +do so, we generate 85 random dual unitaries u 2 U with +real entries, and calculate the spectral form factor +K(t) = 1 +85 +X +u2U +��tr(ut) +��2 . +(82) +In Figure +Figure 9: Spectral form factor of dual unitary. +12 +A then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +where += a| 2 A is the transpose. That is, after one time +step, the particle is in the antipodal location (x = 4), so +the period of the state a0 | i is �t = 2. We have checked +that the same dynamical behaviour happens for any initial +location x 2 Z8 of the particle. In summary, all single- +particle states generate a closed orbit of period �t = 2, +which allows us to construct exact eigenstates via (38). +Because of the existence of simple zero and one-particle +eigenstates, we consider these cases somehow “integrable”. +Next, let us compare the above results with the dynam- +ics of the following two and three-particle states +, +, +, +(79) +generated with a random matrix += a 2 A with real +entries. For each such pure state | ih | we numerically +calculate its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r| ih |T �r . +(80) +And for each such mixed state we calculate the e↵ective +dimension of its support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values as in the second-order Renyi entropy via log2 de↵ = +h2(⇢) = � log2(⇢2). For states ⇢ proportional to a pro- +jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the +dimension of the projector. +The e↵ective dimension of +the above two and three-particle states is plotted in Fig- +ure 8. The corresponding curves attain a much larger value +than those of the zero and one-particle states, which have +de↵(1) = 1, 2; and take a longer time to equilibrate. In +addition, the lack of convergence to an integer value sug- +gests that the corresponding evolutions do not generate a +closed orbit, and hence, do not have simple eigenstates as- +sociated to them. Figure 8 also shows that this behaviour +is very similar to that of a random state | i 2 H8. This +establishes a sharp distinction between one particle or less +on one side, and two particles or more on the other. These +results open the possibility that the dynamics of two or +more particles is quantum-chaotic. However, the results +of the following subsection suggest that the contrary is +true. +1 +11 +1 +1 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +Figure 8: +E↵ective +dimension +of +1,2,3-particle +states. Each curve shows the dimension of the e↵ective +subspace de↵(t) as a function of time t, generated by the +evolution of the tensor-network state next to it. The top +curve is generated by a random state with real entries. +The dynamics is generated by the four-layer circuit (14) +with a random real dual unitary. We observe a qualita- +tive distinction between the single-particle state and the +rest. The di↵erences between the other four states are not +essential, because they vary according to the instance of +random dual unitary. +4.4 +Absence of quantum chaos +First of all, let us see that our method for sampling real +random dual unitaries generates an ensemble qualitatively +similar to that of Haar-random real unitaries (i.e. orthog- +onal matrices) without the duality constraint. In order to +do so, we generate 85 random dual unitaries u 2 U with +real entries, and calculate the spectral form factor +K(t) = 1 +85 +X +u2U +��tr(ut) +��2 . +(82) +In Figure +Figure 9: Spectral form factor of dual unitary. +12 +A then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +where += a| 2 A is the transpose. That is, after one time +step, the particle is in the antipodal location (x = 4), so +the period of the state a0 | i is �t = 2. We have checked +that the same dynamical behaviour happens for any initial +location x 2 Z8 of the particle. In summary, all single- +particle states generate a closed orbit of period �t = 2, +which allows us to construct exact eigenstates via (38). +Because of the existence of simple zero and one-particle +eigenstates, we consider these cases somehow “integrable”. +Next, let us compare the above results with the dynam- +ics of the following two and three-particle states +, +, +, +(79) +generated with a random matrix += a 2 A with real +entries. For each such pure state | ih | we numerically +calculate its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r| ih |T �r . +(80) +And for each such mixed state we calculate the e↵ective +dimension of its support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values as in the second-order Renyi entropy via log2 de↵ = +h2(⇢) = � log2(⇢2). For states ⇢ proportional to a pro- +jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the +dimension of the projector. +The e↵ective dimension of +the above two and three-particle states is plotted in Fig- +ure 8. The corresponding curves attain a much larger value +than those of the zero and one-particle states, which have +de↵(1) = 1, 2; and take a longer time to equilibrate. In +addition, the lack of convergence to an integer value sug- +gests that the corresponding evolutions do not generate a +closed orbit, and hence, do not have simple eigenstates as- +sociated to them. Figure 8 also shows that this behaviour +is very similar to that of a random state | i 2 H8. This +establishes a sharp distinction between one particle or less +on one side, and two particles or more on the other. These +results open the possibility that the dynamics of two or +more particles is quantum-chaotic. However, the results +of the following subsection suggest that the contrary is +true. +1 +11 +1 +1 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +Figure 8: +E↵ective +dimension +of +1,2,3-particle +states. Each curve shows the dimension of the e↵ective +subspace de↵(t) as a function of time t, generated by the +evolution of the tensor-network state next to it. The top +curve is generated by a random state with real entries. +The dynamics is generated by the four-layer circuit (14) +with a random real dual unitary. We observe a qualita- +tive distinction between the single-particle state and the +rest. The di↵erences between the other four states are not +essential, because they vary according to the instance of +random dual unitary. +4.4 +Absence of quantum chaos +First of all, let us see that our method for sampling real +random dual unitaries generates an ensemble qualitatively +similar to that of Haar-random real unitaries (i.e. orthog- +onal matrices) without the duality constraint. In order to +do so, we generate 85 random dual unitaries u 2 U with +real entries, and calculate the spectral form factor +K(t) = 1 +85 +X +u2U +��tr(ut) +��2 . +(82) +In Figure +Figure 9: Spectral form factor of dual unitary. +12 +A then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +where += a| 2 A is the transpose. That is, after one time +step, the particle is in the antipodal location (x = 4), so +the period of the state a0 | i is �t = 2. We have checked +that the same dynamical behaviour happens for any initial +location x 2 Z8 of the particle. In summary, all single- +particle states generate a closed orbit of period �t = 2, +which allows us to construct exact eigenstates via (38). +Because of the existence of simple zero and one-particle +eigenstates, we consider these cases somehow “integrable”. +Next, let us compare the above results with the dynam- +ics of the following two and three-particle states +, +, +, +(79) +generated with a random matrix += a 2 A with real +entries. For each such pure state | ih | we numerically +calculate its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r| ih |T �r . +(80) +And for each such mixed state we calculate the e↵ective +dimension of its support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values as in the second-order Renyi entropy via log2 de↵ = +h2(⇢) = � log2(⇢2). For states ⇢ proportional to a pro- +jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the +dimension of the projector. +The e↵ective dimension of +the above two and three-particle states is plotted in Fig- +ure 8. The corresponding curves attain a much larger value +than those of the zero and one-particle states, which have +de↵(1) = 1, 2; and take a longer time to equilibrate. In +addition, the lack of convergence to an integer value sug- +gests that the corresponding evolutions do not generate a +closed orbit, and hence, do not have simple eigenstates as- +sociated to them. Figure 8 also shows that this behaviour +is very similar to that of a random state | i 2 H8. This +establishes a sharp distinction between one particle or less +on one side, and two particles or more on the other. These +results open the possibility that the dynamics of two or +more particles is quantum-chaotic. However, the results +of the following subsection suggest that the contrary is +true. +1 +11 +1 +1 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +Figure 8: +E↵ective +dimension +of +1,2,3-particle +states. Each curve shows the dimension of the e↵ective +subspace de↵(t) as a function of time t, generated by the +evolution of the tensor-network state next to it. The top +curve is generated by a random state with real entries. +The dynamics is generated by the four-layer circuit (14) +with a random real dual unitary. We observe a qualita- +tive distinction between the single-particle state and the +rest. The di↵erences between the other four states are not +essential, because they vary according to the instance of +random dual unitary. +4.4 +Absence of quantum chaos +First of all, let us see that our method for sampling real +random dual unitaries generates an ensemble qualitatively +similar to that of Haar-random real unitaries (i.e. orthog- +onal matrices) without the duality constraint. In order to +do so, we generate 85 random dual unitaries u 2 U with +real entries, and calculate the spectral form factor +K(t) = 1 +85 +X +u2U +��tr(ut) +��2 . +(82) +In Figure +Figure 9: Spectral form factor of dual unitary. +12 +A then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +where += a| 2 A is the transpose. That is, after one time +step, the particle is in the antipodal location (x = 4), so +the period of the state a0 | i is �t = 2. We have checked +that the same dynamical behaviour happens for any initial +location x 2 Z8 of the particle. In summary, all single- +particle states generate a closed orbit of period �t = 2, +which allows us to construct exact eigenstates via (38). +Because of the existence of simple zero and one-particle +eigenstates, we consider these cases somehow “integrable”. +Next, let us compare the above results with the dynam- +ics of the following two and three-particle states +, +, +, +(79) +generated with a random matrix += a 2 A with real +entries. For each such pure state | ih | we numerically +calculate its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r| ih |T �r . +(80) +And for each such mixed state we calculate the e↵ective +dimension of its support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values as in the second-order Renyi entropy via log2 de↵ = +h2(⇢) = � log2(⇢2). For states ⇢ proportional to a pro- +jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the +dimension of the projector. +The e↵ective dimension of +the above two and three-particle states is plotted in Fig- +ure 8. The corresponding curves attain a much larger value +than those of the zero and one-particle states, which have +de↵(1) = 1, 2; and take a longer time to equilibrate. In +addition, the lack of convergence to an integer value sug- +gests that the corresponding evolutions do not generate a +closed orbit, and hence, do not have simple eigenstates as- +sociated to them. Figure 8 also shows that this behaviour +is very similar to that of a random state | i 2 H8. This +establishes a sharp distinction between one particle or less +on one side, and two particles or more on the other. These +results open the possibility that the dynamics of two or +more particles is quantum-chaotic. However, the results +of the following subsection suggest that the contrary is +true. +1 +11 +1 +1 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +Figure 8: +E↵ective +dimension +of +1,2,3-particle +states. Each curve shows the dimension of the e↵ective +subspace de↵(t) as a function of time t, generated by the +evolution of the tensor-network state next to it. The top +curve is generated by a random state with real entries. +The dynamics is generated by the four-layer circuit (14) +with a random real dual unitary. We observe a qualita- +tive distinction between the single-particle state and the +rest. The di↵erences between the other four states are not +essential, because they vary according to the instance of +random dual unitary. +4.4 +Absence of quantum chaos +First of all, let us see that our method for sampling real +random dual unitaries generates an ensemble qualitatively +similar to that of Haar-random real unitaries (i.e. orthog- +onal matrices) without the duality constraint. In order to +do so, we generate 85 random dual unitaries u 2 U with +real entries, and calculate the spectral form factor +K(t) = 1 +85 +X +u2U +��tr(ut) +��2 . +(82) +n = 4 +(83) +n = 8 +(84) +In Figure +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +A then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +where += a| 2 A is the transpose. That is, after one time +step, the particle is in the antipodal location (x = 4), so +the period of the state a0 | i is �t = 2. We have checked +that the same dynamical behaviour happens for any initial +location x 2 Z8 of the particle. In summary, all single- +particle states generate a closed orbit of period �t = 2, +which allows us to construct exact eigenstates via (38). +Because of the existence of simple zero and one-particle +eigenstates, we consider these cases somehow “integrable”. +Next, let us compare the above results with the dynam- +ics of the following two and three-particle states +, +, +, +(79) +generated with a random matrix += a 2 A with real +entries. For each such pure state | ih | we numerically +calculate its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r| ih |T �r . +(80) +And for each such mixed state we calculate the e↵ective +dimension of its support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values as in the second-order Renyi entropy via log2 de↵ = +h2(⇢) = � log2(⇢2). For states ⇢ proportional to a pro- +jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the +dimension of the projector. +The e↵ective dimension of +the above two and three-particle states is plotted in Fig- +ure 8. The corresponding curves attain a much larger value +than those of the zero and one-particle states, which have +de↵(1) = 1, 2; and take a longer time to equilibrate. In +addition, the lack of convergence to an integer value sug- +gests that the corresponding evolutions do not generate a +closed orbit, and hence, do not have simple eigenstates as- +sociated to them. Figure 8 also shows that this behaviour +is very similar to that of a random state | i 2 H8. This +establishes a sharp distinction between one particle or less +on one side, and two particles or more on the other. These +results open the possibility that the dynamics of two or +more particles is quantum-chaotic. However, the results +of the following subsection suggest that the contrary is +true. +1 +11 +1 +1 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +which has period �t = 1. +If we add a particle at the +boundary location x = 0 by applying the symmetric oper- +ator += a0 2 A0 with a| +0 = a0 then we obtain the state +a0 | i = +0 +1 +�1 +, +(77) +which has the similar evolution +T[1]a0 | i = +0 +, +T[2]T[1]a0 | i = +0 +, +T[3]T[2]T[1]a0 | i = +0 +, +Ta0 | i = +0 +, +(78) +That is, after one time step the particle is in the antipodal +location (x = 4), so the period of the state a0 | i is �t = +2. We have checked that the same dynamical behaviour +happens for any initial location x 2 Z8 of the particle. +In summary, all single-particle states generate a closed +orbit of period �t = 2, which allows us to construct exact +eigenstates via (38). +Next, let us compare the above with the dynamics of +the following two and three-particle states +xxxx +(79) +For each such pure state | ih | we numerically calculate +its time average after t time steps +⇢(t) = +1 +t + 1 +t +X +r=0 +T r⇢ T �r . +(80) +And for each such mixed state we calculate the dimension +of its e↵ective support +de↵(t) = +⇥ +tr⇢2(t) +⇤�1 , +(81) +which takes into account the di↵erent weights of the eigen- +values. In particular, this formula is related to the second- +order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2). +We have seen in (76) that the state with zero particles +| i is an eigenstate of the dynamics, therefore its evolu- +tion generates a one-dimensional subspace de↵(1) = 1. +According to (78), states with one particle ax | i gener- +ate an orbit with period �t = 2, therefore its evolution +generates a two-dimensional subspace de↵(1) = 2 (also +calculated numerically in Figure 8). In both cases, zero +and one particle, the states generate a closed orbit, and +hence, a family of eigenstates. For this reason we say that +this cases are somehow “integrable”. The e↵ective dimen- +sion for the cases of two and three particles is plotted in +Figure 8. The fact that the corresponding curves slowly +converge to a non-integer values suggests that the corre- +sponding evolutions do not close and orbit, and hence, do +not have simple eigenstates associated to them. In addi- +tion, we observe that the e↵ective dimension is larger. +A third class of dynamical behaviour is that produced +by a random state, also shown in Figure 8. In this case, the +Figure 8: E↵ective dimension of 1, 2, 3 particles. +Each line shows the dimension of the e↵ective subspace as +a function of time, generated by the evolution of the state +next to it. We observe a qualitative distinction between +the following three cases: (i) one particle, (ii) two and +three particles, (iii) and a random state. +e↵ective dimension is much larger, and the equilibration +time much longer. So, in summary, we observe three qual- +itatively di↵erent dynamical behaviours for states with (i) +zero or one particle, (ii) two particles or a few more, and +(iii) random states. +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +Figure 8: +E↵ective +dimension +of +1,2,3-particle +states. Each curve shows the dimension of the e↵ective +subspace de↵(t) as a function of time t, generated by the +evolution of the tensor-network state next to it. The top +curve is generated by a random state with real entries. +The dynamics is generated by the four-layer circuit (14) +with a random real dual unitary. We observe a qualita- +tive distinction between the single-particle state and the +rest. The di↵erences between the other four states are not +essential, because they vary according to the instance of +random dual unitary. +4.4 +Absence of quantum chaos +First of all, let us see that our method for sampling real +random dual unitaries generates an ensemble qualitatively +similar to that of Haar-random real unitaries (i.e. orthog- +onal matrices) without the duality constraint. In order to +do so, we generate 85 random dual unitaries u 2 U with +real entries, and calculate the spectral form factor +K(t) = 1 +85 +X +u2U +��tr(ut) +��2 . +(82) +n = 4 +(83) +n = 8 +(84) +In Figure +5 +Lorentz transformations +In this section we define the operators Rl and Ll men- +tioned in Section 2.6, which jointly implement a contrac- +tion and a Lorentz boost towards the right and left re- +spectively. The treatment of Lorentz transformations is +simpler in the infinite chain, so in this section we as- +sume n = 1 and x 2 Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H1 ! H1 are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a 2 A0 +12 +Figure 10: Spectral form factor of the evolution op- +erator T of four layers for the chains with n = 4 and +n = 8. The two plots are very similar, but they look dif- +ferent because the n = 4 shows all the vertical scale while +the n = 8 only shows a narrow range. Both plots are es- +sentially flat with no “dip”, a characteristic of “integrable” +systems with Poisson level statistics. +simpler in the infinite chain, so in this section we as- +sume n = ∞ and x ∈ Z. +Let us now mention a sub- +tle issue. +In the infinite chain, global operators like +T, S, Rl, Ll : H∞ → H∞ are not well-defined. Fortunately, +they have a well-defined adjoint action on any operator +supported on a finite region; for example, T maps a ∈ A0 +onto +TaT † = +a +. +(85) +Therefore, in this section, any global operator H∞ → H∞ +is understood as an adjoint action on the quasi-local alge- +bra. +5.1 +Local Lorentz contractions +The operators Rl, Ll are constructed with the building +blocks Jx, Kx, which we call local Lorentz contractions. +For any even x we define the local Lorentz right-boost +contraction isometry as +Jx = q− 1 +2 · · · +x +x +· · · +(86) +It is a contraction because, as shown in (90), it has a +non-trivial kernel. Also, as can be seen in the picture, in +some sense there are two more bottom legs than top legs +(although both numbers are infinite). +If we act with Jx on the time-translation operator T +13 + +defined in (34) we obtain +JxT = q− 1 +2 · · · +· · · +(87) += q− 1 +2 · · · +· · · += q− 1 +2 · · · +· · · += q− 1 +2 · · · +x +· · · = TJx+2 +which can be summarised by the following equality +TJxT † = Jx−2 , +(88) +SJxS† = Jx+2 . +(89) +The second equality is just the application of the space- +translation operator S. +Using these identities and the +dual-unitary constraints, we can calculate the action of Jx +on a (trace-less) operator ay ∈ Ay at an arbitrary location +y, +JxayJ† +x = +� +� +� +� +� +� +� +ay +if y ≤ x − 2 +Tay−2T † +if y ≥ x + 2 +0 +if y = x ± 1 +η+(a)(x−1,x) +if y = x +, +(90) +where we define the completely-positive map +η+(a) = vx−1Ω+(a)x−1v† +x−1 +(91) += q−1vx−1trx−2(ux−2ax−2u† +x−2)v† +x−1 ∈ Ax−1,x . +Analogously, for any odd x we define the local Lorentz +left-boost contraction as +Kx = q− 1 +2 · · · +x +x +· · · +(92) +By proceeding as in (87) we obtain +TKxT † = Kx+2 , +(93) +SKxS† = Kx+2 . +(94) +The action of Kx on a (trace-less) operator ay ∈ Ay at an +arbitrary location y is +KxayK† +x = +� +� +� +� +� +� +� +Tay+2T † +if y ≤ x − 2 +ay +if y ≥ x + 2 +0 +if y = x ± 1 +η−(a)(x,x+1) +if y = x +, +(95) +where we define completely-positive map +η−(a) = vxΩ−(a)x+1v† +x += q−1vxtrx+2(ux+1ax+2u† +x+1)v† +x ∈ Ax,x+1 . +(96) +5.2 +Global Lorentz contractions +For any even integer l > 0 we define the the (global) +Lorentz right-boost contraction isometry as +Rl = · · · J5l J3l Jl ˜J−l ˜J−3l ˜J−5l · · · +(97) +where we also define ˜Jx = S T −1Jx. The reason for using +˜Jx instead of Jx when x < 0 is that ˜Jx is corrected with a +spacetime translation so that Bl acts trivially around the +origin x ∈ [−(l − 2), l − 2] ⊆ Z. Recall that pure Lorentz +transformations fix the origin (x, t) = (0, 0). A calculation +similar to that in (87) yields +˜Jx = q− 1 +2 · · · +x +x +· · · +(98) +Analogously, for any odd integer l > 0, we define the +the (global) Lorentz left-boost contraction isometry as +Ll = · · · K−5l K−3l K−l ˜Kl ˜K3l ˜K5l · · · +(99) +where we also define ˜Kx = S T −1Kx so that the origin is +fixed. The following lemma specifies the action of Rl and +Ll on any operator of the form a(x, t) = T taxT −t. +Lemma 1. If l is an even positive integer, a ∈ A a local +operator and x a location such that |x − lm| > 1 for all +odd integers m, then +Rla(x, t)R† +l = a +� +x − 2fl(x − 2t), t + fl(x − 2t) +� +, +(100) +where we define the function fl(x) = +� x−l +2l +� ++ 1, plotted +in Figure 11. If l is an odd positive integer, a ∈ A a local +operator and x a location such that |x − lm| > 1 for all +odd integers m, then +Lla(x, t)L† +l = a +� +x − 2fl(x + 2t), t − fl(x + 2t) +� +. +(101) +14 + +-�� +-�� +� +�� +�� +-� +-� +� +� +� +��(�) +Figure 11: Plot of the function fl(x) for l = 4. +We can write the coordinate transformations of the lemma +as +x′ = x − 2fl(x ∓ 2t) +t′ = t ± fl(x ∓ 2t) +� +, +(102) +where the upper and lower signs correspond to the right +and left boosts respectively. In the limit |x| ≫ l we can +use the approximation +fl(x) ≈ x +2l , +(103) +to write the coordinates transformation as +x′ ≈ +� +1 − 1 +l +� +x ± 2 +l t +t′ ≈ +� +1 − 1 +l +� +t ± 1 +2lx +� +. +(104) +Note that, independently of the sign ±, this transforma- +tion preserves Minkowski’s metric up to a scale factor +(2t′)2 − x′2 = +� +1 − 2 +l +� � +(2t)2 − x2� +. +(105) +(Recall that, in this model, the speed of light is c = 2.) We +can also obtain the velocity of the Lorentz boost in (104) +by first, undoing the contraction by dividing (x′, t′) by the +scale factor +� +1 − 1/l, and second, comparing the resulting +transformation to a standard Lorentz boost. This results +in the velocity +v = +±2 +√ +4 − 2l + l2 . +(106) +In Lemma 1, the premise |x − lm| > 1 warrants that Rl +and Ll perform a pure spacetime transformation, leaving +the local operator a(x, t) unaltered (apart from evolving +it in time). On the other hand, when x = lm ± 1 for some +odd m, we have that Rla(x, t)R† +l = 0 as a consequence +of (90). And when x = lm for some odd m, in addition +to a spacetime transformation (x, t) �→ (x′, t′), the local +operator a is processed by the complete positive maps η± +and ˜η± defined in (91) and (108). However, in the limit +l ≫ 1, most locations (x, t) satisfy the premise. +Proof of Lemma 1. In order to prove (100) we need to +write the action of ˜Jx = S T −1Jx on a (trace-less) op- +erator ay ∈ Ay at an arbitrary location y, +˜Jxay ˜J† +x = +� +� +� +� +� +� +� +T †ay+2T +if y ≤ x − 2 +ay +if y ≥ x + 2 +0 +if y = x ± 1 +˜η+(ax) +if y = x +(107) +where +˜η+(a) = u† +x−2Ω+(a)x−1ux−2 +(108) += q−1vx−1trx−2(ux−2ax−2u† +x−2)v† +x−1 ∈ Ax−2,x−1 . +The above has been obtained by applying the translation +S T −1 to the action (90). Importantly, the premise of the +lemma (|x − lm| > 1 for all odd numbers m) implies that +only the first two cases in (90) and (107) are relevant, +which simplifies this proof. +In what follows we analyse the action of Bl on a local +operator ax for the three cases where fl(x) is equal, larger +or smaller than zero, separately. +If fl(x) = 0 then ax +commutes with all operators Jlm and ˜J−lm where m is a +positive odd integer, therefore (97) implies RlaxR† +l = ax. +If fl(x) ≥ 1 then ax commutes with all operators ˜J−lm +and Jlm, except for the Jlm with m ∈ {1, 3, . . . , 2fl(x) − +1}. Note that fl(x) counts the number of non-commuting +factors in Bl. This implies +RlaxR† +l = (J[2fl(x)−1]l · · · Jl)ax(J[2fl(x)−1]l · · · Jl)† += a[x − 2fl(x), fl(x)] , +(109) +where the last equality follows from (90). If fl(x) ≤ −1 +then the operators which do not commute with ax are +˜Jml with m ∈ {2fl(x) + 2, . . . , −3, −1}. Note that |fl(x)| +counts the number of non-commuting factors in Bl. Sim- +larly, we obtain +RlaxR† +l = ( ˜J[2fl(x)+1]l · · · ˜J−l)ax( ˜J[2fl(x)+1]l · · · ˜J−l)† += a[x − 2fl(x), fl(x)] , +(110) +where the last equality follows from (107). +Finally, we invoke (88-89) to obtain the algebraic iden- +tity +Rl(ST) = (ST)Rl , +(111) +which allows to generalise (109) and (110) to the case +a(x, t) with t ̸= 0, +Rla(x, t)R† +l += (BlStT t)a(x − 2t, 0)(BlStT t)† += (StT tBl)ax−2t(BlStT t)† += (StT t)a[x − 2t − 2fl(x − 2t), fl(x − 2t)](StT t)† += a[x − 2fl(x − 2t), t + fl(x − 2t)] . +(112) +This concludes the proof of (100). +15 + +In order to prove (101) we proceed analogously, but this +time, we use the action of ˜Kx = S−1T −1Kx on an arbi- +trary local operator +˜Kxay ˜K† +x = +� +� +� +� +� +� +� +ay +if y ≤ x − 2 +T †ay−2T +if y ≥ x + 2 +0 +if y = x ± 1 +˜η−(ax) +if y = x +(113) +where +˜η−(a) = u† +x+1Ω−(a)x+1ux+1 +(114) += q−1vx−1trx−2(ux−2ax−2u† +x−2)v† +x−1 ∈ Ax+1,x+2 . +Also, we use the fact that (93-94) imply +Ll(S−1T) = (S−1T)Ll , +(115) +to complete the proof. +6 +Outlook +In this work we have introduced conformal QCAs, which +are discrete-spacetime versions of CFTs, and studied their +properties as models of holography. We have obtained sev- +eral results, but we have mostly opened venues for future +research. In what follows we enumerate some of the prob- +lems that will be addressed in future work. +• How much of the CFT phenomenology is covered by +conformal QCAs? Do they have a continuum limit? +• Characterise the algebraic structure of scale and +Lorentz transformations in conformal QCAs. In this +work, the operators implementing these transforma- +tions (C, D, Rl, Ll) have been constructed so that +classical geometries (i.e. tensor-network states) are +mapped to classical geometries, and not superposi- +tions thereof. The reason for this is that it simpli- +fies the visualisation of the dynamics of geometry and +matter. However, if we relax this property, other con- +structions exist which generate a more structured al- +gebra. +• What is the mechanism that produces Poisson level +statistics on the spectrum of a conformal QCA con- +structed with a random dual unitary (which has +Wigner-Dyson level statistics)? +• Our framework allows for calculating the time evolu- +tion of arbitrary (discrete) geometries. However, it is +still not clear whether this dynamics corresponds to +a discrete version of Einstein’s field equations. If this +is the case then conformal QCAs will provide a new +perspective on quantum gravity. +7 +Acknowledgements +I am thankful to Diego Blas, Sougato Bose, Tom Holden- +Dye, Arijeet Pal and Andrea Russo for valuable discus- +sion. This work has been supported by the UK’s Engineer- +ing and Physical Sciences Research Council (grant number +EP/R012393/1). +References +[1] Juan Maldacena. The large n limit of superconformal +field theories and supergravity. 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Quantum effec- +tive action in spacetimes with branes and boundaries. +Physical Review D, 73(6), mar 2006. +[37] Tom Farshi, Jonas Richter, Daniele Toniolo, Arijeet +Pal, and Lluis Masanes. Absence of localization in +two-dimensional clifford circuits. arXiv:2210.10129, +2022. +17 + +t = 0 +t = 0.5 += +t = 1 += +t = 1.5 +t = 2 += +t = 2.5 += +t = 3 += +t = 3.5 += +t = 4 +t = 4.5 += +t = 5 += +1 +Figure 12: Dynamics of flat space with one particle at the corner. Same assumptions (the dual unitary and +a are real) and notation than in Figure 7. Operator inserted at t = .5 is the transpose of that at t = 2, and the same +relation holds for the pairs of times (0, 2.5), (1, 1.5), (3, 4.5), (3.5, 4) and (2.5, 5). The particle reaches the antipodal +point at t = 5, and the period is ∆t = n +2 = 10. +18 + +0 +t = 0 += a +0 +t = .5 +0 +t = 1 +0 +t = 1.5 += +0 +t = 2 += +0 +t = 2.5 += +0 +t = 3 += +0 +t = 3.5 += +0 +t = 4 += +0 +t = 4.5 +0 +t = 5 += +0 +t = 5.5 += +0 +t = 6 += +0 +t = 6.5 += +0 +t = 7 += +0 +t = 7.5 += +0 +t = 8 +1 +Figure 13: Dynamics of toy AdS with one particle. Same assumptions (the dual unitary and a are real) and +notation than Figure 7. Operator inserted at t = 1.5 is the transpose of that at t = 7, and the same relation holds for +the pairs of times (2, 6.5), (2.5, 6), (3, 5.5), (3.5, 5) and (4, 4.5). The particle reaches the antipodal point at t = n +4 = 8, +and the period is ∆t = n +2 = 16. +19 + diff --git a/0NE0T4oBgHgl3EQf_QLA/content/tmp_files/load_file.txt b/0NE0T4oBgHgl3EQf_QLA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5fb7509d3ee07a585433aa5b668a0c124970a0ef --- /dev/null +++ b/0NE0T4oBgHgl3EQf_QLA/content/tmp_files/load_file.txt @@ -0,0 +1,1624 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf,len=1623 +page_content='Discrete holography in dual-unitary circuits Llu´ıs Masanes∗ London Centre for Nanotechnology, University College London, UK Department of Computer Science, University College London, UK January 10, 2023 Abstract We introduce a family of dual-unitary circuits in 1+1 dimensions which constitute a discrete analog of confor- mal field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' These circuits are quantum cellular automata which are invariant under the joint action of Lorentz and scale transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Dual unitaries are four-legged tensors which satisfy the unitarity condition across the time as well as the space direction, a property that makes the model mathematically tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Using dual unitaries too, we construct tensor-network states for our 1+1 model, which are interpreted as spatial slices of curved 2+1 discrete geometries, where the metric distance is defined by the entanglement structure of the state, fol- lowing Ryu-Takayanagi’s prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The dynamics of the circuit induces a natural dynamics on these geome- tries, which we study for flat and anti-de Sitter spaces, and in the presence or absence of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe that the dynamics of spaces with two or more particles differs from that of zero or one, suggesting the presence of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' But this contrasts with the fact that the family of models appears to be non-chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 1 Introduction In 1997 Juan Maldacena proposed a duality between (i) certain theories of quantum gravity with negative cosmo- logical constant in d + 1 spacetime dimensions, and (ii) conformally-symmetric quantum field theories (CFT) in d spacetime dimensions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This duality is also known as the AdS/CFT correspondence, because when the cosmo- logical constant is negative, the ground state of classical gravity is anti-de Sitter space (AdS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Important aspects of the correspondence were soon elaborated by other au- thors [2, 3], and today Maldacena’s paper gathers 22889 citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A central feature of this duality is that the spacetime curvature on the gravity side corresponds to the entangle- ment structure and its dynamics on the CFT side [4–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In [13] this is expressed as holography being a hydrody- namic description of the boundary entanglement with en- tropies as its macroscopic phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This opens the pos- ∗l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='masanes@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='uk sibility of describing the entanglement dynamics of other many-body systems (different than CFT) in terms of ge- ometry with an extra dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' One direction where this exploration has been fruitful is that of discrete hologra- phy, where one considers quantum systems with discrete degrees of freedom (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' spin chains) that are dual to dis- crete geometries [14–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Some advantages of the discrete approach are: mathematical tractability and rigour, more straightforward simulation on classical and quantum com- puters, and more accessible experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this work we present a family of quantum circuits which are invariant under the joint action of Lorentz and scale transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (Recall that invariance under Lorentz transformations alone is impossible on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=') Our circuits are constructed with any given two-site dual- unitary gate and its complex conjugated following the pattern T = · · · · · (1) so that causality is strictly respected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This type of dynam- ics is called quantum cellular automata (QCA) [27–30], and it is the discrete-spacetime version of quantum field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (Recall that in “lattice field theory” time is con- tinuous, which implies the loss of either causality or unitar- ity, both respected by QCAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=') In this work we introduce conformal QCAs in 1+1 dimensions, and leave the general- isation to higher dimensions for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' To our knowl- edge, these are the first QCAs whose evolution operator T has a form of Lorentz and scale invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' An essential in- gredient of these QCAs are dual unitaries: four-legged ten- sors which satisfy the unitarity condition across the space as well as the time direction (7-8), implying the unitarity of circuit (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Dual-unitary circuits have recently attracted a lot of attention because they provide mathematically- tractable models of quantum chaos [31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' After a detailed presentation of conformal QCAs (Sec- tion 2) we explore the holographic properties of these mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In Section 3 we analyse how tensor-network states for 1+1-dimensional QCAs define 2+1 discrete geometries with metric distance defined by the entanglement struc- ture of the state, following Ryu–Takayanagi’s prescrip- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='02825v1 [hep-th] 7 Jan 2023 tion [4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' General Relativity in 2+1 dimensions does not have propagating gravitational degrees of freedom, but it can have non-trivial dynamics for the manifold boundary, which is what we analyse in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that our tensor-network states are constructed with the same dual-unitary tensor as the evolution operator T makes the dynamics of these discrete geometries very natural (see Figure 1), avoiding the need of any duality map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We con- struct tensor networks states representing AdS space, AdS double-sided black hole and thermal AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The discreteness of time in QCAs implies the absence of ground and ther- mal states, which opens the possibility of describing other spaces, like the finite piece of flat space with boundary shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In Section 4 we analyse the dynamics of these spaces in the presence of matter, and observe two very different be- haviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' When there is only one particle, this oscillates from one point of the boundary to its antipodal counter- part, undergoing an orbit with a short period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' That is, the total state evolves similarly in the cases of zero and one particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' On the other hand, when there are two particles or more, the dynamics appears to be of scrambling type, exploring a large space of states which do not close an or- bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The complexity of this dynamics justifies describing these states by their time average, leading to a mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Also, it suggests the possibility that these states contain some probability amplitude of having a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' How- ever, simulations indicate that conformal QCAs do not have quantum-chaotic dynamics [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' That is, even if we construct a circuit (1) with a random dual unitary (which displays Wigner-Dyson level statistics), the par- ticular structure of the evolution operator (1) produces Poissonian level statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This is a very anomalous phe- nomenon that deserves to be explored with more detail in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 2 Conformal QCAs In this section we introduce conformal QCAs and discuss some of their properties, like scale and Lorentz invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='1 Quantum circuits in 1+1 dimensions Consider a chain of n sites labelled by integers x ∈ Zn modulo n, where n is multiple of four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In each site there is a quantum systems with Hilbert space Cq, so the Hilbert space of the chain is Hn = (Cq)⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We denote by A the algebra of matrices acting on Cq, and by Ax,y,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' the algebra of matrices acting on sites x, y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The dynamics of the system is given by the time-translation operator T defined through the circuit T = �� x oddvx ��� x evenux � (2) = · · · u v u v u v u v u v x=−1 0 1 2 x=−1 0 1 2 · · t (3) where the two-site operators ux, vx ∈ Ax,x+1 are unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The action of T on a local operator at the origin a ∈ A0 is TaT † = u† u a v v v† v† ∈ A−1,0,1,2 , (4) reflecting the causality of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The (two-site) space- translation operator S = · · · · · (5) allows for writing the translation-invariance of the dynam- ics as ST = TS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='2 Dual unitaries If we represent the two-site unitary u ∈ A0,1 as a four- legged tensor u = then we can write its transpose as uT = and its conjugation by the swap operator s = as sus† = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Also, we denote complex conjugation with a darker shade u∗ = , so that the Hermitian conjugate is u† = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' With this notation we can say that is unitary if it satisfies the two equivalent conditions = and = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (7) Also, we say that is a dual unitary if, in addition to unitarity, it satisfies the two equivalent conditions = and = , (8) which can be phrased as “unitarity in the space direction”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Dual unitarity implies that, for any traceless local op- erator a ∈ A0, the partial trace of uau† ∈ A0,1 on site 1 vanishes, tr1(u0a0u† 0) = a = a = 10 tr(a) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (9) In other words, the operator uau† ∈ A0,1 is a linear com- bination of terms of the form 10 ⊗ c1 and b0 ⊗ c1 with traceless b, c, but not of the form b0 ⊗ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' That is, all terms in uau† act non-trivially on site 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Returning to our model, if u, v are dual unitaries then a local operator a ∈ Ax on an even site x evolving as T taT −t grows towards the right at maximal speed, which in lat- tice units is c = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The operator T taT −t may also grow towards the left and develop a highly non-local and com- plex structure, but that is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Alternatively, if the initial operator a ∈ Ax is located on an odd site x then T taT −t grows towards the left at maximal speed −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 2 t = 0 0 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 1 0 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 2 0 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 3 0 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 4 0 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 5 0 1 Figure 1: Dynamics of flat space with a boundary of n = 20 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each rectangle represents a tensor-network state of the QCA, where black dots in a blue line are the free legs that correspond to Cq systems, and site x = 0 is marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' At each time step t we depict the state T t|Ψfl⟩ ∈ H20, and at semi-integer times t we depict TevenT ⌊t⌋|Ψfl⟩, where ⌊t⌋ is the largest integer less than or equal to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The sequence produces an orbit of period ∆t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we approximate one of these rectangles by an infinite spatial strip, then this dynamics resembles that of General Relativity, where the width of the spatial strip decreases until it collapses and bounces back (see [35,36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Hence, we see that the even/odd location x ∈ Zn plays the role of a momentum ±2 quantum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In sum- mary, every perturbation in a dual-unitary circuit grows at maximal speed towards the right, the left, or both, as in CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='3 Free particles and quantum chaos In order to simplify the discussion of this subsection (only) we restrict ourselves to circuits (2) with v = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The first thing to do when we are given a dual unitary u = is to obtain the spectral decomposition of the maps Ω+ : A0 → A1 and Ω− : A1 → A0, defined as Ω+(a0) = 1 q tr0(u0a0u† 0) = 1 q a , (10) Ω−(a1) = 1 q tr1(u0a1u† 0) = 1 q a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (11) The eigenvectors of Ω+ with unimodular eigenvalue Ω+(e) = eime satisfy TexT † = eimex+2 , (12) for all even x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Analogously, the eigenvectors of Ω− with unimodular eigenvalue Ω−(e) = eime satisfy TexT † = eimex−2 , (13) for all odd x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the dual-unitary and QCA literature, these operators are respectively called right/left-moving solitons [34] and gliders [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' When acting on a state, these operators can create a free particle with velocity ±2 and quasi-mass m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The eigenvectors Ω±(e) = λe with eigenvalue modulus less than one |λ| < 1 grow under T in a scrambled fashion which fills up all the lightcone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This dynamics displays many signatures of quantum chaos, including the profile of the spectral form factor [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='4 Definition of conformal QCA Let us define the family of circuits introduced and analysed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For any given dual unitary u = we define the following time-translation operator T = · · · 0 0 · · (14) where site x = 0 is marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Note that, for any four local unitaries a, b, c, d, the new dual unitary u′ = a0b1u c0d1 = a b c d (15) defines a new circuit T ′ via (14) which is equal to T up to a local change of basis, T ′ = (· · · a0b1d2c3a4 · · · )T(· · · a0b1d2c3a4 · · · )† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This reminds the structure of a gauge theory, but it is not the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 3 Another property of the structure of T is that it pro- duces a cancellation of the phases of travelling solitons, forcing them all to be massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' That is, if e0 satisfies u0e0u0 = eime1 then TexT † = ex+4 for all even x (and analogously for odd x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Recall that CFTs can only have massless particles - however this is not sufficient to be con- formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the next section we show that T is invariant under scale transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 Scale invariance The discussion in this section requires the size of the chain n to be a multiple of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We start by defining the contrac- tion isometry C : Hn → H n 2 as C = q− n 8 � · · 0 · · � (16) which maps an n-site chain to an n 2 -site chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The par- ticular structure of operators C and T together with the dual unitarity of their building block allows to easily calculate the product CT in a manner that is independent of the choice of : CT = · · · · · = · · · · · = · · · · · = · · · · · (17) Continuing in a similar fashion we obtain CT = · · · · · = · · · · · (18) The above can be synthesised as C2nT 2 2n = TnC2n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (19) where we have added a subindex to the operators to indi- cate the size of the chain where they act on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The above equation tells us that, the action of Cn produces a rescal- ing of space and time by a factor 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' By using the dual unitary constraints (7) and (8) we can calculate the action of C on a local operator a ∈ Ax, CaxC† = � 0 if x = 0, 1, 2, 3 mod 8 a x 2 if x = 4, 5, 6, 7 mod 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (20) This action depends on the position x in a very non- smooth way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, the action of C is smooth on oper- ators having a good field-theory limit Φn(x) = � y∈Zn ϕ(y − x)ay , (21) where ϕ(y) is a smearing function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' a gaussian) centred around the origin and spreaded over a large number of lattice units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case we have CΦn(x)C† ≈ 1 2Φ n 2 ( x 2) , (22) where the subindex of Φn stresses that the field operators on the left- and right-hand sides act on chains of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' It is possible to construct contraction isometries with scale factor different than 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This can be achieved by separating the vectors in (16) by a length different than 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6 Lorentz transformations In this subsection we give a brief summary of Lorentz transformations on conformal QCAs, and refer to Sec- tion 5 for the complete presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Lorentz transfor- mations are more clearly discussed in an infinite chain, so here and in Section 5 we assume n = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In Section 5 we define the isometry Rl : H∞ → H∞ which jointly implements a contraction and a spacetime transformation which resembles a Lorentz boost towards the right, with velocity v parametrised by the positive integer l as v = 2 √ 4 − 2l + l2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (23) These transformations commute with the translations in the diagonal direction (x, t) = (1, 1), Rl(ST) = (ST)Rl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (24) The action of Rl on a local operator a(x, t) := T taxT −t can be informally described as Rla(x, t)Rl = � a(x′, t′) for most (x, t) complicated for a few (x, t) (25) 4 where, in the regime |x| ≫ l, the transformed coordinates (x′, t′) can be written as x′ = � 1 − 1 l � x + 2 l t t′ = � 1 − 1 l � t + 1 2lx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (26) The label “complicated” in (25) stands for a transforma- tion that is not purely spacetime, as in the first case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, note that transformation (26) preserves Minkowski’s met- ric up to a scale factor (ct′)2 − x′2 = � 1 − 2 l � � (ct)2 − x2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (27) (Recall that the speed of light is c = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=') Now, we can remove the scale transformation from (26) by dividing the new coordinates (x′, t′) by the scale factor � 1 − 2/l, so obtaining the pure Lorentz transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Once this is done, we simplify the first equation by imposing x = 0, obtaining x′ � 1 − 2/l = 2/l � 1 − 2/l t = v √ 1 − v2 t , (28) where the second equality follows from the standard form of a Lorentz transformation with velocity v and x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This second equality can be use to isolate v as a function of l and confirm the relation (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In Section 5 we also define the isometry Ll : H∞ → H∞, which jointly implements a contraction and a spacetime transformation which resembles a Lorentz boost towards the left, with velocity v parametrised by the positive inte- ger l as v = −2 √ 4 − 2l + l2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (29) The action of Ll on a local operator a(x, t) := T taxT −t is Lla(x, t)Ll = � a(x′, t′) for most (x, t) complicated for a few (x, t) (30) where, in the regime |x| ≫ l, the transformed coordinates (x′, t′) can be written as x′ = � 1 − 1 l � x − 2 l t t′ = � 1 − 1 l � t − 1 2lx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (31) Note that this transformation also preserves Minkowski’s metric up to the same scale factor � 1 − 2/l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Like with the scale transformations described in the pre- vious subsection, the action of Bl and Ll becomes smooth on operators of the form Φ(x, t) = T tΦ(x)T −t, where the smeared operator Φ(x) is defined in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular we have RlΦ(x, t)R† l ≈ � 1 − 1 l � Φ(x′, t′) (32) for all (x, t), not just “most”, avoiding the “complicated” cases of (25) and (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Naturally, the coordinates (x′, t′) in (32) satisfy (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Interestingly, the conjugated operators R† l and L† l imple- ment a Lorentz boost together with a dilation (instead of a contraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' And in this case, the boost directions are re- versed: R† l is a left boost and L† l a right boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Therefore, the composition R† l Ll produces a Lorentz transformation without a scale transformation, but it has a non-trivial kernel, and hence, it is not an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='7 Two-layer conformal circuits To simplify notation we redefine u as the coarse-grained (dual) unitary u = = (33) where the double-arrow notation encapsulates the new symmetry uT = sus†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We also redefine q so that the Hilbert space of a coarse-grained site has dimension q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Now we can write the evolution operator (14) as a two- layer circuit T = · · · · · (34) In the rest of this work, our starting point is a dual unitary u = with the symmetry uT = sus†, and the time- translation operator (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Note that this dynamics is more general than the coarse-grained four-layer circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 3 Discrete holography In this section we construct tensor-network states for 1+1 QCAs, and show that they can be interpreted as spacial slices of 2+1 discrete geometries with metric distance de- fined by their entanglement structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='1 Tensor-network states and dynamics By using the four-legged tensor , its complex conju- gated and their rotated versions, we can construct tensor- network states for the chain Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' One example of such states for a chain of n = 20 sites is |Ψfl⟩ = 1 q5 −1 0 1 2 , (35) where each black dot in the blue line represents a free leg of the tensor network, and hence, a Cq system of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The label “fl” stands for “flat”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Naturally, the black dots represent the sites of the chain Z20 in the same order, like the marked sites x = −1, 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 When the state evolves via the quantum circuit |Ψfl⟩ → T |Ψfl⟩, the corresponding tensor network also evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' To simplify the calculation of this evolution, it is convenient to separate each of the two layers of the time-translation operator as T = ToddTeven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' With this notation we can write the evolution of |Ψfl⟩ as Teven |Ψfl⟩ = = , (36) and ToddTeven |Ψfl⟩ = = , (37) where here and in the rest of the paper we ignore the normalisation of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Note that the action of T on the tensor network is not fully determined until we know the position of one site, like for example x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The complete evolution of |Ψfl⟩ is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Remarkably, this evolution is cyclic and has a very small period (∆t = 5) when compared with the approximate recurrence time of a typical state in H20, which would be doubly exponential in the size (∼ exp q20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This (flat- space) tensor network can be generalised to any boundary size n multiple of four, with corresponding period ∆t = n 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Of course, there are many other tensor-network states, and we analyse some of them below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Consider a state |Φ⟩ and a positive integer ∆t satisfying T ∆t |Φ⟩ = |Φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If ∆t is the smallest such integer, then they generate the orbit T t |Φ⟩ for t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' , ∆t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (An example of orbit is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=') Given any such orbit we can construct some eigenstates of T as |Φω⟩ = ∆t−1 � t=0 ei 1 ∆t ωt T t |Φ⟩ , (38) T |Φω⟩ = ei 1 ∆t ω |Φω⟩ (39) for ω = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' , ∆t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Hence, it is meaningful to associate to this subspace a dynamical mode of (quasi) energy E = 1 ∆t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (40) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='2 Entanglement geometry The above tensor-network states can be interpreted as 2D spatial geometries, where the metric distance is fixed the a) h = 5 log q b) h = 2 log q c) 2 4 2 1 1 2 4 4 2 3 5 1 1 3 5 3 3 3 5 2 2 6 6 1 Figure 2: Entanglement and geodesics in flat space (tensor network from Figure 1 at t = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figures a) b): the entanglement entropy h between the region consisting of red dots and the rest of the chain is equal to the small- est number of black lines (times log q) that are crossed by a curve enclosing the red dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In very light grey, the position of the gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure c): distance between each location and that with the pink dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each white rhom- bus represents a location in the bulk, which in this case has flat curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Locations in the boundary of the bulk correspond to links in the chain Z20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' entanglement structure of the state, via Ryu-Takayanagi’s prescription [4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This identifies the von Neumann en- tropy of a set of consecutive sites in the chain with the length of the shortest curve beginning and ending at the boundary of the set (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Note that we can apply this prescription to spaces other than AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us obtain the metric distance of the simplest tensor- network state |Ψfl⟩ = 1 q 0 1 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (41) The unitary condition (7) implies that the reduced density matrix of subsystem {0, 1} is proportional to the identity matrix, which implies that |Ψfl⟩ is maximally entangled with respect to the bipartition 01|23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Similarly, the dual- unitary condition (8) implies that |Ψfl⟩ is maximally en- tangled with respect to the bipartition 03|12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This in turn implies that each site (0,1,2 or 3) is maximally entangled with the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This precise entanglement structure is con- tained in the geometry , where each of the 4 triangles represents a location in the surface, and the distance be- tween two locations is given by the number of black lines that are crossed when travelling from one location to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the previous example, the emergent geometry does not contain interior points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 2 depicts the geome- try of a more complex tensor-network state, which has a non-trivial interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked the equality between entropy and distance for all bipartitions, but we have not proven that this geometry uniquely captures the entangle- ment structure of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' But in any case, this geom- etry is special, because it has the same structure as the underlying tensor network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Now, we can use the distance defined in Figure 2 to calculate the length of any curve 6 in the bulk, not necessarily the shortest one connecting a pair boundary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This reveals that the geometry in question is a piece flat space with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the continuum setup (AdS/CFT) there is an exten- sive literature [8–13] addressing the problem of how to ob- tain the bulk geometry given the entanglement structure of the CFT state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the following subsections we discuss the geometry of relevant states in Hn, which generate dis- crete versions of AdS space with and without a black hole, and the double-sided AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='3 Anti-de Sitter state The dilation isometry Dn : Hn → H2n maps a chain of length n (multiple of 4) to a chain of length 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Its par- ticular form is Dn = q−n/8� · · −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 · · � , (42) where we have included the site labels x of the input and output chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Note that the dilation D is the Hermitian conjugate of the contraction C defined in (16), except that C is defined for the four-layer circuit (14) and here we define D for the two-layer circuit (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The toy AdS state is recursively defined as |Ψtoy 4 ⟩ = 1 q 0 1 2 3 , (43) |Ψtoy 2n ⟩ = Dn |Ψtoy n ⟩ , (44) which results in |Ψtoy n ⟩ = D n 2 · · · D16D8D4 � �� � ϱads |Ψtoy 4 ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (45) Note that the size of the chain is n = 2ϱads4, where the pos- itive integer ϱads denotes the number of recursions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This tensor network is depicted in Figure 3, where we can see that each recursion corresponds to a value of the radial coordinate ϱ = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' , ϱads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Also, the intermediate state |Ψtoy 2ϱ4⟩ in the recursion (45) represents a spatial slice of AdS with the radial coordinate restricted to the interval [0, ϱ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' By proceeding as in (36) and (37) we can check that the 4-site toy AdS state (43) is an eigenstate of the evolution operator T4 |Ψtoy 4 ⟩ = |Ψtoy 4 ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (46) Also, by proceeding as in (18) we obtain the equality T 4 2nDn = DnT 2 n , (47) which implies that, when the radial coordinate ϱ decreases by one unit, time slows down by a factor of two: T 4 2n |Ψtoy 2n ⟩ = DnT 2 n |Ψtoy n ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (48) In Section 4 we show that the relationship (48) between time at different radial locations applies to local clocks made of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The state |Ψtoy n ⟩ is not an eigenstate of Tn, it evolves in time through the orbit shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The corre- sponding period can be calculated by using equations (46) and (47), obtaining (Tn)2ϱads+1 |Ψtoy n ⟩ = |Ψtoy n ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (49) That is, the period of this orbit is proportional to the length of the chain ∆t = 2ϱads+1 = n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Again, note that this is much shorter than the recurrence time of a typical state in Hn, which would be doubly exponential in the size (∼ exp qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Using (40) we associate to empty AdS an energy Eads = 2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (50) The geometry of the tensor network of |Ψtoy n ⟩ is not reg- ular along the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For instance, the geodesic distance from a boundary point x to the centre (ϱ = 0) depends on x, as shown in the red lines of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In- terestingly, the T-eigenstate |Ψads n ⟩ = n 2 −1 � t=0 T t |Ψtoy n ⟩ , (51) has a more regular geometry in the large-q limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Specifi- cally, it produces the metric distance ∆s2 Ψads = log2q � −22ϱ∆τ 2 + ∆ϱ2 + 22ϱ ∆θ2 π2 � , (52) to leading order in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (The proof of this fact will be pre- sented elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=') Recall that this distance characterises a discrete geometry, hence, the increments ∆τ, ∆ϱ, ∆θ are discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The radial coordinate ϱ ∈ [0, ϱads] ⊂ Z has already been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The time coordinate τ at the boundary is related to the QCA time via t = log q 2ϱadsτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Note that the proper time defined by (52) reflects the ra- dial dependence implied by (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The angular coordinate θ ∈ [0, 2π] is discretised in 2ϱ4 steps of size ∆θ = 2π2−ϱ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Hence, each step has one unit of proper distance and the proper distance of a complete circle is log q 2ϱ4, the log- arithm of the dimension of the chain in the correspond- ing intermediate recurrence step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Finally, note that the distance (52) strongly resembles AdS’s distance in global coordinates ds2 AdS = α2 � − cosh2ϱ dτ 2 + dϱ2 + sinh2ϱ dθ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (53) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='4 Thermofield double state In the previous subsection we saw that eigenstates produce smoother geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, eigenstates require super- positions of tensor networks, and hence, are less easy to 7 s = 0 s = 3 log q D16 ϱads = 3 D8 ϱ = 2 D4 ϱ = 1 |Ψtoy 4 〉 ϱ = 0 31 0 8 13 18 24 T 2 4 1 Figure 3: Toy AdS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The left figure shows (continuum) AdS tiled with equal-size triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The right figure displays the the tensor network of the toy AdS state for a chain of size n = 32 with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each level of the four recursions is identified by the corresponding value of the radial coordinate ϱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In red, the minimal length paths from boundary locations x = 13, 18 to the centre, and their length s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Which shows that this “toy” version of AdS is very non-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' t = 0 0 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 1 0 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 2 0 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 3 0 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 4 0 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 5 0 t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 6 0 t = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 7 0 t = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 8 0 1 Figure 4: Dynamics of the toy AdS state |Ψtoy 32 ⟩ represented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The position of the chain location x = 0 is represented at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 8 D16 D8 D4 |Ψads 4 〉 0 T 2 4 1 Figure 5: Thermofield double state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This tensor net- work represents a toy version of a two sided black hole in AdS with boundary radius ϱads = 3 and horizon radius ϱh = 0, which implies boundary size n = 32 and horizon size a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Periodic boundary conditions are understood in the two chains and the throat (the piece connecting the two symmetric sides).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The throat has been growing for 2 local time steps, which implies that the QCA has been evolving for t = 2 × 23 = 16 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' visualise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason, in this subsection, we continue using non-eigenstate tensor-network states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The thermofield double (TFD) is a joint state of two identical chains Hn⊗Hn evolving in time via the dynamics T ⊗T ⊺, where T ⊺ is the transpose of T (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This state is characterised by the largest and smallest values of the radial coordinate outside the throat ϱ ∈ [ϱh, ϱads] ⊆ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' These parameters fix the size of each chain to n = 2ϱads4, and the area (length) of the horizon to a log q, where we define a = 2ϱh4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The TFD with n = a is the maximally- entangled state between the two chains |Ψtfd a,a⟩ = � x∈Za |ψ⟩x , (54) |ψ⟩x = 1 √q q � k=1 |k⟩x ⊗ |k⟩x , (55) where |ψ⟩x is the maximally-entangled state between site x of one chain and site x of the other chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' When n > a the TFD can be recursively generated via |Ψtfd 2n,a⟩ = Dn ⊗ D∗ n |Ψtfd n,a⟩ , (56) where D∗ n is the complex conjugate of Dn, defined in (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The TFD can be explicitly written as |Ψtfd n,s⟩ = (D n 2 · · · D2aDa � �� � ϱads−ϱh ) ⊗ (D n 2 · · · D2aDa � �� � ϱads−ϱh )∗ |Ψtfd a,a⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (57) Next, let us analyse the dynamics of the TFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Proceed- ing as in (18) we obtain the identity (T ⊺ 2n)4D∗ n = D∗ n(T ⊺ n)2 , (58) which is not equivalent to (47), although here it produces a similar result: when the radial coordinate ϱ is decreased D16 D8 D4 |Ψads 4 〉 0 T 2 4 1 Figure 6: Thermal AdS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This tensor network rep- resents the toy version of a simple black hole in AdS, for a chain of size n = 32 and horizon area a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The two blue lines correspond to the input and output of the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Periodic boundary conditions are understood in the chain and the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' by one, time slows down by a factor two (T2n ⊗ T ⊺ 2n)4 |Ψtfd 2n,a⟩ = (Dn ⊗ D∗ n)(Tn ⊗ T ⊺ n)2 |Ψtfd n,a⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Now we recover the standard fact that the throat worm- hole grows linearly in time, since the action of T ⊗ T ⊺ cannot be simplified Ta ⊗ T ⊺ a |Ψtfd a,a⟩ = T 2 a ⊗ 1 |Ψtfd a,a⟩ , (59) as also illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Note however, that the propper time in the throat is slower than that on the boundary by an exponential factor 2(ϱads−ϱh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 Thermal anti-de Sitter state If we perform the partial trace on the right chain in the TFD then we obtain the mixed state ρn,a = trright|Ψtfd n,a⟩⟨Ψtfd n,a| = (D n 2 · · · D2aDa � �� � ϱads−ϱh )1a(D n 2 · · · D2aDa � �� � ϱads−ϱh )† , (60) where 1a is the identity acting on Ha (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that Dn are isometries implies that the entropy of ρn,a is a log q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' When n = a the state ρads a,a is time-independent and maximally mixed, which corresponds to infinite tem- perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Like in the previous variants of AdS, here we also have the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' When n > a, when the radial coordinate ϱ decreases by one unit, time slows down by a factor two (T2n)4ρ2n,a(T † 2n)4 = Dn (Tn)2ρn,a(T † n)2D† n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (61) This implies that the evolution of the thermal AdS state undergoes a cycle of period ∆t = 2ϱads−ϱh+1 = 2n a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Hence, we associate to it an energy E Eads = a 4, (62) 9 where we have substituted the energy of pure AdS Eads obtained in (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' At this stage, it is not clear how to interpret this identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 4 Spaces with matter In the previous section we considered tensor-network states (45) constructed with the building blocks and , and interpreted them as curved empty spaces for the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this section we consider bulk spaces with matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We say that a state contains matter (when interpreted as a bulk state) when it cannot be written as a tensor network, or a super-position thereof, constructed with the building blocks and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The states with matter that we analyse are empty spaces with the addition of some (non-building blocks) operators on a small number of links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' When only one link is affected we interpret it as the position of a particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='1 Ambiguity in the position of particles Let us see that, if the dual unitary is chaotic (has no solitons), then there are particle states with a well-defined position for the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us start by considering the (empty) flat space state |Ψfl⟩ = 0 , (63) and apply an arbitrary operator = a0 ∈ A0 at the chain site x = 0 a0 |Ψfl⟩ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (64) We interpret this state as the empty space (63) with one particle at its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, the same sate can also be written as a0 |Ψfl⟩ = , (65) where the two-site operator is defined by = , (66) which then satisfies = , (67) and implies the equality between (64) and (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Analo- gously, the operator = , (68) allows to write the same state (64) as a0 |Ψfl⟩ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (69) And, if we use the operator = , (70) then we can write the same state as a0 |Ψfl⟩ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (71) Clearly, there is a large number of ways of writing the same state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If a is a soliton of , or a linear combination thereof, then the operators , and defined in (66), (68) and (70) act non-trivially on a single site only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Contrary, if is generic then, in addition of having no solitons, we expect that all alternative ways of writing the one- particle state (64) involve operators with terms acting non- trivially in more than one site, like for example = b⊗c+· · · This fact eliminates the ambiguity of the particle position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Therefore we use the following prescription: If a state can be written as a tensor network of the building blocks, and one extra tensor on a single link, then we interpret this state as a space with one particle at the position of mentioned link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This prescription allows us to put a particle at any lo- cation in the bulk (not necessarily the boundary), like for example .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (72) However, In the following subsection we see that this pre- scription cannot be applied to all single-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='2 Dynamics of spaces with one particle We have already seen the dynamics of the empty-space state |Ψfl⟩ in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Now, let us explore what happens when we add one particle a0 |Ψfl⟩ at the boundary site x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 7 shows that the geometry of T ta0 |Ψfl⟩ evolves like that of T t |Ψfl⟩, if we represent the particle 10 t = 0 = a t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 = t = 1 = t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 = T t = 2 = t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 = t = 3 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 t = 4 = T t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 t = 5 1 Figure 7: Dynamics of flat space with one particle - sequence of states at different times t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' At t = 0, initial state of geometry (in grey) and particle (red dot represents arbitrary operator a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' At each half time step t, state of geometry and particle, and below, definition of the operator representing the particle in terms of operators defined in previous time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each green operator is the transpose of the red operator with the same form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' At t = 2 and t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 the operators shrink - this requires the dual unitary and the operator a to be real, because then operator equalities at t = 2 and t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 are equivalent to those at t = 1 and t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' At t = 5, the geometry returns to its original state but the particle is in the antipodal position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Hence the period is ∆t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' with certain inserted operators, also defined in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This evolution has a very short period (∆t = 10) when the dual unitary and the operator are real = , a = a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (73) (The same holds if, instead of real, the operator is symmet- ric a = a⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=') This implies that the subsequent operators = = (74) are real too;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' and it makes the operator equalities at t = 2 and t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 the transposition of equalities at t = 1 and t = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 respectively, rendering them equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This implies the shrinking of operators at t = 2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5, and a return to the original a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Note that the recurrence time of a0 |Ψfl⟩ ∈ Hn is t = n 2 , while that of a typical state in Hn is t ∼ exp qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 7 also shows that some intermediate states can- not be written with a single-site tensor representing the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Hence, as mentioned above, the position of the particle in these states is not well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' When the particle ax is located at other points of the boundary of flat space, the corresponding state ax |Ψfl⟩ ex- periences an evolution similar to that with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Specif- ically, for any x, the particle reaches the antipodal point at t = n 4 , and the period of the evolution is ∆t = n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fig- ure 12 shows the evolution of the same flat space when the initial location of the particle is a corner (x = −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 13 shows the dynamics of toy AdS with a particle at the boundary location x = 0, that is a0 |Ψtoy 32 ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case we also observe that the particle reaches the antipo- dal point at t = n 4 and the recurrence happens at t = n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, contrary to flat space, certain initial positions of the particle (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' a9 |Ψtoy 32 ⟩) generate a state with recur- rence time much longer than ∆t = n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the following section we observe that two-particle states also give rise to this second type of dynamics, with longer recurrence times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='3 One vs two-particle dynamics In this subsection we compare the dynamics of states with different numbers of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We do so by numerically obtaining the dimension of the effective subspace explored by the evolution of each such state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This is done with the dynamics of the four-layer circuit (14) generated by a randomly sampled dual unitary with real entries and local dimension q = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We use the four-layer circuit to get rid of the constraint = and, in this way, enlarge the size of the set of dual unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This fact increases the similarity between the properties of different instances of the random dual unitary, due to concentration of measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For the sake of simplicity we perform the above- described comparison with the small empty-space state |Ψ⟩ = 0 1 −1 ∈ H8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (75) We calculate its dynamics by proceeding as in (36) and (37) but with the four-layer circuit T = T[4]T[3]T[2]T[1] 11 described in (14), obtaining the cycle T[1] |Ψ⟩ = 0 , T[2]T[1] |Ψ⟩ = 0 , T[3]T[2]T[1] |Ψ⟩ = 0 , T |Ψ⟩ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (76) We observe that the period is ∆t = 1, and so, |Ψ⟩ is an eigenstate of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying an operator = a ∈ A then we obtain the state a0 |Ψ⟩ = 0 1 −1 , (77) which has the similar evolution T[1]a0 |Ψ⟩ = 0 , T[2]T[1]a0 |Ψ⟩ = 0 , T[3]T[2]T[1]a0 |Ψ⟩ = 0 , Ta0 |Ψ⟩ = 0 , (78) where = a⊺ ∈ A is the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' That is, after one time step, the particle is in the antipodal location (x = 4), so the period of the state a0 |Ψ⟩ is ∆t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x ∈ Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period ∆t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Because of the existence of simple eigenstates, we consider the zero and one-particle subspace “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above results with the dynam- ics of the following two and three-particle states , , , (79) generated with a random matrix = a ∈ A with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For each such pure state |Ψ⟩⟨Ψ| we numerically calculate its time average after t time steps ρ(t) = 1 t + 1 t � r=0 T r|Ψ⟩⟨Ψ|T −r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the effective dimension of its support deff(t) = � trρ2(t) �−1 , (81) which takes into account the different weights of the eigen- values, as in the second-order Renyi entropy via log2 deff = h2(ρ) = − log2 ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For example, if a states ρ is propor- tional to a projector ρ2 ∝ ρ then deff is equal to the di- mension of the projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The effective dimension of the above two and three-particle states is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe that the corresponding curves attain a much larger value than those of the zero and one-particle states, 20 40 60 80 100 120 140 5 10 15 20 25 30 35 1 11 1 1 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 Figure 8: Effective dimension of 1,2,3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each curve shows the dimension deff(t) of the effective subspace as a function of time t, generated by the evolution of the tensor-network state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The top curve is gener- ated by a random state with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The dynamics is generated by the four-layer circuit (14) with a real random dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the single-particle state and the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The differences be- tween the other four states are not essential, because they vary according to the instance of random dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' which have deff(∞) = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' and take a longer time to equi- librate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addition, the lack of convergence to an integer value suggests that the corresponding evolutions do not generate a closed orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 8 also shows that the behaviour of the states (79) is similar to that of a random state in H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This establishes a sharp distinction between one particle or less on one side, and two particles or more on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' These results open the possibility that the dynamics in the two or more particles subspace is quantum-chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, the results of the following subsection suggest that the contrary is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='4 Absence of quantum chaos First of all, let us see that our method for sampling ran- dom dual unitaries generates an ensemble qualitatively similar to that of Haar-random unitaries (with no dual- ity constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Actually, since we generate random dual unitaries with real entries, we need to compare these with Haar-random orthogonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In order to do so, we generate a set U of 85 random dual unitaries u ∈ U with real entries, and we calculate the spectral form factor KU(t) = 1 |U| � u∈U ��tr(ut) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (82) The result are the red points in Figure 9, which are con- trasted with the form factor of the orthogonal group KSO(d)(t) = � 2t − t log2(1 − 2t/d) if t < d 2d − t log2 � 2t+d 2t−d � if t ≥ d , (83) 12 5 10 15 2 4 6 8 10 12 A then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) where = a| 2 A is the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' That is, after one time step, the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single- particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Because of the existence of simple zero and one-particle eigenstates, we consider these cases somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above results with the dynam- ics of the following two and three-particle states , , , (79) generated with a random matrix = a 2 A with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r| ih |T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the e↵ective dimension of its support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values as in the second-order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For states ⇢ proportional to a pro- jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the dimension of the projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimension of the above two and three-particle states is plotted in Fig- ure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The corresponding curves attain a much larger value than those of the zero and one-particle states, which have de↵(1) = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' and take a longer time to equilibrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addition, the lack of convergence to an integer value sug- gests that the corresponding evolutions do not generate a closed orbit, and hence, do not have simple eigenstates as- sociated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 8 also shows that this behaviour is very similar to that of a random state | i 2 H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This establishes a sharp distinction between one particle or less on one side, and two particles or more on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' These results open the possibility that the dynamics of two or more particles is quantum-chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, the results of the following subsection suggest that the contrary is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 1 11 1 1 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 Figure 8: E↵ective dimension of 1,2,3-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each curve shows the dimension of the e↵ective subspace de↵(t) as a function of time t, generated by the evolution of the tensor-network state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The top curve is generated by a random state with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The dynamics is generated by the four-layer circuit (14) with a random real dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualita- tive distinction between the single-particle state and the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The di↵erences between the other four states are not essential, because they vary according to the instance of random dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 9: Spectral form factor of dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='4 Absence of quantum chaos First of all, let us see that our method for sampling real random dual unitaries generates an ensemble qualitatively similar to that of Haar-random real unitaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' orthog- onal matrices) without the duality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In order to do so, we generate 85 random dual unitaries u 2 U with real entries, and calculate the spectral form factor K(t) = 1 85 X u2U ��tr(ut) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (82) In Figure 12 A then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) where = a| 2 A is the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' That is, after one time step, the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single- particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Because of the existence of simple zero and one-particle eigenstates, we consider these cases somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above results with the dynam- ics of the following two and three-particle states , , , (79) generated with a random matrix = a 2 A with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r| ih |T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the e↵ective dimension of its support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values as in the second-order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For states ⇢ proportional to a pro- jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the dimension of the projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimension of the above two and three-particle states is plotted in Fig- ure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The corresponding curves attain a much larger value than those of the zero and one-particle states, which have de↵(1) = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' and take a longer time to equilibrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addition, the lack of convergence to an integer value sug- gests that the corresponding evolutions do not generate a closed orbit, and hence, do not have simple eigenstates as- sociated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 8 also shows that this behaviour is very similar to that of a random state | i 2 H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This establishes a sharp distinction between one particle or less on one side, and two particles or more on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' These results open the possibility that the dynamics of two or more particles is quantum-chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, the results of the following subsection suggest that the contrary is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 1 11 1 1 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 Figure 8: E↵ective dimension of 1,2,3-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each curve shows the dimension of the e↵ective subspace de↵(t) as a function of time t, generated by the evolution of the tensor-network state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The top curve is generated by a random state with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The dynamics is generated by the four-layer circuit (14) with a random real dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualita- tive distinction between the single-particle state and the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The di↵erences between the other four states are not essential, because they vary according to the instance of random dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 9: Spectral form factor of dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='4 Absence of quantum chaos First of all, let us see that our method for sampling real random dual unitaries generates an ensemble qualitatively similar to that of Haar-random real unitaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' orthog- onal matrices) without the duality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In order to do so, we generate 85 random dual unitaries u 2 U with real entries, and calculate the spectral form factor K(t) = 1 85 X u2U ��tr(ut) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (82) In Figure 12 Figure 9: Spectral form factor of dual unitary with real entries and local dimension q = 3, averaged over 85 random instances (red dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Spectral form factor of the orthogonal group SO(d) of dimension d = q2 = 9 (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Both plots display the initial “dip” and are qualita- tively similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This signals the presence of quantum chaos in random dual unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' with d = q2 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe in Figure 9 that the two form factors are qualitatively similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, both dis- play the so called “dip” for t < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This behaviour signals the presence of quantum chaos in random dual unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' With the above mentioned 85 instances u ∈ U, we con- struct 85 instances of the four-layer evolution operator T for the chains with n = 4 and n = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Using formula (82) but replacing u by T, we calculate the correspond- ing form factors and plot them in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We ob- serve that both form factors are essentially flat, with no “dip”, a behaviour characteristic of Poisson level statistics, which appears in “integrable” systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This is very sur- prising, because both evolution operators are constructed with unitaries u which, as discussed above, display quan- tum chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Hence, we conclude that the special structure of the conformal circuit (14) cancels the chaos present in the building block .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' It is proven in [37] that the asymptotic value of the form factor is K(∞) = � E g2 E , (84) where E are the eigenvalues of the evolution operator and gE the corresponding degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the absence of degen- eracies (gE = 1) we have K(∞) = qn, which is the stan- dard value in generic models including random unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, the two approximately constant plots in Fig- ure 10 show values for K(t) much larger than the Hilbert- space dimension qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This signals that the spectrum of T is highly structured in a way that leads to large degener- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This structure will be studied in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is 50 100 150 500 1000 1500 2000 100 200 300 400 500 600 700 500000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='0×106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5×106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='0×106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5×106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='0×106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5×106 A then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) where = a| 2 A is the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' That is, after one time step, the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single- particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Because of the existence of simple zero and one-particle eigenstates, we consider these cases somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above results with the dynam- ics of the following two and three-particle states , , , (79) generated with a random matrix = a 2 A with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r| ih |T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the e↵ective dimension of its support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values as in the second-order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For states ⇢ proportional to a pro- jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the dimension of the projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimension of the above two and three-particle states is plotted in Fig- ure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The corresponding curves attain a much larger value than those of the zero and one-particle states, which have de↵(1) = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' and take a longer time to equilibrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addition, the lack of convergence to an integer value sug- gests that the corresponding evolutions do not generate a closed orbit, and hence, do not have simple eigenstates as- sociated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 8 also shows that this behaviour is very similar to that of a random state | i 2 H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This establishes a sharp distinction between one particle or less on one side, and two particles or more on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' These results open the possibility that the dynamics of two or more particles is quantum-chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, the results of the following subsection suggest that the contrary is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 1 11 1 1 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 Figure 8: E↵ective dimension of 1,2,3-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each curve shows the dimension of the e↵ective subspace de↵(t) as a function of time t, generated by the evolution of the tensor-network state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The top curve is generated by a random state with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The dynamics is generated by the four-layer circuit (14) with a random real dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualita- tive distinction between the single-particle state and the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The di↵erences between the other four states are not essential, because they vary according to the instance of random dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='4 Absence of quantum chaos First of all, let us see that our method for sampling real random dual unitaries generates an ensemble qualitatively similar to that of Haar-random real unitaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' orthog- onal matrices) without the duality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In order to do so, we generate 85 random dual unitaries u 2 U with real entries, and calculate the spectral form factor K(t) = 1 85 X u2U ��tr(ut) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (82) In Figure Figure 9: Spectral form factor of dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 12 A then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) where = a| 2 A is the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' That is, after one time step, the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single- particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Because of the existence of simple zero and one-particle eigenstates, we consider these cases somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above results with the dynam- ics of the following two and three-particle states , , , (79) generated with a random matrix = a 2 A with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r| ih |T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the e↵ective dimension of its support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values as in the second-order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For states ⇢ proportional to a pro- jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the dimension of the projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimension of the above two and three-particle states is plotted in Fig- ure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The corresponding curves attain a much larger value than those of the zero and one-particle states, which have de↵(1) = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' and take a longer time to equilibrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addition, the lack of convergence to an integer value sug- gests that the corresponding evolutions do not generate a closed orbit, and hence, do not have simple eigenstates as- sociated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 8 also shows that this behaviour is very similar to that of a random state | i 2 H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This establishes a sharp distinction between one particle or less on one side, and two particles or more on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' These results open the possibility that the dynamics of two or more particles is quantum-chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, the results of the following subsection suggest that the contrary is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 1 11 1 1 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 Figure 8: E↵ective dimension of 1,2,3-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each curve shows the dimension of the e↵ective subspace de↵(t) as a function of time t, generated by the evolution of the tensor-network state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The top curve is generated by a random state with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The dynamics is generated by the four-layer circuit (14) with a random real dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualita- tive distinction between the single-particle state and the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The di↵erences between the other four states are not essential, because they vary according to the instance of random dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='4 Absence of quantum chaos First of all, let us see that our method for sampling real random dual unitaries generates an ensemble qualitatively similar to that of Haar-random real unitaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' orthog- onal matrices) without the duality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In order to do so, we generate 85 random dual unitaries u 2 U with real entries, and calculate the spectral form factor K(t) = 1 85 X u2U ��tr(ut) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (82) In Figure Figure 9: Spectral form factor of dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 12 A then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) where = a| 2 A is the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' That is, after one time step, the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single- particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Because of the existence of simple zero and one-particle eigenstates, we consider these cases somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above results with the dynam- ics of the following two and three-particle states , , , (79) generated with a random matrix = a 2 A with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r| ih |T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the e↵ective dimension of its support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values as in the second-order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For states ⇢ proportional to a pro- jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the dimension of the projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimension of the above two and three-particle states is plotted in Fig- ure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The corresponding curves attain a much larger value than those of the zero and one-particle states, which have de↵(1) = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' and take a longer time to equilibrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addition, the lack of convergence to an integer value sug- gests that the corresponding evolutions do not generate a closed orbit, and hence, do not have simple eigenstates as- sociated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 8 also shows that this behaviour is very similar to that of a random state | i 2 H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This establishes a sharp distinction between one particle or less on one side, and two particles or more on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' These results open the possibility that the dynamics of two or more particles is quantum-chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, the results of the following subsection suggest that the contrary is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 1 11 1 1 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 Figure 8: E↵ective dimension of 1,2,3-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each curve shows the dimension of the e↵ective subspace de↵(t) as a function of time t, generated by the evolution of the tensor-network state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The top curve is generated by a random state with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The dynamics is generated by the four-layer circuit (14) with a random real dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualita- tive distinction between the single-particle state and the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The di↵erences between the other four states are not essential, because they vary according to the instance of random dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='4 Absence of quantum chaos First of all, let us see that our method for sampling real random dual unitaries generates an ensemble qualitatively similar to that of Haar-random real unitaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' orthog- onal matrices) without the duality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In order to do so, we generate 85 random dual unitaries u 2 U with real entries, and calculate the spectral form factor K(t) = 1 85 X u2U ��tr(ut) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (82) In Figure Figure 9: Spectral form factor of dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 12 A then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) where = a| 2 A is the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' That is, after one time step, the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single- particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Because of the existence of simple zero and one-particle eigenstates, we consider these cases somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above results with the dynam- ics of the following two and three-particle states , , , (79) generated with a random matrix = a 2 A with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r| ih |T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the e↵ective dimension of its support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values as in the second-order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For states ⇢ proportional to a pro- jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the dimension of the projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimension of the above two and three-particle states is plotted in Fig- ure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The corresponding curves attain a much larger value than those of the zero and one-particle states, which have de↵(1) = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' and take a longer time to equilibrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addition, the lack of convergence to an integer value sug- gests that the corresponding evolutions do not generate a closed orbit, and hence, do not have simple eigenstates as- sociated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 8 also shows that this behaviour is very similar to that of a random state | i 2 H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This establishes a sharp distinction between one particle or less on one side, and two particles or more on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' These results open the possibility that the dynamics of two or more particles is quantum-chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, the results of the following subsection suggest that the contrary is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 1 11 1 1 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 Figure 8: E↵ective dimension of 1,2,3-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each curve shows the dimension of the e↵ective subspace de↵(t) as a function of time t, generated by the evolution of the tensor-network state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The top curve is generated by a random state with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The dynamics is generated by the four-layer circuit (14) with a random real dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualita- tive distinction between the single-particle state and the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The di↵erences between the other four states are not essential, because they vary according to the instance of random dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='4 Absence of quantum chaos First of all, let us see that our method for sampling real random dual unitaries generates an ensemble qualitatively similar to that of Haar-random real unitaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' orthog- onal matrices) without the duality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In order to do so, we generate 85 random dual unitaries u 2 U with real entries, and calculate the spectral form factor K(t) = 1 85 X u2U ��tr(ut) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (82) In Figure Figure 9: Spectral form factor of dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 12 A then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) where = a| 2 A is the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' That is, after one time step, the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single- particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Because of the existence of simple zero and one-particle eigenstates, we consider these cases somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above results with the dynam- ics of the following two and three-particle states , , , (79) generated with a random matrix = a 2 A with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r| ih |T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the e↵ective dimension of its support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values as in the second-order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For states ⇢ proportional to a pro- jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the dimension of the projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimension of the above two and three-particle states is plotted in Fig- ure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The corresponding curves attain a much larger value than those of the zero and one-particle states, which have de↵(1) = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' and take a longer time to equilibrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addition, the lack of convergence to an integer value sug- gests that the corresponding evolutions do not generate a closed orbit, and hence, do not have simple eigenstates as- sociated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 8 also shows that this behaviour is very similar to that of a random state | i 2 H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This establishes a sharp distinction between one particle or less on one side, and two particles or more on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' These results open the possibility that the dynamics of two or more particles is quantum-chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, the results of the following subsection suggest that the contrary is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 1 11 1 1 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 Figure 8: E↵ective dimension of 1,2,3-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each curve shows the dimension of the e↵ective subspace de↵(t) as a function of time t, generated by the evolution of the tensor-network state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The top curve is generated by a random state with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The dynamics is generated by the four-layer circuit (14) with a random real dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualita- tive distinction between the single-particle state and the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The di↵erences between the other four states are not essential, because they vary according to the instance of random dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='4 Absence of quantum chaos First of all, let us see that our method for sampling real random dual unitaries generates an ensemble qualitatively similar to that of Haar-random real unitaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' orthog- onal matrices) without the duality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In order to do so, we generate 85 random dual unitaries u 2 U with real entries, and calculate the spectral form factor K(t) = 1 85 X u2U ��tr(ut) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (82) n = 4 (83) n = 8 (84) In Figure 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 A then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) where = a| 2 A is the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' That is, after one time step, the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single- particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Because of the existence of simple zero and one-particle eigenstates, we consider these cases somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above results with the dynam- ics of the following two and three-particle states , , , (79) generated with a random matrix = a 2 A with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r| ih |T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the e↵ective dimension of its support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values as in the second-order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For states ⇢ proportional to a pro- jector ⇢2 / ⇢ the e↵ective dimension de↵ is equal to the dimension of the projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimension of the above two and three-particle states is plotted in Fig- ure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The corresponding curves attain a much larger value than those of the zero and one-particle states, which have de↵(1) = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' and take a longer time to equilibrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addition, the lack of convergence to an integer value sug- gests that the corresponding evolutions do not generate a closed orbit, and hence, do not have simple eigenstates as- sociated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Figure 8 also shows that this behaviour is very similar to that of a random state | i 2 H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This establishes a sharp distinction between one particle or less on one side, and two particles or more on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' These results open the possibility that the dynamics of two or more particles is quantum-chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, the results of the following subsection suggest that the contrary is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 1 11 1 1 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 which has period �t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we add a particle at the boundary location x = 0 by applying the symmetric oper- ator = a0 2 A0 with a| 0 = a0 then we obtain the state a0 | i = 0 1 �1 , (77) which has the similar evolution T[1]a0 | i = 0 , T[2]T[1]a0 | i = 0 , T[3]T[2]T[1]a0 | i = 0 , Ta0 | i = 0 , (78) That is, after one time step the particle is in the antipodal location (x = 4), so the period of the state a0 | i is �t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have checked that the same dynamical behaviour happens for any initial location x 2 Z8 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In summary, all single-particle states generate a closed orbit of period �t = 2, which allows us to construct exact eigenstates via (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Next, let us compare the above with the dynamics of the following two and three-particle states xxxx (79) For each such pure state | ih | we numerically calculate its time average after t time steps ⇢(t) = 1 t + 1 t X r=0 T r⇢ T �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (80) And for each such mixed state we calculate the dimension of its e↵ective support de↵(t) = ⇥ tr⇢2(t) ⇤�1 , (81) which takes into account the di↵erent weights of the eigen- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In particular, this formula is related to the second- order Renyi entropy via log2 de↵ = h2(⇢) = � log2(⇢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have seen in (76) that the state with zero particles | i is an eigenstate of the dynamics, therefore its evolu- tion generates a one-dimensional subspace de↵(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' According to (78), states with one particle ax | i gener- ate an orbit with period �t = 2, therefore its evolution generates a two-dimensional subspace de↵(1) = 2 (also calculated numerically in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In both cases, zero and one particle, the states generate a closed orbit, and hence, a family of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For this reason we say that this cases are somehow “integrable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The e↵ective dimen- sion for the cases of two and three particles is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The fact that the corresponding curves slowly converge to a non-integer values suggests that the corre- sponding evolutions do not close and orbit, and hence, do not have simple eigenstates associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In addi- tion, we observe that the e↵ective dimension is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A third class of dynamical behaviour is that produced by a random state, also shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this case, the Figure 8: E↵ective dimension of 1, 2, 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each line shows the dimension of the e↵ective subspace as a function of time, generated by the evolution of the state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualitative distinction between the following three cases: (i) one particle, (ii) two and three particles, (iii) and a random state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' e↵ective dimension is much larger, and the equilibration time much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' So, in summary, we observe three qual- itatively di↵erent dynamical behaviours for states with (i) zero or one particle, (ii) two particles or a few more, and (iii) random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 Figure 8: E↵ective dimension of 1,2,3-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Each curve shows the dimension of the e↵ective subspace de↵(t) as a function of time t, generated by the evolution of the tensor-network state next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The top curve is generated by a random state with real entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The dynamics is generated by the four-layer circuit (14) with a random real dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We observe a qualita- tive distinction between the single-particle state and the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The di↵erences between the other four states are not essential, because they vary according to the instance of random dual unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='4 Absence of quantum chaos First of all, let us see that our method for sampling real random dual unitaries generates an ensemble qualitatively similar to that of Haar-random real unitaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' orthog- onal matrices) without the duality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In order to do so, we generate 85 random dual unitaries u 2 U with real entries, and calculate the spectral form factor K(t) = 1 85 X u2U ��tr(ut) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (82) n = 4 (83) n = 8 (84) In Figure 5 Lorentz transformations In this section we define the operators Rl and Ll men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='6, which jointly implement a contrac- tion and a Lorentz boost towards the right and left re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The treatment of Lorentz transformations is simpler in the infinite chain, so in this section we as- sume n = 1 and x 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' H1 are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a 2 A0 12 Figure 10: Spectral form factor of the evolution op- erator T of four layers for the chains with n = 4 and n = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The two plots are very similar, but they look dif- ferent because the n = 4 shows all the vertical scale while the n = 8 only shows a narrow range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Both plots are es- sentially flat with no “dip”, a characteristic of “integrable” systems with Poisson level statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' simpler in the infinite chain, so in this section we as- sume n = ∞ and x ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Let us now mention a sub- tle issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the infinite chain, global operators like T, S, Rl, Ll : H∞ → H∞ are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Fortunately, they have a well-defined adjoint action on any operator supported on a finite region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' for example, T maps a ∈ A0 onto TaT † = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (85) Therefore, in this section, any global operator H∞ → H∞ is understood as an adjoint action on the quasi-local alge- bra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='1 Local Lorentz contractions The operators Rl, Ll are constructed with the building blocks Jx, Kx, which we call local Lorentz contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' For any even x we define the local Lorentz right-boost contraction isometry as Jx = q− 1 2 · · · x x · · (86) It is a contraction because, as shown in (90), it has a non-trivial kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Also, as can be seen in the picture, in some sense there are two more bottom legs than top legs (although both numbers are infinite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If we act with Jx on the time-translation operator T 13 defined in (34) we obtain JxT = q− 1 2 · · · · · (87) = q− 1 2 · · · · · = q− 1 2 · · · · · = q− 1 2 · · · x · · = TJx+2 which can be summarised by the following equality TJxT † = Jx−2 , (88) SJxS† = Jx+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (89) The second equality is just the application of the space- translation operator S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Using these identities and the dual-unitary constraints, we can calculate the action of Jx on a (trace-less) operator ay ∈ Ay at an arbitrary location y, JxayJ† x = � � � � � � � ay if y ≤ x − 2 Tay−2T † if y ≥ x + 2 0 if y = x ± 1 η+(a)(x−1,x) if y = x , (90) where we define the completely-positive map η+(a) = vx−1Ω+(a)x−1v† x−1 (91) = q−1vx−1trx−2(ux−2ax−2u† x−2)v† x−1 ∈ Ax−1,x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Analogously, for any odd x we define the local Lorentz left-boost contraction as Kx = q− 1 2 · · · x x · · (92) By proceeding as in (87) we obtain TKxT † = Kx+2 , (93) SKxS† = Kx+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (94) The action of Kx on a (trace-less) operator ay ∈ Ay at an arbitrary location y is KxayK† x = � � � � � � � Tay+2T † if y ≤ x − 2 ay if y ≥ x + 2 0 if y = x ± 1 η−(a)(x,x+1) if y = x , (95) where we define completely-positive map η−(a) = vxΩ−(a)x+1v† x = q−1vxtrx+2(ux+1ax+2u† x+1)v† x ∈ Ax,x+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (96) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='2 Global Lorentz contractions For any even integer l > 0 we define the the (global) Lorentz right-boost contraction isometry as Rl = · · · J5l J3l Jl ˜J−l ˜J−3l ˜J−5l · · · (97) where we also define ˜Jx = S T −1Jx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The reason for using ˜Jx instead of Jx when x < 0 is that ˜Jx is corrected with a spacetime translation so that Bl acts trivially around the origin x ∈ [−(l − 2), l − 2] ⊆ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Recall that pure Lorentz transformations fix the origin (x, t) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' A calculation similar to that in (87) yields ˜Jx = q− 1 2 · · · x x · · (98) Analogously, for any odd integer l > 0, we define the the (global) Lorentz left-boost contraction isometry as Ll = · · · K−5l K−3l K−l ˜Kl ˜K3l ˜K5l · · · (99) where we also define ˜Kx = S T −1Kx so that the origin is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The following lemma specifies the action of Rl and Ll on any operator of the form a(x, t) = T taxT −t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If l is an even positive integer, a ∈ A a local operator and x a location such that |x − lm| > 1 for all odd integers m, then Rla(x, t)R† l = a � x − 2fl(x − 2t), t + fl(x − 2t) � , (100) where we define the function fl(x) = � x−l 2l � + 1, plotted in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If l is an odd positive integer, a ∈ A a local operator and x a location such that |x − lm| > 1 for all odd integers m, then Lla(x, t)L† l = a � x − 2fl(x + 2t), t − fl(x + 2t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (101) 14 �� �� � �� �� � � � � � ��(�) Figure 11: Plot of the function fl(x) for l = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We can write the coordinate transformations of the lemma as x′ = x − 2fl(x ∓ 2t) t′ = t ± fl(x ∓ 2t) � , (102) where the upper and lower signs correspond to the right and left boosts respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In the limit |x| ≫ l we can use the approximation fl(x) ≈ x 2l , (103) to write the coordinates transformation as x′ ≈ � 1 − 1 l � x ± 2 l t t′ ≈ � 1 − 1 l � t ± 1 2lx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (104) Note that, independently of the sign ±, this transforma- tion preserves Minkowski’s metric up to a scale factor (2t′)2 − x′2 = � 1 − 2 l � � (2t)2 − x2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (105) (Recall that, in this model, the speed of light is c = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=') We can also obtain the velocity of the Lorentz boost in (104) by first, undoing the contraction by dividing (x′, t′) by the scale factor � 1 − 1/l, and second, comparing the resulting transformation to a standard Lorentz boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This results in the velocity v = ±2 √ 4 − 2l + l2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (106) In Lemma 1, the premise |x − lm| > 1 warrants that Rl and Ll perform a pure spacetime transformation, leaving the local operator a(x, t) unaltered (apart from evolving it in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' On the other hand, when x = lm ± 1 for some odd m, we have that Rla(x, t)R† l = 0 as a consequence of (90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' And when x = lm for some odd m, in addition to a spacetime transformation (x, t) �→ (x′, t′), the local operator a is processed by the complete positive maps η± and ˜η± defined in (91) and (108).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, in the limit l ≫ 1, most locations (x, t) satisfy the premise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In order to prove (100) we need to write the action of ˜Jx = S T −1Jx on a (trace-less) op- erator ay ∈ Ay at an arbitrary location y, ˜Jxay ˜J† x = � � � � � � � T †ay+2T if y ≤ x − 2 ay if y ≥ x + 2 0 if y = x ± 1 ˜η+(ax) if y = x (107) where ˜η+(a) = u† x−2Ω+(a)x−1ux−2 (108) = q−1vx−1trx−2(ux−2ax−2u† x−2)v† x−1 ∈ Ax−2,x−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The above has been obtained by applying the translation S T −1 to the action (90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Importantly, the premise of the lemma (|x − lm| > 1 for all odd numbers m) implies that only the first two cases in (90) and (107) are relevant, which simplifies this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In what follows we analyse the action of Bl on a local operator ax for the three cases where fl(x) is equal, larger or smaller than zero, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If fl(x) = 0 then ax commutes with all operators Jlm and ˜J−lm where m is a positive odd integer, therefore (97) implies RlaxR† l = ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If fl(x) ≥ 1 then ax commutes with all operators ˜J−lm and Jlm, except for the Jlm with m ∈ {1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' , 2fl(x) − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Note that fl(x) counts the number of non-commuting factors in Bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This implies RlaxR† l = (J[2fl(x)−1]l · · · Jl)ax(J[2fl(x)−1]l · · · Jl)† = a[x − 2fl(x), fl(x)] , (109) where the last equality follows from (90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If fl(x) ≤ −1 then the operators which do not commute with ax are ˜Jml with m ∈ {2fl(x) + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' , −3, −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Note that |fl(x)| counts the number of non-commuting factors in Bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Sim- larly, we obtain RlaxR† l = ( ˜J[2fl(x)+1]l · · · ˜J−l)ax( ˜J[2fl(x)+1]l · · · ˜J−l)† = a[x − 2fl(x), fl(x)] , (110) where the last equality follows from (107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Finally, we invoke (88-89) to obtain the algebraic iden- tity Rl(ST) = (ST)Rl , (111) which allows to generalise (109) and (110) to the case a(x, t) with t ̸= 0, Rla(x, t)R† l = (BlStT t)a(x − 2t, 0)(BlStT t)† = (StT tBl)ax−2t(BlStT t)† = (StT t)a[x − 2t − 2fl(x − 2t), fl(x − 2t)](StT t)† = a[x − 2fl(x − 2t), t + fl(x − 2t)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' (112) This concludes the proof of (100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 15 In order to prove (101) we proceed analogously, but this time, we use the action of ˜Kx = S−1T −1Kx on an arbi- trary local operator ˜Kxay ˜K† x = � � � � � � � ay if y ≤ x − 2 T †ay−2T if y ≥ x + 2 0 if y = x ± 1 ˜η−(ax) if y = x (113) where ˜η−(a) = u† x+1Ω−(a)x+1ux+1 (114) = q−1vx−1trx−2(ux−2ax−2u† x−2)v† x−1 ∈ Ax+1,x+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Also, we use the fact that (93-94) imply Ll(S−1T) = (S−1T)Ll , (115) to complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 6 Outlook In this work we have introduced conformal QCAs, which are discrete-spacetime versions of CFTs, and studied their properties as models of holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' We have obtained sev- eral results, but we have mostly opened venues for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In what follows we enumerate some of the prob- lems that will be addressed in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' How much of the CFT phenomenology is covered by conformal QCAs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Do they have a continuum limit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Characterise the algebraic structure of scale and Lorentz transformations in conformal QCAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' In this work, the operators implementing these transforma- tions (C, D, Rl, Ll) have been constructed so that classical geometries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' tensor-network states) are mapped to classical geometries, and not superposi- tions thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The reason for this is that it simpli- fies the visualisation of the dynamics of geometry and matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, if we relax this property, other con- structions exist which generate a more structured al- gebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' What is the mechanism that produces Poisson level statistics on the spectrum of a conformal QCA con- structed with a random dual unitary (which has Wigner-Dyson level statistics)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Our framework allows for calculating the time evolu- tion of arbitrary (discrete) geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' However, it is still not clear whether this dynamics corresponds to a discrete version of Einstein’s field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' If this is the case then conformal QCAs will provide a new perspective on quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 7 Acknowledgements I am thankful to Diego Blas, Sougato Bose, Tom Holden- Dye, Arijeet Pal and Andrea Russo for valuable discus- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' This work has been supported by the UK’s Engineer- ing and Physical Sciences Research Council (grant number EP/R012393/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' References [1] Juan Maldacena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The large n limit of superconformal field theories and supergravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' International Journal of Theoretical Physics, 38(4):1113–1133, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Nesterov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Quantum effec- tive action in spacetimes with branes and boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Physical Review D, 73(6), mar 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' [37] Tom Farshi, Jonas Richter, Daniele Toniolo, Arijeet Pal, and Lluis Masanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Absence of localization in two-dimensional clifford circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='10129, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 17 t = 0 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 = t = 1 = t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 t = 2 = t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 = t = 3 = t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 = t = 4 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 = t = 5 = 1 Figure 12: Dynamics of flat space with one particle at the corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Same assumptions (the dual unitary and a are real) and notation than in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Operator inserted at t = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 is the transpose of that at t = 2, and the same relation holds for the pairs of times (0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5), (1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5), (3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5, 4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The particle reaches the antipodal point at t = 5, and the period is ∆t = n 2 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 18 0 t = 0 = a 0 t = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 1 0 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 = 0 t = 2 = 0 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 = 0 t = 3 = 0 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 = 0 t = 4 = 0 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 0 t = 5 = 0 t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 = 0 t = 6 = 0 t = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 = 0 t = 7 = 0 t = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 = 0 t = 8 1 Figure 13: Dynamics of toy AdS with one particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Same assumptions (the dual unitary and a are real) and notation than Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' Operator inserted at t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5 is the transpose of that at t = 7, and the same relation holds for the pairs of times (2, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5, 6), (3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5, 5) and (4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' The particle reaches the antipodal point at t = n 4 = 8, and the period is ∆t = n 2 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQf_QLA/content/2301.02825v1.pdf'} diff --git a/1NE1T4oBgHgl3EQfRgP7/content/tmp_files/2301.03055v1.pdf.txt b/1NE1T4oBgHgl3EQfRgP7/content/tmp_files/2301.03055v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..589cfee4b2fadfac059e5858fc6400fa12532bc8 --- /dev/null +++ b/1NE1T4oBgHgl3EQfRgP7/content/tmp_files/2301.03055v1.pdf.txt @@ -0,0 +1,4884 @@ +Spectral estimates for free boundary minimal +surfaces via Montiel–Ros partitioning methods +Alessandro Carlotto, Mario B. Schulz, David Wiygul +Abstract +We adapt and extend the Montiel–Ros methodology to compact manifolds with boundary, +allowing for mixed (including oblique) boundary conditions and also accounting for the action of +a finite group G together with an additional twisting homomorphism σ: G → O(1). We then +apply this machinery in order to obtain quantitative lower and upper bounds on the growth rate +of the Morse index of free boundary minimal surfaces with respect to the topological data (i. e. +the genus and the number of boundary components) of the surfaces in question. In particular, we +compute the exact values of the equivariant Morse index and nullity for two infinite families of +examples, with respect to their maximal symmetry groups, and thereby derive explicit two-sided +linear bounds when the equivariance constraint is lifted. +1 Introduction +Despite a profusion of constructions of free boundary minimal surfaces in the Euclidean unit ball +B3 over the course of the past decade ([14–16,24,31] via optimization of the first Steklov eigenvalue, +[4,25,26] via min-max methods for the area functional, and [6,11,18–20,22] via gluing methods), +many basic questions about the space of such surfaces remain open. The reader is referred to +[12, 13, 27] for recent overviews of the field. In particular, so far it is only for the rotationally +symmetric examples, planar discs through the origin and critical catenoids, that the exact value of +the Morse index is actually known (see [8,36,38]). The present manuscript is the first in a series of +works aimed at shedding new light on this fundamental invariant, which (also due to its variational +content, and thus to its natural connection with min-max theory, cf. [28–30] and references therein) +has acquired great importance within geometric analysis. +Partly motivated by the corresponding conjectures concerning closed minimal hypersurfaces in +manifolds of positive Ricci curvature (cf. [1, 33]), five years ago the first-named author proved +with Ambrozio and Sharp a universal lower bound for the index of any free boundary minimal +surface in any mean-convex subdomain Ω of R3 in terms of the topological data of the surface under +consideration. Specifically, it was shown in [2] that the following estimate holds: +index(Σ) ≥ 1 +3(2g + b − 1) +(1.1) +where Σ is any free boundary minimal surface in Ω, and g, b denote respectively its genus and the +number of its boundary components. It is then a natural, and by now well-known question, whether +such a lower bound can be complemented by an affine upper bound or whether – instead – it is +1 +arXiv:2301.03055v1 [math.DG] 8 Jan 2023 + +1 Introduction +A. Carlotto, M. B. Schulz, D. Wiygul +conceivable to have a superlinear growth rate of the index with respect to g and b. In this article we +show that there are in fact infinite families of free boundary minimal surfaces in B3 whose index is +bounded from above (and below) by explicit affine functions of the topological data. More broadly, +we embed such a result in a network of index estimates that in turn build on a generalization of the +fundamental Montiel–Ros methodology – as first presented in [32] – that is of independent interest +and wider applicability. +In general terms, we shall be concerned here with proving effective estimates for (part of) the +spectrum of Schrödinger-type operators on bounded Lipschitz domains of Riemannian manifolds, +combined with mixed boundary conditions, that will be – on disjoint portions of the boundary in +question – of Dirichlet or Robin (oblique) type. Summarizing and oversimplifying things to the +extreme, the number of eigenvalues of any such operator below a given threshold can be estimated +by suitably partitioning the domain into finitely many subdomains, provided one adjoins Dirichlet +boundary conditions in the interior boundaries when aiming for lower bounds, and Neumann +boundary conditions in the interior boundaries for upper bounds instead. We refer the reader to +Section 2 for the setup of our problem together with our standing assumptions, and to the first part +of Section 3 (specifically to Theorem 3.1, and Corollary 3.2) for precise statements. +In fact, often times (yet not always) the partitions mentioned above naturally relate to the underlying +symmetries of the problem in question, which is in particular the case for some of the classes of +free boundary minimal surfaces in B3 that have so far been constructed. With this remark in +mind, a peculiar (and, a posteriori, fundamental) feature of our work is the development of the +Montiel–Ros methodology in the presence of the action of a group G together with an additional +twisting homomorphism σ: G → O(1), in the terms explained in Section 2.4. This allows, for +instance, to explicitly and transparently study how the Morse index of a given free boundary minimal +surface depends on the symmetries one imposes, namely to look at the “functor” (G, σ) → indσ +G(T), +where T denotes the index (Jacobi) form of the surface in question. As apparent even from the +simplest examples we shall discuss, this perspective turns out to be very natural and effective in +tackling the geometric problems we are interested in. +With this approach, lower bounds are sometimes relatively cheap to obtain. One way they can +derived is from ambient Killing vector fields, once it is shown that the associated (scalar-valued) +Jacobi field on the surface under consideration vanishes along the (interior) boundary of any domain +of the chosen partition, which in practice amounts to suitably designing the partition and picking the +Killing field given the geometry of the problem. We present one simple yet paradigmatic such result +in Proposition 4.2, which concerns free boundary minimal surfaces with pyramidal or prismatic +symmetry in B3. Instead, upper bounds are often a lot harder to obtain and shall typically rely on +finer information than the sole symmetries of the scene one deals with. Said otherwise, one needs to +know how (i. e. by which method) the surface under study has been obtained. +We will develop here a detailed analysis of the Morse index of the two families of free boundary +minimal surfaces we constructed in our recent, previous work [6]. Very briefly, using gluing methods +of essentially PDE–theoretic character, we obtained there a sequence Σ−K0∪B2∪K0 +m +of surfaces having +genus m, three boundary components and antiprismatic symmetry group Am+1, and a sequence +Ξ−K0∪K0 +n +of surfaces having genus zero, n + 2 boundary components and prismatic symmetry group +Pn. As we described at length in Section 7 therein, with data (cf. Table 2 and Table 3) and +heuristics, numerical simulations for the Morse index of the surfaces in the former sequence display +a seemingly “erratic” behaviour, as such values do not align on the graph of any affine function, nor +seem to exhibit any obvious periodic pattern. This is a rather unexpected behaviour (by comparison +2 + +1 Introduction +A. Carlotto, M. B. Schulz, D. Wiygul +e. g. with other families of examples, say in the round three-dimensional sphere, see [21]), which +obviously calls for a careful study that we carry through in Section 5 of the present article. In +particular, we establish the following statement: +Theorem 1.1 (Index estimates for Σ−K0∪B2∪K0 +m +and Ξ−K0∪K0 +n +). There exist m0, n0 > 0 such that +for all integers m > m0 and n > n0 the Morse index and nullity of the free boundary minimal +surfaces Σ−K0∪B2∪K0 +m +, Ξ−K0∪K0 +n +⊂ B3 satisfy the bounds +2m + 1 ≤ ind(Σ−K0∪B2∪K0 +m +), +ind(Σ−K0∪B2∪K0 +m +) + nul(Σ−K0∪B2∪K0 +m +) ≤ 10m + 10, +2n + 2 ≤ ind(Ξ−K0∪K0 +n +), +ind(Ξ−K0∪K0 +n +) + nul(Ξ−K0∪K0 +n +) ≤ 8n. +To the best of our knowledge, this is the very first upper bound obtained for the Morse index of a +sequence of free boundary minimal surfaces in the Euclidean unit ball B3. In fact, the upper bound +in this “absolute estimate” follows quite easily by combining the “relative estimate”, associated to +the equivariant Morse index of these surfaces (with respect to their respective maximal symmetry +groups) with the aforementioned Proposition 3.1. +The next statement thus pertains to such +equivariant bounds, for which we do obtain equality, thus settling part of Conjecture 7.7 (iv) and +Conjecture 7.9 (iv) of [6]. We stress that neither family is constructed variationally, and thus +there is actually no cheap index bound one can extract from the design methodology itself; on +the contrary, this statement indicates a posteriori that the families of surfaces in question may in +principle be constructed (even in a non-asymptotic regime) by means of min-max schemes generated +by 2-parameter sweepouts, modulo the well-known problem of fully controlling the topology in the +process (cf. [4]). +Theorem 1.2 (Equivariant index and nullity of Σ−K0∪B2∪K0 +m +and Ξ−K0∪K0 +n +). There exist m0, n0 > 0 +such that for all integers m > m0 and n > n0 the equivariant Morse index and nullity of the free +boundary minimal surfaces Σ−K0∪B2∪K0 +m +, Ξ−K0∪K0 +n +⊂ B3 satisfy +indAm+1(Σ−K0∪B2∪K0 +m +) = 2, +nulAm+1(Σ−K0∪B2∪K0 +m +) = 0, +indPn(Ξ−K0∪K0 +n +) = 2, +nulPn(Ξ−K0∪K0 +n +) = 0. +The main idea behind the proof of these results, or – more precisely – for the upper bounds can +only be explained by recalling, in a few words, how the surfaces in question have been constructed. +Following the general methodology of [17], one first considers a singular configuration, that is a +formal union of minimal surfaces in B3 (not necessarily free boundary), then its regularization +– which needs the use of (wrapped) periodic minimal surfaces in R3, to desingularize near the +divisors, and controlled interpolation processes between the building blocks in play – and, thirdly +and finally, the perturbation of such configurations to exact minimality (at least for some values of +the parameters), while also ensuring proper embeddedness and accommodating the free boundary +condition. Here we first get a complete understanding of the index and nullities of the building +blocks, for the concrete cases under consideration in Section 5. In somewhat more detail, the +analysis of the Karcher–Scherk towers (the periodic building blocks employed in either construction) +exploits, in a substantial fashion, the use of the Gauss map, which allows one to rephrase the initial +geometric question into as one for the spectrum of simple elliptic operators of the form ∆gS2 + 2 +on suitable (typically singular, i. e. spherical triangles, wedges or lunes) subdomains of round S2, +3 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +with mixed boundary conditions, and possibly subject to additional symmetry requirements. The +analysis of the other building blocks – disks and asymmetric catenoidal annuli – is more direct, +although, in the latter case, trickier than it may first look (see e. g. Lemma 5.8). +Once that preliminary analysis is done, we then prove that, corresponding to the (local) geometric +convergence results (that are implied by the very gluing methodology) there are robust spectral +convergence results that serve our scopes. However, a general challenge in the process is that +gluing constructions typically have transition regions where different scales interact with each +another: in our constructions of the sequences Σ−K0∪B2∪K0 +m +and Ξ−K0∪K0 +n +such regions occur between +the catenoidal annuli K0 (as well as the disk B2 in the former case) and the wrapped Karcher– +Scherk towers, roughly at distances between m−1 and m−1/2 (respectively n−1 and n−1/2) from the +equatorial S1. As a result, we need to deal with delicate scale-picking arguments, an ad hoc study of +the geometry of such regions (cf. Lemma 5.21) and – most importantly – prove the corresponding +uniform bounds for eigenvalues and eigenfunctions (collected in Lemma 5.25), which allow to rule +out pathologic concentration phenomena, thereby leading to the desired conclusions. +Acknowledgements. +The authors wish to express their sincere gratitude to Giada Franz for a +number of conversations on themes related to those object of the present manuscript. This project +has received funding from the European Research Council (ERC) under the European Union’s +Horizon 2020 research and innovation programme (grant agreement No. 947923). The research of +M. S. was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) +under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics– +Geometry–Structure, and the Collaborative Research Centre CRC 1442, Geometry: Deformations +and Rigidity. Part of this article was finalized while A. C. was visiting the ETH-FIM, whose support +and excellent working conditions are gratefully acknowledged. +2 Notation and standing assumptions +2.1 Boundary value problems for Schrödinger operators on Lipschitz domains +Let Ω be a Lipschitz domain of a smooth, compact d-dimensional manifold M with (possibly empty) +boundary ∂M, by which we mean here a nonempty, open subset of M whose boundary is everywhere +locally representable as the graph of a Lipschitz function. We do not require – at least in general – +Ω to be connected, and we admit the case Ω = M (where Ω denotes the closure of Ω in M), when +of course ∂Ω = ∂M, the boundary of the ambient manifold in question. Throughout this article we +will in fact assume d ≥ 2. +We are going to study the spectrum of a given Schrödinger operator on Ω subject to boundary +conditions and, sometimes, symmetry constraints. Such symmetry constraints will be encoded in +terms of equivariance with respect to a certain group action, which we shall specify at due place. +The Schrödinger operator +∆g + q +is determined by the data of a given smooth Riemannian metric g on Ω and a given smooth (i. e. +C∞) function q: Ω → R. To avoid ambiguities, we remark here that a function (or tensor field) on +4 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +Ω smooth if it is the restriction of a smooth tensor field on M or – equivalently – on a relatively +open set containing Ω. +The boundary conditions are specified by another smooth function r: Ω → R and a decomposition +∂Ω = ∂DΩ ∪ ∂NΩ ∪ ∂RΩ +(2.1) +where the sets on the right-hand side are the closures of pairwise disjoint open subsets ∂DΩ, ∂NΩ, +and ∂RΩ of ∂Ω. +Somewhat more specifically, we will consider the spectrum of the operator ∆g + q subject to the +Dirichlet, Neumann, and Robin conditions +� +� +� +� +� +� +� +u = 0 +on ∂DΩ, +du(ηΩ +g ) = 0 +on ∂NΩ, +du(ηΩ +g ) = ru +on ∂RΩ, +(2.2) +where ηΩ +g is the almost-everywhere defined outward unit normal induced by g on ∂Ω. +It is obviously the case that the Neumann boundary conditions can be regarded as a special case of +their inhomogenous counterpart, however it is convenient – somewhat artificially – to distinguish +them in view of the later applications we have in mind, to the study of the Morse index of free +boundary minimal surfaces. +2.2 Sobolev spaces and traces +To pose the problem precisely we introduce the Sobolev space H1(Ω, g) consisting of all real-valued +functions in L2(Ω, g) which have a weak g-gradient whose pointwise g-norm is also in L2(Ω, g); then +H1(Ω, g) is a Hilbert space equipped with the inner product +⟨u, v⟩H1(Ω,g) := +� +Ω +�uv + g(∇gu, ∇gv) +� dH d(g), +integrating with respect to the d-dimensional Hausdorff measure induced by g. (We say a function +u ∈ L1 +loc(Ω, g) has a weak g-gradient ∇gu if ∇gu is a measurable vector field on Ω with pointwise +g norm in L1 +loc(Ω, g) and +� +Ω g(X, ∇gu) dH d(g) = − +� +Ω u divg X dH d(g) for every smooth vector +field X on Ω of relatively compact support, where divg X is the g divergence of X; ∇gu is uniquely +defined whenever it exists, modulo vector fields vanishing almost everywhere.) +Under our assumptions on ∂Ω we have a bounded trace map H1(Ω, g) → L2(∂Ω, g), extending +the restriction map C1(Ω) → C0(∂Ω). (The Hilbert space L2(∂Ω, g) is defined using either the +(d−1)-dimensional Hausdorff measure H d−1(g) induced by g or, equivalently, the almost-everywhere +defined volume density induced by g on ∂Ω.) In fact, we have not only boundedness of this map +but also the stronger inequality +∥u|∂Ω∥L2(∂Ω,g) ≤ C(Ω, g) +� +ϵ∥u∥H1(Ω,g) + C(ϵ)∥u∥L2(Ω,g) +� +(2.3) +for all u ∈ H1(Ω, g), all ϵ > 0, some C(Ω, g) independent of u and ϵ, and some C(ϵ) independent +of u and (Ω, g). (This can be deduced, for example, by inspecting the proof of Theorem 4.6 in [9]: +5 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +specifically, we can apply the Cauchy–Schwarz inequality (weighting with ϵ, as standard) to the +inequality immediately above the line labeled (⋆ ⋆ ⋆) on page 158 of the preceding reference, whose +treatment of Lipschitz domains in Euclidean space is readily adapted to our setting.) +For each C ∈ {D, N, R}, indicating one of the boundary conditions we wish to impose, by composing +the preceding trace map with the restriction L2(∂Ω, g) → L2(∂CΩ, g) , since ∂CΩ is open in ∂Ω, we +also get a trace map ·|∂C : H1(Ω, g) → L2(∂CΩ, g). In practice we will consider traces on just ∂DΩ +and ∂RΩ. Considering the condition on ∂DΩ we will then define +H1 +∂DΩ(Ω, g) := {u ∈ H1(Ω, g) : u|∂DΩ = 0}, +that is obviously to be understood in the sense of traces, in the terms we just described, and we +remark that (2.3) also clearly holds with ∂Ω on the left-hand side replaced by ∂RΩ (or by ∂DΩ or +∂NΩ, but we have no need of the inequality in these cases). +2.3 Bilinear forms and their eigenvalues and eigenspaces +Corresponding to the above data we define the bilinear form T = T[Ω, g, q, r, ∂DΩ, ∂NΩ, ∂RΩ] by +T : H1 +∂DΩ(Ω, g) × H1 +∂DΩ(Ω, g) → R +(u, v) �→ +� +Ω +� +g(∇gu, ∇gv) − quv +� +dH d(g) − +� +∂RΩ +ruv dH d−1(g). +(2.4) +Then T is symmetric, bounded, and coercive as encoded in the following three equations respectively: +∀u, v ∈ H1 +∂DΩ(Ω, g) +T(u, v) = T(v, u), +∀u ∈ H1 +∂DΩ(Ω, g) +T(u, u) ≤ +�1 + C(Ω, g, q, r) +�∥u∥2 +H1(Ω,g), +(2.5) +∀u ∈ H1 +∂DΩ(Ω, g) +T(u, u) ≥ 1 +2∥u∥2 +H1(Ω,g) − C(Ω, g, q, r)∥u∥2 +L2(Ω,g), +(2.6) +where, for (2.5) and (2.6), one can take C(Ω, g, q, r) = ∥q∥C0(Ω) + C(Ω, g)∥r∥C0(∂RΩ), thanks to +the trace inequality (2.3). From these three properties and the Riesz representation theorem for +Hilbert spaces it follows that for some constant Λ = Λ(Ω, g, q, r) > 0 there exists a linear map +R: L2(Ω, g) → H1 +∂DΩ(Ω, g) such that T(Rf, v) + Λ⟨Rf, ιv⟩L2(Ω,g) = ⟨f, ιv⟩L2(Ω,g) for all functions +f ∈ L2(Ω, g) and v ∈ H1 +∂DΩ(Ω, g), where we have introduced the inclusion map ι: H1 +∂DΩ(Ω, g) → +L2(Ω, g). +(Of course, if f is smooth then standard elliptic interior regularity results ensures that u is as well +smooth on Ω and there satisfies the equation −(∆g + q − Λ)u = f in a classical pointwise sense.) +Since the inclusion H1(Ω, g) �→ L2(Ω, g) is compact (see for example Section 7 of Chapter 4 of +[37]) and of course the inclusion of the closed subspace H1 +∂DΩ(Ω, g) �→ H1(Ω, g) is bounded, the +aforementioned maps ι: H1 +∂DΩ(Ω, g) → L2(Ω, g) and the composite ιR: L2(Ω, g) → L2(Ω, g) are also +both compact operators. Furthermore, to confirm that ιR is symmetric we simply note that (by +appealing to the equation defining the operator R, with Rf1 and Rf2 in place of v) +⟨f2, ιRf1⟩L2(Ω,g) = T(Rf2, Rf1) + Λ⟨ιRf2, ιRf1⟩L2(Ω,g) += T(Rf1, Rf2) + Λ⟨ιRf1, ιRf2⟩L2(Ω,g) = ⟨f1, ιRf2⟩L2(Ω,g) +6 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +for all f1, f2 ∈ L2(Ω, g). That being clarified, to improve readability we will from now on refrain +from explicitly indicating the inclusion map ι in our equations. +With slight abuse of language, in the setting above we call λ ∈ R an eigenvalue of T if there exists +a nonzero u ∈ H1 +∂DΩ(Ω, g) such that +∀v ∈ H1 +∂DΩ(Ω, g) +T(u, v) = λ⟨u, v⟩L2(Ω,g), +(2.7) +and we call any such u an eigenfunction of T with eigenvalue λ. (We caution that the notions of +eigenfunctions and eigenvalues depend not only on T but also on the underlying metric g; for the +sake of convenience we choose to suppress the latter dependence from our notation.) +Hence, as a consequence of the key facts we presented before this definition, one can prove by well- +known arguments the existence of a discrete spectrum for the “shifted” elliptic operator (∆g +q)−Λ +subject to the very same boundary conditions (2.2). As a straightforward corollary, by accounting +for the shift, we obtain the following conclusions for T: +• the set of eigenvalues of T is discrete in R and bounded below, +• for each eigenvalue of T the corresponding eigenspace has finite dimension, +• there exists an Hilbertian basis {ej}∞ +j=1 for L2(Ω, g) consisting of eigenfunctions of T, +• and {ej}∞ +j=1 has dense span in H1 +∂DΩ(Ω, g). +(To avoid ambiguities, we remark that the phrase Hilbertian basis refers to a countable, complete +orthonormal system for the Hilbert space in question.) For each integer i ≥ 1 we write λi (T) for +the ith eigenvalue of T (listed with repetitions in nondecreasing order, in the usual fashion). There +holds the usual min-max characterization +λi (T) = min +� +max +� T(w, w) +∥w∥2 +L2(Ω,g) +: 0 ̸= w ∈ W +� +: W ⊂ +subspace +H1 +∂DΩ(Ω, g), dim W = i +� +. +(2.8) +Next, for any t ∈ R we let E=t(T) denote the (possibly trivial) linear span, in H1 +∂DΩ(Ω, g), of the +eigenfunctions of T with eigenvalue t, and, more generally, for any t ∈ R and any binary relation ∼ +on R (in practice <, ≤, >, ≥, or =) we set +E∼t(T) := ClosureL2(Ω,g) +� +Span +�� +s∼t +E=s(T) +�� +and we denote the corresponding orthogonal projection by +π∼t +T : L2(Ω, g) → E∼t(T). +That is, the space E∼t(T) has been defined to be the closure in L2(Ω, g) of the span of all +eigenfunctions of T having eigenvalue λ such that λ ∼ t. Of course E∼t(T) is a subspace of +H1 +∂DΩ(Ω, g) – in particular – whenever the former has finite dimension. Taking ∼ to be equality +clearly reproduces the originally defined space E=t(T). +For future use observe that the above spectral theorem for T implies +(E∼t(T))⊥L2(Ω,g) = E̸∼t(T), +E, ≥, +(2.9) +7 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +and +∀u ∈ H1 +∂DΩ(Ω, g) ∩ +� +E≤t(T) ∪ E≥t(T) +� +T(u, u) = t∥u∥2 +L2(Ω,g) ⇒ u ∈ E=t(T), +throughout which t is any real number (not necessarily an eigenvalue of T) and where in the first +equality of (2.9) ∼ is any relation on R and ̸∼ its negation (so that {s ̸∼ t} = R \ {s ∼ t} for any +t ∈ R). +Index and nullity. +In the setting above, and under the corresponding standing assumption, we +shall define the non-negative integers +ind(T) := dim E<0(T) +and +nul(T) := dim E=0(T), +called, respectively, the index and nullity of T. Such invariants will be of primary interest in our +applications. +2.4 Group actions +Let G be a finite group of smooth diffeomorphisms of M, each restricting to a surjective isometry of +(Ω, g). Then, as for any group of diffeomorphism of Ω, we have the standard (left) action of G on +functions on Ω via pullback: +(φ, u) �→ u ◦ φ−1 = φ−1∗u +for all φ ∈ G, u: Ω → R. +We say that a function u is G-invariant if it is invariant under this action: equivalently u ◦ φ = u for +all φ ∈ G. +We can also twist this action by orthogonal transformations on the fiber R: given in addition to G +a group homomorphism σ: G → O(1) = {−1, 1}, we define the action +(φ, u) �→ σ(φ)(u ◦ φ−1) = σ(φ)φ−1∗u +for all φ ∈ G, u: Ω → R, +and we call a function (G, σ)-invariant if it is invariant under this action. Obviously the above +standard action (φ, u) �→ u ◦ φ−1 is recovered by taking the trivial homomorphism σ ≡ 1. We +also comment that one could of course replace R by C and correspondingly O(1) by U(1) (and in +the preceding sections instead work with Sobolev spaces over C) though we restrict attention to +real-valued functions in this article. +Since, by virtue of our initial requirement, G is a group of isometries of (Ω, g), the above twisted +action yields a unitary representation of G in L2(Ω, g), i. e. a group homomorphism +�σ: G → O +�L2(Ω, g) +� +φ �→ σ(φ)φ−1∗ +(2.10) +whose target are the global isometries of L2(Ω, g); we note that the same conclusions hold true with +H1(Ω, g) in place of L2(Ω, g). The corresponding subspaces of (G, σ)-invariant functions, in L2(Ω, g) +8 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +or H1(Ω, g), are readily checked to be closed, and thus Hilbert spaces themselves. That said, we +define the orthogonal projection +πG,σ : L2(Ω, g) → L2(Ω, g) +u �→ +1 +|G| +� +φ∈G +�σ(φ)u. +(2.11) +Here |G| is the order of G, which – we recall – is assumed throughout to be finite. The image of +L2(Ω, g) under πG,σ thus consists of (G, σ)-invariant functions. +Remark 2.1. One could lift the finiteness assumption, say by allowing G to be a compact Lie group, +requiring σ to be continuous, and replacing the finite average in (2.11) with the average over G with +respect to its Haar measure (which reduces to the former for finite G). However, with a view towards +our later applications, in this article we will content ourselves with the finiteness assumption, which +allows for a lighter exposition. +Henceforth we make the additional assumptions that G globally (i. e. as sets) preserves each of ∂DΩ, +∂NΩ, and ∂RΩ, and that q and r are both G-invariant. Each element of �σ(G) then preserves also +H1 +∂DΩ(Ω, g) and the bilinear form T, and the projection πG,σ commutes with the projection π∼t +T , +for any t ∈ R and binary relation ∼ on R (as above). In particular πG,σ preserves each eigenspace +E=t(T) of T, and more generally the space +E∼t +G,σ(T) := πG,σ(E∼t(T)) +(2.12) +is a subspace of E∼t(T). +For each integer i ≥ 1 we can then define λG,σ +i +(T), the ith (G, σ)-eigenvalue of T, to be the ith +eigenvalue of T having a (G, σ)-invariant eigenfunction (by definition nonzero), counting with +multiplicity as before; equivalently one can work with spaces of (G, σ)-invariant functions and derive +the analogous conclusions as in Subsection 2.3 directly in that setting. +Remark 2.2. We explicitly note, for the sake of completeness, that under no additional assumptions +on the group G and the homomorphism σ it is possible that the space of (G, σ)-invariant functions +be finite dimensional (possibly even of dimension zero). This type of phenomenon happens, for +instance, when every point of the manifold M is a fixed point of a σ-odd isometry. In this case, +all conclusions listed above still hold true, but need to be understood with a bit of care: the +corresponding sequence of eigenvalues λG,σ +1 +(T) ≤ λG,σ +2 +(T) ≤ . . . will in fact just be a finite sequence, +consisting say of I(G, σ) elements, counted with multiplicity as usual; we shall formally convene that +λG,σ +i +(T) = +∞ for i > I(G, σ). That being said, we also remark that this phenomenon patently +does not occur for the Jacobi form of the two sequences of free boundary minimal surfaces we +examine in Sections 4 and 5. +In this equivariant framework we still have the corresponding min-max characterization +λG,σ +i +(T) = min +� +max +� T(w, w) +∥w∥2 +L2(Ω,g) +: 0 ̸= w ∈ W +� +: W ⊂ +subspace +πG,σ +�H1 +∂DΩ(Ω, g) +�, dim W = i +� +. +(2.13) +9 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +We also define the (G, σ)-index and (G, σ)-nullity +indσ +G(T) := dim E<0 +G,σ(T) +and +nulσ +G(T) := dim E=0 +G,σ(T) +of T. Obviously we can recover E∼t(T), λi (T), and the standard index and nullity by taking G to +be the trivial group. As mentioned in the introduction, we reiterate that it is one of the goals of the +present article to study, for fixed g and T, how these numbers (index indσ +G(T) and nullity nulσ +G(T)) +depend on G and σ. +Terminology. +For the sake of brevity, we shall employ the phrase admissible data to denote any +tuple (Ω, g, q, r, ∂DΩ, ∂NΩ, ∂RΩ, G, σ) satisfying all the standing assumptions presented up to now. +We digress briefly to highlight two important special cases, which warrant additional notation. +Example 2.3 (Actions of order-2 groups). When |G| = 2, there are precisely two homomorphisms +G → O(1). Considering such homomorphisms, and the corresponding (G, σ)-invariant functions, we +may define G-even or G-odd functions. Hence, we may call ind+ +G and ind− +G the G-even and G-odd +index, and likewise for the nullity. Clearly, we always have +� +� +� +ind(T) = ind+ +G(T) + ind− +G(T), +nul(T) = nul+ +G(T) + nul− +G(T). +(2.14) +Example 2.4 (Actions of self-congruences of two-sided hypersurfaces). Suppose, momentarily, that +(M, g) is isometrically embedded (as a codimension-one submanifold) in a Riemannian manifold +(N, h), that the set Ω be connected and assume further that the normal bundle of M over Ω is +trivial. Then we can pick a unit normal ν on Ω and thereby identify – as usual – sections of the +normal bundle of M|Ω with functions on Ω. With this interpretation of functions on Ω in mind and +G now a finite group of diffeomorphisms of N that map Ω onto itself (as a set), and everywhere on +Ω preserve the ambient metric h meaning that φ∗h = h for any φ ∈ G, we have a natural action +given by +(φ, u) �→ sgnν(φ)(u ◦ φ−1) +for all φ ∈ G, u: Ω → R, +where sgnν(φ) := h(φ∗ν, ν) is a constant in O(1) = {1, −1}. We shall further assume that the action +of G on Ω is faithful, meaning that only the identity element fixes Ω pointly; this assumption is +always satisfied in our applications. +In this context we continue to say that a function u: Ω → R is G-invariant if u = u ◦ φ for all φ ∈ G, +and we say rather that u is G-equivariant if u = sgnν(φ)u ◦ φ for all φ ∈ G (that is, noting the +identity sgnν(φ) = sgnν(φ−1), provided u is invariant under the sgnν-twisted G action). +Similarly, in this context, we set +indG(T) := indsgnν +G +(T) +and +nulG T := nulsgnν +G +(T), +(2.15) +which we may refer to as simply the G-equivariant index and G-equivariant nullity of T. We point +out that we are abusing notation in the above definitions in that, on the right-hand side of each, +in place of G we mean really the group, isomorphic to G by virtue of the faithfulness assumption, +obtained by restricting each element of G to Ω, and in place of sgnν we mean really the corresponding +homomorphism, well-defined by the faithfulness assumption, on this last group of isometries of Ω. +We now return to the more general assumptions on G preceding this paragraph. +10 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +2.5 Subdomains +Suppose that Ω1 ⊂ Ω is another Lipschitz domain of M (cf. Figure 1). We shall define +∂intΩ1 := ∂Ω1 ∩ Ω, +∂extΩ1 := ∂Ω1 \ ∂intΩ1, +∂Dint +D +Ω1 := (∂extΩ1 ∩ ∂DΩ) ∪ ∂intΩ1, +∂Nint +D +Ω1 := ∂extΩ1 ∩ ∂DΩ, +∂Dint +N +Ω1 := ∂extΩ1 ∩ ∂NΩ, +∂Nint +N +Ω1 := (∂extΩ1 ∩ ∂NΩ) ∪ ∂intΩ1, +∂Dint +R +Ω1 := ∂extΩ1 ∩ ∂RΩ, +∂Nint +R +Ω1 := ∂extΩ1 ∩ ∂RΩ. +(2.16) +In this way we prepare to pose two different sets of boundary conditions on Ω1, whereby, roughly +speaking, in both cases ∂Ω1 inherits whatever boundary condition is in effect on ∂Ω wherever the +two meet (corresponding to ∂extΩ1) and the two sets of conditions are distinguished by placing +either the Dirichlet or the Neumann condition on the remainder of the boundary (corresponding to +∂intΩ1). Naturally associated to these two sets of conditions are the bilinear forms +T Dint +Ω1 +:= T[Ω1, g, q, r, ∂Dint +D +Ω1, ∂Dint +N +Ω1, ∂Dint +R +Ω1], +T Nint +Ω1 +:= T[Ω1, g, q, r, ∂Nint +D +Ω1, ∂Nint +N +Ω1, ∂Nint +R +Ω1], +(2.17) +defined, respectively, on the Sobolev spaces H1 +∂Dint +D +Ω1(Ω1, g) and H1 +∂Nint +D +Ω1(Ω1, g). +Recalling (G, σ) from above, with the tacit understanding that (Ω, g, q, r, ∂DΩ, ∂NΩ, ∂RΩ, G, σ) is +admissible, we further assume that each element of G maps Ω1 onto itself; since G preserves Ω +and respects the decomposition (2.1), it follows that it also respects the decompositions (2.16). +Somewhat abusively, we shall write �σ and πG,σ not only for the maps (2.10) and (2.11) but also +for their counterparts with Ω replaced by Ω1, which are well-defined under our assumptions. The +spaces E∼t +G,σ(T Dint +Ω1 ) and E∼t +G,σ(T Nint +Ω1 ) as in (2.12), are then also well-defined. +∂RΩ +∂DΩ +∂DΩ +∂NΩ +∂NΩ +Ω1 +∂extΩ1 ∩ ∂NΩ +∂extΩ1 ∩ ∂DΩ +∂extΩ1 ∩ ∂RΩ +∂intΩ1 +Figure 1: Example of a Lipschitz domain Ω with subdomain Ω1. +11 + +3 Fundamental tools +A. Carlotto, M. B. Schulz, D. Wiygul +3 Fundamental tools +3.1 Index and nullity bounds in the style of Montiel and Ros +Recalling the notation and assumptions of Section 2, suppose now that we have not only Ω1 ⊂ Ω as +above, but also (open) Lipschitz subdomains Ω1, . . . , Ωn ⊂ Ω which are pairwise disjoint, each of +which satisfies the same assumptions as Ω1 in Section 2.5, and whose closures cover Ω. In particular, +we assume that each element of the group G maps each subdomain Ωi onto itself. We assume +further that G acts transitively on the connected components of Ω and note that this last condition +is always satisfied in the important special case that Ω is connected. +Proposition 3.1 (Montiel–Ros bounds on the number of eigenvalues below a threshold). With +assumptions as in the preceding paragraph and notation as in Section 2, the following inequalities +hold for any t ∈ R +(i) dim E 0 less than the injectivity radius of (M, g), say δ0, we +are given a Lipschitz domain Ωδ ⊂ Ω such that (Ωδ, g, q, r, ∂DΩ, ∂NΩ, ∂RΩ, G, σ) are also admissible +data (with suitable restrictions of tensors and functions tacitly understood), and whose complement +Kδ := Ω \ Ωδ satisfies +� +p∈S +Bf1(δ)(p) ⊂ Kδ ⊂ +� +p∈S +Bf2(δ)(p) +(3.8) +for some finite set of points S ⊂ Ω and monotone functions f1, f2 : [0, δ0[ → R≥0 such that +limδ→0 f2(δ) = 0. Consider the sets as in (2.16) with Ωδ in lieu of Ω1 as well as the associated +bilinear form +T Dint +Ωδ +:= T[Ωδ, g, q, r, ∂Dint +D +Ωδ, ∂Dint +N +Ωδ, ∂Dint +R +Ωδ]. +Then for each integer i ≥ 1 +λG,σ +i +� +T Dint +Ωδ +� +≥ λG,σ +i +(T) , +(3.9) +and we have +lim +δ→0 λG,σ +i +� +T Dint +Ωδ +� += λG,σ +i +(T) . +(3.10) +The conclusion simply relies on the fact that points have null W 1,s-capacity in Rn for 1 ≤ s ≤ n +and so, in particular, have null W 1,2-capacity in Rn for any n ≥ 2; for the sake of completeness, +we provide a self-contained argument focusing on the case of surfaces (d = 2), where a logarithmic +cutoff trick is required, and omit the simpler modifications for d ≥ 3. +Proof. Given any uδ, vδ ∈ H1 +∂Dint +D +Ωδ(Ωδ), postulated to be (G, σ)-invariant, it is standard to note +that their extensions by 0, say uδ, vδ respectively, belong to H1 +∂DΩ(Ω), that such functions are +themselves (G, σ)-invariant, and for any δ ∈ (0, δ0) there hold ⟨uδ, vδ⟩L2(Ωδ,g) = ⟨uδ, vδ⟩L2(Ω,g) +and T Dint +Ωδ (uδ, uδ) = T(uδ, uδ). Hence, it follows at once from the variational characterization of +eigenvalues, (2.13), that for each integer i ≥ 1 we have indeed λG,σ +i +� +T Dint +Ωδ +� +≥ λG,σ +i +(T), which is +the first claim. Appealing again to the domain monotonicity, it actually suffices to check (3.10) +in the case when Kδ is in fact a union of metric balls, namely when we have equality in (3.8), for +f1 = f2. To simplify the notation we can (without loss of generality, up to reparametrization) +assume in fact f2(δ) = δ for any δ in the assumed domain. That said, given any u, v ∈ H1 +∂DΩ(Ω), +19 + +3 Fundamental tools +A. Carlotto, M. B. Schulz, D. Wiygul +(G, σ)-invariant, and δ > 0 (small as in the statement) one can simply define uδ = uϕδ, vδ = vϕδ +where (for r := dg(p, q) and p ∈ S) we set +ϕδ(q) = +� +� +� +� +� +� +� +0 +if r ≤ δ3/4 +3 − 4log r +log δ +if δ3/4 ≤ r ≤ δ1/2 +1 +otherwise. +It is then clear that uδ, vδ ∈ H1 +∂Dint +D +Ωδ(Ωδ), that such functions are (G, σ)-invariant, and, in addition, +lim +δ→0 T Dint +Ωδ (uδ, uδ) = T(u, u), lim +δ→0⟨uδ, vδ⟩L2(Ωδ,g) = ⟨u, v⟩L2(Ω,g). +Hence, again appealing to (2.13), we must conclude +lim sup +δ→0 +λG,σ +i +� +T Dint +Ωδ +� +≤ λG,σ +i +(T) . +(3.11) +whence, combining this inequality with the one above, the conclusion follows. +Corollary 3.10. Given the setting and the assumptions of Proposition 3.9, we have +lim +δ→0 indσ +G(T Dint +Ωδ ) = indσ +G(T). +3.4 Conformal change in dimension two +In this section we suppose, in addition to the assumptions above, that d = dim M = 2 and that +we are given a smooth, strictly positive, G-invariant function ρ on Ω. Note that the above bilinear +form T of (2.4) is invariant under scaling, namely under the simultaneous transformations g �→ ρ2g, +q �→ ρ−2q and r �→ ρ−1r: +T +�Ω, ρ2g, ρ−2q, ρ−1r, ∂DΩ, ∂NΩ, ∂RΩ +� = T +�Ω, g, q, r, ∂DΩ, ∂NΩ, ∂RΩ +� +with the corresponding domains H1 +∂DΩ(Ω, ρ2g) and H1 +∂DΩ(Ω, g) agreeing as sets of functions and +having equivalent norms. This claim needs a clarification: the standard H1-norms of H1 +∂DΩ(Ω, ρ2g) +and H1 +∂DΩ(Ω, g) are only equivalent up to constants that depend on the extremal (inf and sup) +values of the conformal factor ρ. +In general, the eigenvalues (as defined in Subsection 2.3) will be affected by the conformal scaling, +and yet the index and nullity are nonetheless invariant when this operation is performed: +Proposition 3.11 (Invariance of index and nullity under conformal change in dimension two). +With assumptions as in the preceding paragraph +indσ +G(T, ρ2g) = indσ +G(T, g) +and +nulσ +G(T, ρ2g) = nulσ +G(T, g). +Proof. By definition u ∈ E=0 +G,σ(T, g) if and only if u is (G, σ)-invariant and T(u, v) = 0 for all +v ∈ H1 +∂DΩ(Ω, g) (and likewise if each g is replaced by ρ2g), so the nullity equality is clear. For the +index, because we can reverse the roles of g and ρ2g by replacing ρ with ρ−1, it suffices to check that +the claim holds with ≥ in place of =. This follows at once from the min-max characterization (2.13) +applied to the (G, σ)-eigenvalues of (T, ρ2g), by considering the “competitor” subspace E<0 +G,σ(T, g) +in the minimization problem therein, for i = indσ +G(T, g). +20 + +4 Free boundary minimal surfaces in the ball: a first application +A. Carlotto, M. B. Schulz, D. Wiygul +4 Free boundary minimal surfaces in the ball: a first application +From now on, we specialize our study to the case when Ω = M is a properly embedded free +boundary minimal surface, henceforth denoted by Σ, of the closed unit ball B3 := {(x, y, z) ∈ R3 : +x2 + y2 + z2 ≤ 1} in Euclidean space (R3, gR3). Observe that, by the maximum principle, every +embedded free boundary minimal surface is properly embedded. +As anticipated in the introduction, our task here will be to obtain quantitative estimates on the +Morse index of free boundary minimal surfaces, hence our Schrödinger operator is the Jacobi (or +stability) operator on Σ acting on functions subject to the Robin condition +du(ηΣ +gR3) = u +on ∂Σ, +(4.1) +namely: q = |AΣ|2, the squared norm of the second fundamental form of Σ, and ∂DΣ = ∂NΣ = ∅, +∂RΣ = ∂Σ, r = 1. Correspondingly, as our bilinear form T we will consider the index (or stability +or Jacobi) form of Σ, which we will denote by QΣ. We define the index and nullity of Σ in the usual +way, setting +ind(Σ) := ind(QΣ) +and +nul(Σ) := nul(QΣ), +and we likewise define the G-equivariant index and nullity of Σ, indG(Σ) and nulG(Σ) in the sense +of (2.15), when given a group G < O(3) of symmetries of Σ one considers the associated sign +homomorphism. More generally, we will also study the (G, σ)-index and (G, σ)-nullity of Σ, indσ +G(Σ) +and nulσ +G(Σ), when given a group G and, further, a homomorphism σ: G → O(1) (thus, in either +case, these expressions are to be understood by replacing Σ by QΣ). +It has already been mentioned above how general lower bounds for the index, linear in the topological +data (genus and number of boundary components), have been obtained in [2], and by Sargent in +[34] in the special case when the ambient manifold is a convex body in Euclidean R3. We begin this +section by presenting an alternative lower bound (Proposition 4.2 below) in terms of symmetries, +which, though much less general in nature, nevertheless yields sharper lower bounds for many of the +known examples (in terms of the coefficients describing the linear growth rate as a function of the +topological data). Before proceeding, we pause to explain some notation we will find convenient. +Cylindrical coordinates and wedges. +We equip R3 (and so, by restriction, also B3) with standard +– up to labeling – cylindrical coordinates (r, θ, z), so that the point with cylindrical coordinates +(r0, θ0, z0) has Cartesian coordinates (x, y, z) = (r0 cos θ0, r0 sin θ0, z0). For our purposes it will be +convenient to allow arbitrary real values for θ. Given real numbers α ≤ β, we also define the closed +wedge +W β +α := {(r cos θ, r sin θ, z) : r ≥ 0, θ ∈ [α, β], z ∈ R}, +(4.2) +with the half-plane W α +α accommodated as a degenerate wedge. In particular, our convention implies +{θ = α} = W α +α ∪ W α+π +α+π . +21 + +4 Free boundary minimal surfaces in the ball: a first application +A. Carlotto, M. B. Schulz, D. Wiygul +Notation for symmetries. +Given a plane Π ⊂ R3 through the origin, we write RΠ ∈ O(3) for +reflection through Π. Similarly, given a directed line ξ ⊂ R3 through the origin and an angle θ ∈ R, +we write Rθ +ξ for rotation about ξ through angle α in the usual right-handed sense. Typically we will +be interested not exclusively in such a rotation Rθ +ξ but rather in the cyclic subgroup it generates, +with the result that it will never really be important to associate a direction to ξ. Given symmetries +T1, . . . , Tn ∈ O(3), we write ⟨T1, . . . , Tn⟩ for the subgroup they generate. +The order-2 groups generated by reflections through planes will figure repeatedly in the sequel +(beginning with the following proposition), so for succinctness of notation, given a plane Π ⊂ R3 +through the origin, we agree to set Π := ⟨RΠ⟩. In such context, consistently with the general +convention we defined above, we will employ the apex + (respectively: −) to denote functions that +are even (respectively: odd) with respect to the reflection through Π. Similarly (but less frequently), +if ξ is a line through the origin in R3, we will write ξ for the order-2 group generated by reflection +Rξ through ξ (equivalently rotation through angle π in either sense about ξ). +We also pause to name the following three subgroups of O(3), which will be realized as subgroups of +the symmetry groups of the examples we study below and which partly pertain to the statement of +the next proposition: for each integer k ≥ 1 we set +Yk := +� +R{θ=− π +2k }, R{θ= π +2k } +� +(pyramidal group of order 2k), +Pk := +� +R{θ=− π +2k }, R{θ= π +2k }, R{z=0} +� +(prismatic group of order 4k), +Ak := +� +R{θ= π +2k }, Rπ +{y=z=0} +� +(antiprismatic group of order 4k). +(4.3) +Note in particular that we have Yk = Pk ∩ Ak. +Remark 4.1. The above three groups are so named because they are the (maximal) symmetry groups +of, respectively, a right pyramid, prism, or antiprism over a regular k-gon. See e. g. Section 2 of [6] +for pictures and further details, but we caution that the above definition of the subgroup Pk differs +slightly from that given in [6]: the two subgroups are conjugate to one another via rotation through +angle π/(2k) about the z-axis. +With this terminology and notation in place, we can then proceed with the aforementioned lower +index bound, which illustrates the Montiel–Ros methodology as developed in Section 3 and is +interesting in its own right. +Proposition 4.2 (Index lower bounds under pyramidal and prismatic symmetry; cf. [7,21]). Let +Σ be a connected, embedded free boundary minimal surface in B3. Assume that Σ is not a disc +or critical catenoid, that Σ is invariant under reflection through a plane Π1, and that Σ is also +invariant under rotation through an angle α ∈ ]0, 2π[ about a line ξ ⊂ Π1. Then α is a rational +multiple of 2π, there is a largest integer k ≥ 2 such that rotation about ξ through angle 2π/k is also +a symmetry of Σ, and +(i) ind(Σ) ≥ 2k − 1, +(ii) ind− +Π1(Σ) ≥ k − 1, and +(iii) if Σ is additionally invariant under reflection through a plane Π⊥ orthogonal to ξ, then in fact +ind+ +Π⊥(Σ) ≥ 2k − 1. +22 + +4 Free boundary minimal surfaces in the ball: a first application +A. Carlotto, M. B. Schulz, D. Wiygul +Note that the symmetries assumed in the preamble of Proposition 4.2 generate, up to conjugacy in +O(3), the group Yk from (4.3), while one instead obtains (again up to conjugacy) the group Pk by +adjoining the additional symmetry assumed in item (iii). +The proof below is an abstraction and transplantation to the free boundary setting of some index +lower bounds obtained in the course of [21] and drawing on ideas from [32]. The estimates ultimately +depend on a lower bound on the number of nodal domains of a suitable Jacobi field, which was +also the basis for earlier index estimates (of complete minimal surfaces in R3 and closed minimal +surfaces in S3) established by Choe in [7]. +Proof. By excluding the discs and critical catenoids we ensure that Σ is not S1-invariant about +ξ, implying the claim on α and the existence of the rotational symmetry about ξ through angle +of the form 2π/k, as follows. First, if the cyclic subgroup generated by rotation about ξ through +angle α were not finite, then it would be dense in the SO(2) subgroup of rotations about ξ, but the +symmetry group of Σ is closed in O(3); yet, as already observed, our assumptions ensure that Σ has +no SO(2) symmetry subgroup. Thus α must be a rational multiple of 2π, as claimed. Now let β be +the least angle in ]0, 2π[ through which rotation about ξ is generated by the assumed rotational +symmetry through angle α, and let k be the least positive integer such that kβ ≥ 2π. Then rotation +through angle kβ − 2π, which lies in [0, β[, is also generated by the assumed rotational symmetry. +The presumed minimality of β then forces β = 2π/k. +By composing the assumed symmetries, it follows that Σ is also invariant under reflection through +each of the k − 1 planes Π2, . . . , Πk containing ξ and there meeting Π1 at angle an integer multiple +of π/k. Now suppose Π ∈ {Πi}k +i=1. We necessarily have Π∩Σ ̸= ∅ (for example since Π separates B3 +into two components and is a plane of symmetry for Σ, which is assumed to be connected). Because +Π is a plane of symmetry and Σ is embedded, these two surfaces must intersect either orthogonally +or tangentially, but in the latter case Σ must be a disc, which possibility we have excluded by +assumption; consequently, the intersection is orthogonal. Moreover, by the symmetries each of the +2k components W1, . . . , W2k of B3 \ �k +i=1 Πi then has nontrivial intersection Ωi := Σ ∩ Wi with Σ. +Note that the members of the family {Ωi}2k +i=1 are pairwise isometric and each is connected. (Indeed, +Σ is itself connected, so any two points in any single Ωi can be joined by some path in Σ, but this +path can leave Ωi only through the latter’s intersection with planes of symmetry, so we can always +produce a path connecting the two points that is entirely contained in Ωi, by repeated reflection and +replacement, if necessary.) Furthermore, each Ωi has Lipschitz boundary contained in S2 ∪ �k +i=1 Πi, +because the intersection of Σ with either S2 and any of the planes Π1, . . . , Πk is orthogonal (thus +transverse), and exactly k of the Ωi lie on each side of Π1. +Next, letting κξ be a choice of (scalar-valued) Jacobi field on Σ induced by the rotations about ξ +and again using the fact that Σ is not rotationally symmetric (and so, in particular, not planar +either), we conclude that κξ vanishes on Σ ∩ �k +i=1 Πi (because of the aforementioned orthogonality) +but does not vanish identically on any Ωi. As a result, imposing, for each i, the Robin condition +(4.1) on S2 ∩ ∂Ωi and the Dirichlet condition on ∂Ωi ∩ �k +i=1 Πi, the corresponding nullity of Ωi is at +least 1. An appeal to item (i) of Corollary 3.2 now completes the proof. Specifically: +• for our claim (i) we consider the partition of Σ into the 2k domains Ω1, . . . , Ω2k, and take G +to be the trivial group; +23 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +• for our claim (ii) we consider the partition of Σ into the k domains Ω1 ∪ Ω2, . . . , Ω2k−1 ∪ Ω2k +(i. e. we join pairs of adjacent domains), take G = ⟨RΠ1⟩ to be the group with two elements +(as in Example 2.3) and the homomorphism determined by σ(RΠ1) = −1 (thereby imposing +odd symmetry); +• for our claim (iii) we consider the partition of Σ into the 2k domains Ω1 . . . , Ω2k, take G = ⟨RΠ⊥⟩ +to be the group with two elements and the homomorphism determined by σ(RΠ⊥) = +1 +(thereby imposing even symmetry). +Thereby the proof is complete. +5 Effective index estimates for two sequences of examples +5.1 Review of the construction and lower index bounds +Like we have already alluded to in the introduction, in [6] two families of embedded free boundary +minimal surfaces in B3 were constructed by desingularizing (in the spirit of [17]) the configurations +−K0 ∪ K0 and −K0 ∪ B2 ∪ K0, where K0 is the intersection with B3 of a certain catenoidal annulus +having axis of symmetry {x = y = 0} and meeting ∂B3 (not orthogonally) along the equator ∂B2 +and orthogonally along one additional circle of latitude at height h > 0. +Proposition 5.1 (Existence and basic properties of K0). There exists a minimal annulus K0 +which is properly embedded in B3 and intersects the unit sphere ∂B3 exactly along the equator +∂0K0 := ∂B3 ∩ {z = 0} and orthogonally along a circle of latitude at height z = h ≈ 0.87028 which +we denote by ∂⊥K0 := ∂K0 \ ∂0K0. Moreover, K0 coincides with the surface of revolution of the +graph of r: [0, h] → ]0, 1[ given by r(ζ) = (1/a) cosh(aζ − s) for suitable a ≈ 2.3328 and s ≈ 1.4907. +Proof. The existence of K0 is proven in [6, Lemma 3.3]. For the numerical values of a, h and s we +refer to [6, Remark 3.9]. +That being said, these are (somewhat simplified) versions of the main existence results we proved in +[6]. +Theorem 5.2 (Desingularizations of −K0 ∪K0 [6]). For each sufficiently large integer n there exists +in B3 a properly embedded free boundary minimal surface Ξ−K0∪K0 +n +that has genus 0, exactly n + 2 +boundary components and is invariant under the prismatic group Pn from (4.3). Moreover Ξ−K0∪K0 +n +converges to −K0 ∪ K0 in the sense of varifolds, with unit multiplicity, and smoothly away from the +equator, as n → ∞. +Theorem 5.3 (Desingularizations of −K0 ∪ B2 ∪ K0 [6]). For each sufficiently large integer m +there exists in B3 a properly embedded free boundary minimal surface Σ−K0∪B2∪K0 +m +that has genus m, +exactly 3 boundary components and is invariant under the antiprismatic group Am+1 from (4.3). +Moreover Σ−K0∪B2∪K0 +m +converges to −K0 ∪ B2 ∪ K0 in the sense of varifolds, with unit multiplicity, +and smoothly away from the equator, as m → ∞. +24 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +Proposition 5.4 (Lower bounds by symmetry on the index of the examples of [6]). There exist +n0, m0 > 0 such that we have the following index estimates for all integers n > n0 and m > m0 +ind+ +{z=0}(Ξ−K0∪K0 +n +) ≥ 2n − 1 +and +ind(Σ−K0∪B2∪K0 +m +) ≥ 2m + 1. +Proof. As stated in Theorem 5.2, Ξ−K0∪K0 +n +is invariant under the action of the prismatic group +Pn which is generated by the reflections through the vertical planes {θ = −π/(2n)} and {θ = +π/(2n)} and through the horizontal plane {z = 0}. As a composition of the first two reflections, +Pn also contains the rotation by angle 2π/n about the vertical axis ξ0 = {r = 0}. Applying +Proposition 4.2 (iii) with k = n, ξ = ξ0, Π1 = {θ = π/(2n)} and Π⊥ = {z = 0} we obtain +ind+ +{z=0}(Ξ−K0∪K0 +n +) ≥ 2n − 1. +Similarly, Theorem 5.3 states that Σ−K0∪B2∪K0 +m +is invariant under the action of the antiprismatic +group Am+1 which contains the reflection through the vertical plane {θ = π/(2(m + 1))} and also +the rotation by angle 2π/(m + 1) about the vertical axis ξ0. Applying Proposition 4.2 (i) then yields +ind(Σ−K0∪B2∪K0 +m +) ≥ 2m + 1. +In terms of topological data, the previous proposition (compared to [2]) provides a coefficient 2 +for the growth rate of the Morse index of Ξ−K0∪K0 +n +(respectively: Σ−K0∪B2∪K0 +m +) with respect to the +number of boundary components (respectively: of the genus), modulo an additive term. In fact, the +lower bound on the Morse index of Ξ−K0∪K0 +n +can be further improved via the following observation, +which pertains the odd contributions to the index instead (again with respect to reflections across +the {z = 0} plane in R3); incidentally this is also an example of application of Proposition 3.1 to a +collection of domains that are not pairwise isometric. +Proposition 5.5. There exists n0 > 0 such that we have the following index estimates for all +integers n > n0 +ind− +{z=0}(Ξ−K0∪K0 +n +) ≥ 3. +Proof. Let Π1 denote a vertical plane of symmetry, passing through the origin, of the surface +Ξ−K0∪K0 +n +(which, we recall, has prismatic symmetry Pn), let ξ be the line obtained as intersection of +such a plane with {z = 0} and let finally Π2 = ξ⊥ be the vertical plane, again passing through the +origin, that is orthogonal to Π1. Consider on Ξ−K0∪K0 +n +the function κξ = Kξ · ν where Kξ is the +Killing vector field associated to rotations around ξ (oriented either way) and ν is a choice of the +unit normal to the surface in question. Clearly, the flow of Kξ generates a curve of free boundary +minimal surfaces around Ξ−K0∪K0 +n +, hence the function κξ lies in the kernel of the Jacobi operator of +Ξ−K0∪K0 +n +and satisfies the natural Robin boundary condition along the free boundary. Furthermore, +concerning its nodal set, we first note it contains the curves Ξ−K0∪K0 +n +∩ {z = 0}, and Ξ−K0∪K0 +n +∩ Π1. +We also claim that, for any sufficiently large n, the function κξ changes sign along the connected arc +Ξ−K0∪K0 +n +∩ Π+ +2 ∩ {z ≥ z0} +(5.1) +where Π+ +2 denote either of the half-planes determined by Π1 on Π2 and z0 > 0 is any sufficiently +small value (as we are about to describe, stressing that we can choose it independently of n). Since +25 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +ξ +r +z +K0 +−K0 +h +Figure 2: Nodal domains of the function induced by rotations around the symmetry axis ξ. +one has smooth convergence of Ξ−K0∪K0 +n +to −K0 ∪ K0 as n → ∞ away from the equator, it suffices +to verify an analogous claim for K0. In fact, it then follows from an explicit calculation that the +function induced by rotations around the symmetry axis ξ (the analogue of κξ on K0) has opposite +signs on the two endpoints of the arc K0 ∩ Π+ +2 (see Figure 2, right image), and so – assuming +without loss of generality it is negative on the equatorial point – by continuity there exists z0 > 0 +such that the same function is also strictly negative at all points of K0 ∩ Π+ +2 at height z0 ∈ [0, z0]. +In particular, we can indeed choose one such value z0 ∈ (0, z0) once and for all. +Hence, appealing to the aforementioned smooth convergence, by the intermediate value theorem +there must be a point along the arc (5.1) where κξ vanishes. Now, standard results about the +structure of the nodal sets of eigenfunctions of Schrödinger operators ensure that such a zero is +not isolated, but is either a regular point of a smooth curve or a branch point out of which finitely +many smooth arcs emanate. In either case, combining all facts above we must conclude that on +Ξ−K0∪K0 +n +∩ {z ≥ 0} the function κξ has at least four nodal domains, and thus an application of +Theorem 3.1 with t = 0, G = ⟨RΠ⟩ for Π = {z = 0} and σ(RΠ) = −1 ensures the conclusion. +Remark 5.6. Note that the very same argument would lead, when applied with no equivariance +constraint at all (i. e. when G is the trivial group) to the conclusion that for any sufficiently large n +the index of Ξ−K0∪K0 +n +is bounded from below by 7, which however is a lot worse than the bound +provided by combining Proposition 5.4 with Proposition 5.5. Furthermore, we note that one can +show that the function κξ has exactly 8 nodal domains and not more, as visualized in Figure 2. +Remark 5.7. Concerning the sharpness of the estimate given in Proposition 5.5, we note that +numerical simulations of K0 with fixed lower boundary ∂0K0 and upper boundary ∂⊥K0 constraint +to the unit sphere indicate that it has in fact index equal to 3. Roughly speaking, one negative +direction comes from “pinching” the catenoidal neck and the other two negative directions correspond +to “translations” of ∂⊥K0 on the northern hemisphere. +26 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +The rest of this section is aimed at obtaining upper bounds on the Morse index of our examples, +which is a more delicate task and one that relies crucially not only on the symmetries of the +surfaces in question but also on the way they were actually constructed (which we encode in suitable +convergence results). +5.2 Equivariant index and nullity of the models +For upper bounds we will exploit the regionwise convergence of the two families to the models glued +together in their construction. Therefore we first study the index and nullity on these models. +Equivariant index and nullity of K0. +We begin with a summary of the properties of the minimal +annulus K0 we will need. Let ∂0K0 = ∂K0 ∩ {z = 0} and ∂⊥K0 = ∂K0 \ ∂0K0 be as introduced in +Proposition 5.1 so that ∂⊥K0 is the boundary component along which K0 meets the sphere ∂B3 +orthogonally. Referring to equation (2.4), we define +QK0 +N := T +� +K0, gK0, q := +��AK0��2, r := 1, ∂DK0 := ∅, ∂NK0 := ∂0K0, ∂RK0 := ∂⊥K0 +� +(where we abuse notation in that by K0 we really mean its topological interior) to be the Jacobi +form of K0 subject to the natural geometric Robin condition (4.1) on ∂⊥K0 and to the Neumann +condition on ∂0K0. Clearly, for each k ≥ 1 the pyramidal group Yk from (4.3) preserves K0 and +each of its boundary components individually. +Lemma 5.8 (Yk-equivariant index and nullity of K0). With notation as above, for each sufficiently +large integer k +indYk(QK0 +N ) = 1 +and +nulYk(QK0 +N ) = 0. +Proof. We shall start by recalling [6, Lemma 4.4], which states that when imposing the Dirichlet +condition on ∂0K0 and the Robin condition on ∂⊥K0, then the Jacobi operator acting on Yk- +equivariant functions on K0 is invertible provided that k is sufficiently large, which means that the +equivariant nullity vanishes in this case. Considering the coordinate function u = z on K0, which +is harmonic, satisfies the Dirichlet condition on ∂0K0 and the Robin condition on ∂⊥K0, it is also +evident that the equivariant index is at least 1 in this case (cf. [6, Lemma 7.2]). This implies that +when instead the Neumann condition is imposed on ∂0K0, the equivariant index is again at least 1. +Below we prove that it is exactly 1 and the equivariant nullity is exactly 0 in the Neumann case +by showing that the second eigenvalue is strictly positive. (We note here, incidentally, that this +information also proves that the equivariant index is also exactly 1 in the case that a Dirichlet +condition is imposed on ∂0K0.) +Let a, h, s > 0 and r(ζ) = (1/a) cosh(aζ − s) be as in Proposition 5.1. In particular, we have +(r′)2 + 1 = cosh2(aζ − s). Thus, when K0 is parametrized as a surface of revolution in terms of the +coordinates (θ, ζ) with profile function r(ζ), the metric gK0 and the squared norm of the second +fundamental form AK0 on K0 are given by +gK0 = +�(r′)2 + 1 +� dζ2 + r2 dθ2, +|AK0|2 = +(−r′′)2 +((r′)2 + 1)3 + +1 +((r′)2 + 1)2r2 = +a2 + a−2 +cosh4(aζ − s). +27 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +The outward unit conormal along ∂⊥K0 = K0 ∩ {ζ = h} is given by +ηK0 = +1 +� +(r′)2(h) + 1∂ζ = +1 +cosh(ah − s)∂ζ = +1 +ar(h)∂ζ. +Assume, for the sake of a contradiction, that λ2 = λYk,sgn +2 +≤ 0, where we are considering the +spectrum of the Jacobi operator of K0 acting on Yk-equivariant functions (cf. Example 2.4), and +subject to the boundary conditions described above. Then, by first invoking the Courant nodal +domain theorem as in the proof of [6, Lemma 4.4] we may assume that the associated eigenfunction +u2 is rotationally symmetric provided that k is sufficiently large, i. e. u2 only depends on ζ and not +on θ. +That said, let u be a function on K0 which is rotationally symmetric, i. e. constant in θ. Then +∆K0u = +1 +cosh2(aζ − s) +∂2u +∂ζ2 , +and we shall consider the Jacobi operator J = ∆K0 + |AK0|2 and the eigenvalue problem +� +� +� +� +� +� +� +Ju = −λu +u′(0, ·) = 0 +(Neumann condition on ∂0K0) +u′(h, ·) = cosh(ah − s) u(h, ·) +(Robin condition on ∂⊥K0) +Since u2 must change sign, there exists z0 ∈ ]0, h[ such that u2(z0) = 0. Multiplying the eigenvalue +equation +∂2u2 +∂ζ2 + +a2 + a−2 +cosh2(aζ − s)u2 = −λ2u2 cosh2(aζ − s) +(5.2) +with u2 and integrating from ζ = 0 to ζ = z0, we obtain +� z0 +0 +−λ2u2 +2 cosh2(aζ − s) dζ = − +� z0 +0 +|u′ +2|2 dζ + +� z0 +0 +a2 + a−2 +cosh2(aζ − s)u2 +2 dζ. +Since u(z0) = 0, we can obtain the Poincaré-type inequality +� z0 +0 +|u2(ζ)|2 dζ = +� z0 +0 +��� +� ζ +z0 +u′ +2(t) dt +��� +2 +dζ ≤ +� z0 +0 +(z0 − ζ) +� z0 +ζ +|u′ +2(t)|2 dt dζ ≤ z2 +0 +2 +� z0 +0 +|u′ +2(ζ)|2 dζ. +Hence, +� z0 +0 +−λ2u2 +2 cosh2(aζ − s) dζ ≤ +� z0 +0 +� +a2 + a−2 +cosh2(aζ − s) − 2 +z2 +0 +� +u2 +2 dζ. +The right-hand side is negative if +z0 < +� +2 +a2 + a−2 ≈ 0.5962 +and so, in this case, we conclude λ2 > 0, a contradiction. +28 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +Integrating the eigenvalue equation (5.2) instead from ζ = z0 to ζ = h and recalling the Robin +condition u′(h) = cosh(ah − s)u(h) along ∂⊥K0 we obtain the alternative estimate +� h +z0 +−λ2u2 +2 cosh2(aζ − s) dζ = |u(h)|2 cosh(ah − s) − +� h +z0 +|u′ +2|2 dζ + +� h +z0 +a2 + a−2 +cosh2(aζ − s)u2 +2 dζ +≤ +� +(h − z0) cosh(ah − s) − 1 +� � h +z0 +|u′ +2|2 dζ + +� h +z0 +a2 + a−2 +cosh2(aζ − s)u2 +2 dζ +≤ +� +a2 + a−2 + +2 +(h − z0)2 +� +(h − z0) cosh(ah − s) − 1 +�� � h +z0 +u2 +2 dζ +provided that (h − z0) cosh(ah − s) − 1 < 0. Now the right-hand side is negative if z0 > 0.4443. +Since the intervals [0, 0.5962] and [0.4443, h] intersect, we anyway obtain a contradiction. Thus, we +confirm the claim λ2 > 0, as desired. +Observing (as we have already done in the previous proof) that any eigenfunction “generating” the +index in Lemma 5.8 is rotationally invariant, we have the following obvious corollary (which in fact +can conversely be used to prove the lemma, with the aid of Proposition 3.8). In the statement GK0 +denotes the subgroup of O(3) preserving K0. Note that GK0 consists of rotations about the z-axis +and reflections through planes containing the z-axis. In particular GK0 is isomorphic to O(2), and +each element of GK0 preserves either choice of unit normal of K0. +Corollary 5.9 (Fully equivariant index and nullity of K0). With notation as above and recalling +the comments immediately preceding Proposition 3.8, there holds +indGK0(QK0 +N ) = 1 +and +nulGK0(QK0 +N ) = 0. +Equivariant index and nullity of B2. +The analysis for the flat disc B2 (featured in the construction +of just one of the families) is trivial, and the conclusions are as follows; in the statement we write QB2 +N +for the index form of B2 as a minimal surface with boundary in (R3, gR3) subject to the Neumann +boundary condition, namely +QB2 +N := T +� +B2, gB2, 0, 0, ∅, ∂NB2 := ∂B2, ∅ +� +. +Lemma 5.10 ((Am+1-equivariant) index and nullity of B2). With notation as above, +ind(QB2 +N ) = 0 +and +nul(QB2 +N ) = 1. +Moreover, for each integer m ≥ 0 the antiprismatic group Am+1 preserves B2 and +indAm+1(QB2 +N ) = nulAm+1(QB2 +N ) = 0. +Proof. The first line of equalities is clear, since the Jacobi operator on B2 is simply the standard +Laplacian, whose Neumann kernel is spanned by the constants (to rule out index one can just appeal +to the Hopf boundary point lemma). The invariance of B2 under each Am+1 is obvious, and the +proof is then completed by the observation that the constants are not Am+1-equivariant (for any +m ≥ 0). +29 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +From Proposition 5.10 we immediately obtain, analogously to Corollary 5.9 from Proposition 5.8, the +following corollary. In the statement O(2) refers to the group of intrinsic isometries of B2 (extended +to isometries of R2), rather than to some subgroup of O(3), and we write 1 and det for respectively +the trivial and determinant homomorphisms O(2) → O(1). The (O(2), 1)-invariant functions on B2 +are thus the radial functions, while the space of (O(2), det)-invariant functions is trivial. +Corollary 5.11 (Indices and nullities of B2 under O(2) actions). With notation as above we have +ind1 +O(2)(QB2 +N ) = 0, +nul1 +O(2)(QB2 +N ) = 1, +inddet +O(2)(QB2 +N ) = nuldet +O(2)(QB2 +N ) = 0. +Equivariant index and nullity of MΞ and MΣ. +We recall how, away from the equator S1, the +surfaces Ξ−K0∪K0 +n +and Σ−K0∪B2∪K0 +m +are constructed as graphs over (subsets of) −K0 ∪ K0 and +−K0 ∪ B2 ∪ K0. In the vicinity of S1 the surfaces are instead modeled on certain singly periodic +minimal surfaces that belong to a family discovered by Karcher [23] and generalize the classical +singly periodic minimal surfaces of Scherk [35]. We now summarize the key properties of such +models, to the extent needed later. +Proposition 5.12 (Desingularizing models). There exist in R3 complete, connected, properly +embedded minimal surfaces MΞ and MΣ having the following properties, which uniquely determine +the surfaces up to congruence: +(i) MΞ and MΣ are periodic in the y direction with period 2π and the corresponding quotient +surfaces have genus zero. +(ii) MΞ and MΣ are invariant under R{x=0}, R{y=π/2}, and R{y=−π/2}. +(iii) MΞ is invariant under R{z=0} and MΣ under R{y=z=0}. +(iv) MΞ has four ends and MΣ has six ends, all asymptotically planar. +(v) Each of MΞ and MΣ has an end contained in {x ≤ 0} ∩ {z ≥ 0} whose asymptotic plane +intersects {z = 0} at the same angle ω0 > 0 at which K0 intersects B2, and MΣ has additionally +{z = 0} as an asymptotic plane. +(vi) MΞ +fb := MΞ ∩ {x ≤ 0} ∩ {|y| ≤ π/2} and MΣ +fb := MΣ ∩ {x ≤ 0} ∩ {|y| ≤ π/2} are connected +free boundary minimal surfaces in the half slab {x ≤ 0} ∩ {|y| ≤ π/2}, with MΞ +fb invariant +under R{z=0} and MΣ +fb invariant under R{y=z=0} (cf. Figure 3). +(vii) Each of MΞ +fb \ {z = 0} and MΣ +fb \ {y = z = 0} has exactly two connected components. +(viii) MΞ has no umbilics, while the set of umbilic points of MΣ is {(0, nπ, 0) : n ∈ Z}. +(ix) The Gauss map νΞ of MΞ restricted to the closure of either component of MΞ +fb \ {z = 0} is +a bijection onto a solid spherical triangle with all sides geodesic segments of length π/2 (in +other words: a quarter hemisphere), less a point in the interior of one side. +(x) The Gauss map νΣ of MΣ restricted to the closure of either component of MΣ +fb \ {y = z = 0} +is a bijection onto a spherical lune of dihedral angle π/2 (in other words: a half hemisphere), +less one vertex and a point in the interior of one side. +30 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +x +y +z +ω0 +π +y +z +x +ω0 +π +Figure 3: The minimal surfaces MΞ +fb (left) and MΣ +fb (right) as defined in Proposition 5.12 (vi). +We refer the reader to Section 3 and Appendix A of [6] for further details and a fine analysis of +the properties of both surfaces in question. The free boundary minimal surfaces MΞ +fb and MΣ +fb are +visualized in Figure 3. +Next, we want to examine the index and nullity of MΞ +fb and MΣ +fb as free boundary minimal surfaces +in the half slab {x ≤ 0} ∩ {|y| ≤ π/2}. Because the boundary of such a domain is piecewise planar, +the corresponding Robin condition associated with the index forms of these surfaces is in fact +homogeneous (Neumann). +Let us prove an ancillary result. We will observe (in the proof of Lemma 5.16, to follow shortly) that +by virtue of the behavior of the Gauss maps described in Proposition 5.12 the analysis of the index +and nullity of MΞ +fb and MΣ +fb reduces to the following index and nullity computations for boundary +value problems on suitable Lipschitz domains of S2. +Lemma 5.13 (Index and nullity of ∆gS2 + 2 on images of Gauss maps of MΞ +fb and MΣ +fb). Set +ΩΞ +S2 := S2 ∩ {x > 0} ∩ {y > 0} ∩ {z > 0}, +ΩΣ +S2 := S2 ∩ {x > 0} ∩ {y > 0}. +Then we have the following indices and nullities, where the final row holds for any ζ ∈ ]−1, 1[ and, +throughout, T is the bilinear form (2.4) with Ω as indicated, g = gS2 the round metric, q = 2 (so +associated to the Schrödinger operator ∆gS2 + 2), ∂RΩ = ∅, ∂DΩ as indicated, and ∂NΩ = ∂Ω \ ∂DΩ: +Ω +∂DΩ +ind(T) +nul(T) +ΩΞ +S2 +∅ +1 +0 +{z = 0} +0 +1 +ΩΣ +S2 +∅ +1 +1 +{x = 0} +0 +1 +{x = 0} ∩ {z > ζ} +1 +0 +(5.3) +31 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +s +s +s +s +τ (1) +τ (4) +τ (3) +τ (2) +x +z +R1 +W1 +W1(s) +W2 +W2(s) +W3 +W3(s) +W4 +W4(s) +C +ω0 +Figure 4: A view of MΞ(s). +Proof. By Lemma 3.5 we can fill in the first four rows by identifying the index and nullity of +∆gS2 + 2 on the entire sphere subject to appropriate symmetries, the relevant spherical harmonics +being simply the restrictions of affine functions on R3. Lemma 3.5 is not directly applicable to +the final row, but by the min-max characterization (2.13) of eigenvalues the ith eigenvalue for the +bilinear form specified in the that row must lie between the ith eigenvalues of the forms specified in +the two preceding rows (≥ that of the third row and ≤ that of the fourth); moreover, the unique +continuation principle implies that both inequalities must be strict (> and <). The entries of the +final row now follow, concluding the proof. +We shall fix components of MΞ +fb \ {z = 0} and MΣ +fb \ {y = z = 0} and write ΩΞ and ΩΣ for their +respective interiors: it follows from Proposition 5.12 that νΞ|ΩΞ and νΣ|ΩΣ are diffeomorphisms +onto their images, which we can and will identify with, respectively, the triangle ΩΞ +S2 and lune ΩΣ +S2 +of Lemma 5.13, and in particular +{x = 0} ∩ ∂ΩΞ +S2 = νΞ({x = 0} ∩ ∂ΩΞ), +{y = 0} ∩ ∂ΩΞ +S2 = νΞ({y = ±π/2} ∩ ∂ΩΞ), +{z = 0} ∩ ∂ΩΞ +S2 = νΞ({z = 0} ∩ ∂ΩΞ), +and +{x = 0} ∩ ∂ΩΣ +S2 = νΣ(({x = 0} ∪ {y = z = 0}) ∩ ∂ΩΣ), +{y = 0} ∩ ∂ΩΣ +S2 = νΣ({y = ±π/2} ∩ ∂ΩΣ). +In what follows, recalling e. g. that the index of a minimal surface, when finite, can be computed by +exhaustion (cf. [10]) we conveniently introduce this notation, which pertains certain truncations of +MΞ, MΣ, MΞ +fb, and MΣ +fb. To do so, we first fix R1 > 0 large enough such that MΞ \ {x2 + z2 = R2 +1} +consists of five connected components, one component C in {y2 + z2 < R2 +1} and four components +32 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +W1, W2, W3, W4 in the complement, each of which is a graph over (a subset of) an asymptotic half +plane (see Figure 4). For each Wi let τ (i) be a unit vector parallel to the asymptotic half plane of +Wi, perpendicular to the y-axis (the axis of periodicity), and directed away from ∂Wi toward the +corresponding end, namely (up to relabeling) +τ (1) = (cos ω0, 0, sin ω0) = −τ (3), +τ (2) = (− cos ω0, 0, sin ω0) = −τ (4), +where we recall that ω0 > 0 is the angle at which K0 intersects B2. Now, given s > R1, we define +the truncations +Wi(s) := Wi ∩ {τ (i) · (x, y, z) ≤ s}, +MΞ(s) := C ∪ +4� +i=1 +Wi(s), +MΣ(s) analogously (for six ends), +MΞ +−(s) := MΞ(s) ∩ {x ≤ 0}, +MΣ +−(s) := MΣ(s) ∩ {x ≤ 0}, +(5.4) +MΞ +fb(s) := MΞ(s) ∩ MΞ +fb, +MΣ +fb(s) := MΣ(s) ∩ MΣ +fb. +For each ϵ, ϵ′ > 0 we then set +ΩΞ(ϵ) := ΩΞ ∩ MΞ +fb(ϵ−1), +ΩΣ(ϵ, ϵ′) := ΩΣ ∩ MΣ +fb(ϵ−1) ∩ {x2 + y2 + z2 > ϵ′}, +truncating ΩΞ and ΩΣ at (affine) distance ϵ−1 and excising from ΩΣ a disc with radius +√ +ϵ′ and +center at the umbilic (0, 0, 0); similarly MΣ +fb(ϵ−1, ϵ′) := MΣ +fb(ϵ−1) ∩ {x2 + y2 + z2 > ϵ′}. We then +in turn set ΩΞ +S2(ϵ) := νΞ�ΩΞ(ϵ) +� ⊂ ΩΞ +S2 as well as ΩΣ +S2(ϵ, ϵ′) := νΣ�ΩΣ(ϵ, ϵ′) +� ⊂ ΩΣ +S2. As a direct +consequence of Lemma 5.13 and Proposition 3.9 we get what follows. +Corollary 5.14. In the setting above, consider for any ϵ, ϵ′ > 0 the Schrödinger operator ∆gS2 + 2 +on the domains given, respectively, by ΩΞ +S2(ϵ) and ΩΣ +S2(ϵ, ϵ′) and subject to any of the boundary +conditions specified in the table (5.3), where the boundary is contained, respectively, in ∂ΩΣ +S2 and +∂ΩΞ +S2 and subject to Dirichlet conditions elsewhere. In other words, let T Ξ be either bilinear form +corresponding to the top two rows of (5.3), let T Σ be any bilinear form corresponding to the bottom +three rows of (5.3), and consider also the bilinear forms +T Ξ +ϵ := (T Ξ)Dint +ΩΞ +S2(ϵ) = T +� +ΩΞ +S2(ϵ), gS2, 2, 0, ∂DΩΞ +S2 ∪ (∂ΩΞ +S2(ϵ) \ ∂ΩΞ +S2), ∂NΩΞ +S2, ∅ +� +T Σ +ϵ,ϵ′ := (T Σ)Dint +ΩΣ +S2(ϵ,ϵ′) = T +� +ΩΣ +S2(ϵ, ϵ′), gS2, 2, 0, ∂DΩΣ +S2 ∪ (∂ΩΣ +S2(ϵ, ϵ′) \ ∂ΩΣ +S2), ∂NΩΣ +S2, ∅ +� +using the notation (2.17). Then there exists ϵ0 > 0 such that for all 0 < ϵ, ϵ′ < ϵ0 +ind(T Ξ +ϵ ) = ind(T Ξ) +and +ind(T Σ +ϵ,ϵ′) = ind(T Σ). +In particular, we can derive these geometric conclusions: +Corollary 5.15 (Index of MΞ +fb and MΣ +fb). We have the following even and odd indices for MΞ +fb and +MΣ +fb. +S +G +ind+ +G(S) +ind− +G(S) +MΞ +fb +{z = 0} +1 +0 +MΣ +fb +{y = z = 0} +1 +1 +33 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +Proof. We will verify (as a sample) the even index asserted in the second row of the table; the other +claims are checked in the same fashion. The Gauss map of a minimal surface in R3 is (anti)conformal +away from its umbilics, with conformal factor (one half of) the pointwise square of the norm of +its second fundamental form, so by Proposition 3.11, for each ϵ, ϵ′ > 0, the index of ΩΣ +S2(ϵ, ϵ′) with +the foregoing boundary conditions (as in Corollary 5.14, according to the third row of the table +in Lemma 5.13) agrees also with the index of ΩΣ(ϵ, ϵ′) subject to the corresponding boundary +conditions. By Lemma 3.5 this last index agrees with the {y = z = 0}-even index of MΣ +fb(ϵ−1, ϵ′) +subject to the Dirichlet condition along the excisions and the Neumann condition everywhere else. +Hence, thanks to Corollary 5.14, such a value of the index is equal to 1 for any sufficiently small +ϵ, ϵ′. We now conclude, first letting ϵ′ → 0 and appealing to Proposition 3.9 to control the effect of +the excision near (0, 0, 0), and then appealing to the aforementioned characterization of the Morse +index via exhaustions, that MΣ +fb indeed has {y = z = 0}-index 1. +For use in the following subsection we fix a smooth cutoff function Ψ: [0, ∞[ → [0, 1] that is +constantly 1 on {x ≤ 1} and constantly 0 on {x ≥ 2}, and we define on MΞ and MΣ the functions +and metrics +ψΞ := (Ψ ◦ |x|)|MΞ, +ρΞ := +� +ψΞ + 1 +2 +���AMΞ��� +2 +(1 − ψΞ), +hΞ := (ρΞ)2gMΞ, +ψΣ := (Ψ ◦ |x|)|MΣ, +ρΣ := +� +ψΣ + 1 +2 +���AMΣ��� +2 +(1 − ψΣ), +hΣ := (ρΣ)2gMΣ. +(5.5) +Note that ρΞ is invariant under R{z=0}, ρΣ under R{y=z=0}, and both are invariant under R{x=0}, +R{y=−π/2}, and R{y=π/2}. It is natural to associate to MΞ +fb, regarded as a free boundary minimal +surface in the slab {x ≤ 0} ∩ {|y| ≤ π/2}, the stability form QMΞ +fb, defined at least on smooth +functions of compact support by +QMΞ +fb(u, v) := +� +MΞ +fb +gMΞ�∇gMΞu, ∇gMΞv +� dH 2(gMΞ) − +� +MΞ +fb +���AMΞ��� +2 +gMΞuv dH 2(gMΞ). +From the identity +QMΞ +fb(u, v) = +� +MΞ +fb +hΞ�∇hΞu, ∇hΞv +� dH 2(hΞ) − +� +MΞ +fb +���AMΞ��� +2 +hΞuv dH 2(hΞ) +and the manifest boundedness of +��AMΞ��2 +hΞ = (ρΞ)−2��AMΞ��2 +gMΞ we see that QMΞ +fb is in fact well-defined +on H1(MΞ +fb, hΞ). Likewise, the analogously defined QMΣ +fb is well-defined on H1(MΣ +fb, hΣ). +We now point out that we can identify the interiors of MΞ +fb and MΣ +fb under respectively the metrics +hΞ and hΣ as Lipschitz domains as in the setting of Section 2. Concretely, we first consider the +Riemannian quotients � +MΞ and � +MΣ of (MΞ, hΞ) and (MΣ, hΣ) under a fundamental period. Then +� +MΞ is diffeomorphic to S2 with four points removed and � +MΣ to S2 with six points removed. By +virtue of (5.5) and the behavior of the Gauss maps as outlined in Proposition 5.12, we can in fact +choose the last two diffeomorphisms so that they are isometries on neighborhoods of the punctures. +In this way we obtain smooth Riemannian compactifications. By composing the defining projection +of each tower onto its quotient by a fundamental period with the corresponding embedding into the +compactification we identify (via isometric embedding) the interior of MΞ +fb under hΞ and the interior +34 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +of MΣ +fb under hΣ with Lipschitz domains � +MΞ +fb and � +MΣ +fb in the two respective compactifications, and +we likewise identify ∂MΞ +fb and ∂MΣ +fb with subsets of ∂� +MΞ +fb and ∂� +MΣ +fb respectively. Of course, the role +of the “ambient manifold” for such Lipschitz domains is played respectively by the Riemannian +manifolds (S2, hΞ) and (S2, hΣ); here, with slight abuse of notation, we have tacitly extended the +metrics in question across the four and six punctures respectively. +Next, recalling the definition of T from (2.4), we define the bilinear form +Q� +MΞ +fb := T +� +� +MΞ +fb, hΞ, q = (ρΞ)−2���AMΞ��� +2 +gMΞ, r = 0, ∂D� +MΞ +fb = ∅, ∂N� +MΞ +fb = ∂� +MΞ +fb, ∂R� +MΞ +fb = ∅ +� +, +where (as we shall do generally in the sequel for functions defined on MΞ or MΣ, without further +comment) for the potential we tacitly interpret the right-hand side as a function on � +MΞ +fb; we define +Q� +MΣ +fb in analogous fashion. We then have (cf. Section 3.4) the equalities +Q� +MΞ +fb = QMΞ +fb on H1(MΞ +fb, hΞ) +and +Q� +MΣ +fb = QMΣ +fb on H1(MΣ +fb, hΣ). +(5.6) +Lemma 5.16 (Index and nullity of Q� +MΞ +fb and Q� +MΣ +fb). With definitions as in the preceding paragraph +we have the following indices and nullities. +S +G +ind+ +G(QS) +nul+ +G(QS) +ind− +G(QS) +nul− +G(QS) +� +MΞ +fb +{z = 0} +1 +0 +0 +1 +� +MΣ +fb +{y = z = 0} +1 +1 +1 +0 +Proof. The first row follows from a direct application of Proposition 3.11 in conjunction with the +first two rows of the table in Lemma 5.13. Indeed, in this case there are no umbilic points in play +(for, recall, MΞ has no umbilic points) and the Gauss map furnishes an (anti)conformal map from the +compactified quotient onto S2. For MΣ, however, the corresponding conformal factor degenerates at +the umbilic at (0, 0, 0), as all of its translates. Nevertheless, aided by Lemma 3.5 and Corollary 3.10 +we can verify the indices in the second row in much the same fashion, applying Proposition 3.11 on +suitable subdomains (obtained by removing smaller and smaller neighborhoods of the origin). +For the nullities, however, we employ an ad hoc argument, since one cannot expect an analogue +of the aforementioned Corollary 3.10 to hold true in general. That said, we observe first that the +translations in the z direction induce a nontrivial, smooth, bounded, ({y = z = 0}, +)-invariant +(scalar-valued) Jacobi field on MΣ which readily implies it to define an element of H1(MΣ +fb, hΣ). This +shows, in view of (5.6), that the nullities in question are at least the values indicated in the table. +On the other hand, (appealing to Lemma 3.5 for the regularity) each element, say u: � +MΣ +fb → R, of +the eigenspace with eigenvalue zero corresponding to the nullities in question is smooth and bounded. +If we restrict it to ΩΣ ⊂ � +MΣ +fb and consider the precomposition with the inverse of the Gauss map +(which, let us recall, yields an (anti)conformal diffeomorphism νMΣ : ΩΣ → ΩΣ +S2), then the resulting +function u0 := u◦(νMΣ)−1 satisfies (∆gS2 +2)u0 = 0 and so we get an element contributing to nul(T) +where T is as encoded in the third (respectively: the fifth) row of the table (5.3) when starting from +the ({y = z = 0}, +)-invariant (respectively: the ({y = z = 0}, −)-invariant) problem on � +MΣ +fb. +It is clear that one thereby gets injective maps of vector spaces, and so from Lemma 5.13 +nul+ +G(Q� +MΣ +fb) ≤ 1, +nul− +G(Q� +MΣ +fb) ≤ 0 +35 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +which in particular implies that such maps are, a posteriori, linear isomorphisms, and thus completes +the proof. +When we wish to consider the sets MΞ +fb(s) and MΣ +fb(s) endowed respectively with the metrics hΞ +and hΣ, we shall denote them by � +MΞ +fb(s) and � +MΣ +fb(s). Recalling the notation of Subsection 2.5, we +further define +Q� +MΞ +fb(s) +D +:= +� +Q� +MΞ +fb +�Dint +� +MΞ +fb(s) +and +Q� +MΞ +fb(s) +N +:= +� +Q� +MΞ +fb +�Nint +� +MΞ +fb(s). +(5.7) +In short, we are adjoining respectively Dirichlet or Neumann boundary conditions along the cuts. +Lemma 5.17 (Spectra of Q� +MΞ +fb(s) and Q� +MΣ +fb(s)). For each integer i ≥ 1 +lim +s→∞ λ{z=0},± +i +� +Q� +MΞ +fb(s) +D +� += lim +s→∞ λ{z=0},± +i +� +Q� +MΞ +fb(s) +N +� += λ{z=0},± +i +� +Q� +MΞ +fb +� +, +lim +s→∞ λ{y=z=0},± +i +� +Q� +MΣ +fb(s) +D +� += lim +s→∞ λ{y=z=0},± +i +� +Q� +MΣ +fb(s) +N +� += λ{y=z=0},± +i +� +Q� +MΣ +fb +� +, +for any consistent choice of + or − on both sides of each equality. +Proof. We will write down the proof of the two equalities in the first line for the + choice, as the +remaining cases can be proved in the same way. First note that Proposition 3.9 gives us +lim +s→∞ λ{z=0},+ +i +� +Q� +MΞ +fb(s) +D +� += λ{z=0},+ +i +� +Q� +MΞ +fb +� +. +Using the min-max characterization (2.13) of eigenvalues we then also get +lim sup +s→∞ λ{z=0},+ +i +� +Q� +MΞ +fb(s) +N +� +≤ lim sup +s→∞ λ{z=0},+ +i +� +Q� +MΞ +fb(s) +D +� += λ{z=0},+ +i +� +Q� +MΞ +fb +� +. +The key step now in establishing +lim inf +s→∞ λ{z=0},+ +i +� +Q� +MΞ +fb(s) +N +� +≥ λ{z=0},+ +i +� +Q� +MΞ +fb +� +(which completes the proof) is to construct a family of (appropriately symmetric) linear extension +operators Es : H1(� +MΞ +fb(s)) → H1(� +MΞ +fb) uniformly bounded in s, assuming s ≥ s0 for some universal +s0 > 0. With these extensions in hand it is straightforward, for example, to adapt the argument for +(3.6) in the proof of Proposition 3.8. +We now construct the Es extension operators. By the imposed symmetry (in the case under +discussion even reflection through {z = 0}) and by taking s large enough, it suffices to specify the +extension on a single end W, a graph over a subset of the corresponding asymptotic plane Π (with +τ the corresponding defining vector, recalling the notation preceding (5.4)). Let ϖ: W → Π be +the associated projection. By partitioning the given function using appropriately chosen smooth +cutoff functions (fixed independently of s), it in fact suffices to consider the extension problem for +a function v ∈ H1(W ∩ MΞ +fb(s), hΞ) such that the support of ϖ∗v is compactly contained in the +rectangle (expressed in the notation of (5.4)) +{0 < τ · (x, y, z) ≤ s} ∩ {−π ≤ 2y ≤ π}. +36 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +We can extend ϖ∗v via even reflection through the s side of the above rectangle, thereby obtaining an +extension of v to an element of H1(W, hΞ). The asymptotic convergence of W to Π, the monotonic +decay of ρΞ along W toward ∞, and the conformal invariance (in the current two-dimensional +setting) of the Dirichlet energy ensure that this extension has the desired properties. +5.3 Deconstruction of the surfaces and regionwise geometric convergence +We first take a moment to briefly review the constructions of the surfaces from [6]. First (cf. +[6, Section 3]), an approximate minimal surface in B3, called the initial surface, whose boundary +is contained in ∂B3 and which meets ∂B3 exactly orthogonally, is fashioned by hand, via suitable +interpolations, from the models (K0, MΞ or MΣ, and for Σ−K0∪B2∪K0 +m +also B2). Second (cf. [6, +Section 5]), the final exact solution is identified as the normal graph of a small function over +the approximate solution. For what pertains this second step we wish only to highlight that the +assignment of graph to function is made using not the usual Euclidean metric gR3 but instead +an O(3)-invariant metric (fixed once and for all, independently of the data n or m) conformally +Euclidean, and called the auxiliary metric. On a neighborhood of the origin this metric agrees +exactly with the Euclidean one, while on a neighborhood of ∂B3 = S2 it agrees exactly with +the cylindrical metric on S2 × R; this last property and the orthogonality of the intersection of +the initial surface with ∂B3 ensure that the boundary of the resulting graph is also in ∂B3. We +will write �Ξ−K0∪K0 +n +and �Σ−K0∪B2∪K0 +m +for the initial surfaces and ϖΞ +n : Ξ−K0∪K0 +n +→ �Ξ−K0∪K0 +n +and +ϖΣ +m : Σ−K0∪B2∪K0 +m +→ �Σ−K0∪B2∪K0 +m +for the nearest-point projections under the above auxiliary metric. +Turning to the first step, actually (because of the presence of a cokernel) one constructs for each +given n or m not just a single initial surface but a (continuous) one-parameter family of them. In +the construction this parameter is treated as an unknown and is determined only in the second +step, simultaneously with the defining function for the final surface. Here, however, we can take the +construction for granted and accordingly speak of a single initial surface, whose defining parameter +value is some definite (though not explicit) function of n or m as appropriate. Nevertheless we must +explain that this parameter enters the construction at the level of the building blocks, except for +B2, which is unaffected, as follows. First, the catenoidal annulus K0 is just one in a family Kϵ (cf. +the beginning of Subsection 3.1 in [6]) of such annuli, all rotationally symmetric about the z-axis, +depending smoothly on ϵ. The details are not critical here, but each Kϵ is the intersection with B3 +of a complete catenoid with axis the z-axis, and Kϵ meets S2 at two circles of latitude, the upper +one a circle of orthogonal intersection and the lower one the circle at height z = ϵ. Similarly, from +MΞ and MΣ we define, by explicit graphical deformation, families which here we will call MΞ +δ and +MΣ +δ (cf. the beginning of Subsection 3.2 of [6]). These deformations are the identity on the “cores” +of MΞ and MΣ and smoothly transition to translations on the ends, in the z-direction, up or down +depending on the end, and through a displacement determined by δ. Importantly, all the MΞ +δ and +MΣ +δ have the same symmetries as MΞ and MΣ respectively. Now the datum n determines building +blocks MΞ +δΞ(n) and KϵΞ(n), while the datum m determines building blocks MΣ +δΣ(m), KϵΣ(m), and B2. +We next define maps ΦΞ +n and ΦΣ +m ([6, (3.37)]) from neighborhoods of +1 +nMΞ +δΞ(n) ∩ {x ≤ 0} and +1 +m+1MΣ +δΣ(m) ∩ {x ≤ 0} respectively into B3, so as to “wrap” the cores of these surfaces around +the equator S1 approximately isometrically but to take their asymptotic half planes (in {x ≤ 0}) +onto ±KϵΞ(n) in the first case and onto ±KϵΣ(m) and B2 in the second. Thus, just referring to the +family Ξ−K0∪K0 +n +for the sake of brevity, we truncate 1 +nMΞ +δΞ(n) by intersecting with {x ≥ −n3/4} and +37 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +then apply ΦΞ +n. The image is embedded (for n large enough) and contained in the ball, in fact +contained in a tubular neighborhood of S1 with radius of order n−1/4. Near the two truncation +boundary components the surface is a small graph over either ±KϵΞ(n). We smoothly cut off the +defining function in a 1 +n-neighborhood of the boundary to make the surface exactly catenoidal +there and then extend using these annuli on the other side of the truncation boundary all the way +to ∂B3. The result is our initial surface �Ξ−K0∪K0 +n +. The initial surface �Σ−K0∪B2∪K0 +m +is constructed +analogously, now also smoothly transitioning from the middle truncation boundary to coincide with +B2 on neighborhood of the origin. In what follows we will distill those objects and ancillary results +that are needed for the spectral convergence theorems we will prove in Section 5.4. +Decompositions. +Recalling (5.4) for the definition of the below domains, our construction in [6] +provides, in particular, smooth maps +ϕMΞ +n : MΞ +−(n5/8) → Ξ−K0∪K0 +n +, +ϕMΣ +m : MΣ +−((m + 1)5/8) → Σ−K0∪B2∪K0 +m +, +which are smooth coverings of their images. For all 0 < s ≤ √n or, respectively, 0 < s ≤ √m + 1 +we in turn define +MΞ +n (s) := ϕMΞ +n �MΞ +−(s) +� ⊂ Ξ−K0∪K0 +n +, +MΣ +m(s) := ϕMΣ +m�MΣ +−(s) +� ⊂ Σ−K0∪B2∪K0 +m +. +In practice, in addition to the upper bound required on s, we will be interested only in s greater +than a universal constant set by MΞ and MΣ: in essence we want to truncate far enough out +(in the domain) that near and beyond the truncation boundary the surface is already the graph +of a small function over the asymptotic planes. In a typical application to follow we will take s +large in absolute terms and then take n or m large with respect to s, so we will not always repeat +either restriction. When they do hold, Ξ−K0∪K0 +n +\ MΞ +n (s) consists of two connected components and +Σ−K0∪B2∪K0 +m +\ MΣ +m(s) consists of three, and we define +KΞ +n (s) := the closure of the component of Ξ−K0∪K0 +n +\ MΞ +n (s) on which z is maximized, +KΣ +m(s) := the closure of the component of Σ−K0∪B2∪K0 +m +\ MΣ +m(s) on which z is maximized, +BΣ +m(s) := the closure of the component of Σ−K0∪B2∪K0 +m +\ MΣ +m(s) that contains the origin. +Observe that each MΞ +n (s) is invariant under R{z=0}, that the interiors of MΞ +n (s), KΞ +n (s), and +R{z=0}KΞ +n (s) are pairwise disjoint, and that the last three regions cover Ξ−K0∪K0 +n +. In particular, +considering the interior of such sets, one thereby determines a candidate partition for the application +of Proposition 3.1. Similarly, MΣ +m(s) and BΣ +m(s) are invariant under R{y=z=0}; the interiors of MΣ +m(s), +BΣ +m(s), KΣ +m(s), and R{y=z=0}KΣ +m(s) are pairwise disjoint, also such four surfaces cover Σ−K0∪B2∪K0 +m +. +We agree to distinguish the choices s = √n and s = √m + 1 by omission of the parameter value: +MΞ +n := MΞ +n (√n), +KΞ +n := KΞ +n (√n), +MΣ +m := MΣ +m( +√ +m + 1), +KΣ +m := KΣ +m( +√ +m + 1), +BΣ +m := BΣ +m( +√ +m + 1), +as visualized in Figure 5. We also define the dilated truncations (cf. Figure 6) +MΞ +fb,n(s) := n +� +MΞ +n (s) ∩ W π/(2n) +−π/(2n) +� += nϕMΞ +n (MΞ +fb(s)), +MΞ +fb,n := MΞ +fb,n(√n), +MΣ +fb,m(s) := (m + 1) +� +MΣ +m(s) ∩ W π/(2(m+1)) +−π/(2(m+1)) +� += (m + 1)ϕMΣ +m(MΣ +fb(s)), +MΣ +fb,m := MΣ +fb,m( +√ +m + 1), +38 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +where the notation for wedges has been given in (4.2), and finally introduce the transition regions +ΛΞ +n(s) := MΞ +fb,n \ MΞ +fb,n(s), +ΛΣ +m(s) := MΣ +fb,m \ MΣ +fb,m(s). +Geometric estimates. +Before proceeding, we declare the following abbreviated notation for the +metrics and second fundamental forms on MΞ +fb,n and MΣ +fb,m (induced by their inclusions in (R3, gR3)): +gΞ +n := gMΞ +fb,n, +gΣ +m := gMΣ +fb,m, +AΞ +n := AMΞ +fb,n, +AΣ +m := AMΣ +fb,m. +In analogy with (5.5) we first write ψΞ +n, ψΣ +m for the unique functions on MΞ +fb,n, MΣ +fb,m such that +ψΞ = +� +n ◦ ϕMΞ +n +�∗ ψΞ +n, +ψΣ = +� +(m + 1) ◦ ϕMΣ +m +�∗ ψΣ +m +and then in turn define +ρΞ +n := +� +ψΞn + 1 +2 +��AΞn +��2 +gΞ +n(1 − ψΞn) + e−2n, +hΞ +n := (ρΞ +n)2gΞ +n, +ρΣ +m := +� +ψΣ +m + 1 +2 +��AΣm +��2 +gΣ +m(1 − ψΣ +m) + e−2m, +hΣ +m := (ρΣ +m)2gΣ +m. +(5.8) +The terms e−2n and e−2m above are included to ensure the conformal factors vanish nowhere. For +the sake of brevity, and consistently with the notation adopted in the previous subsections, we set +� +MΞ +fb,n := (MΞ +fb,n, hΞ +n), +� +MΞ +fb,n(s) := (MΞ +fb,n(s), hΞ +n) +� +MΣ +fb,m := (MΣ +fb,m, hΣ +m), +� +MΣ +fb,m(s) := (MΣ +fb,m(s), hΣ +m), +so that � +MΞ +fb,n and � +MΣ +fb,m and their truncations � +MΞ +fb,n(s) ⊂ MΞ +fb,n and MΣ +fb,m(s) ⊂ MΣ +fb,m are always +equipped with the conformal metrics hΞ +n and hΣ +m, rather than gΞ +n and gΣ +m. +Lemma 5.18 (Convergence of MΞ +fb,n(s) and MΣ +fb,m(s)). For every s > 0 there exists ms > 0 such +that for every integer m > ms +(i) the region MΣ +fb,m(s) is defined and is the diffeomorphic image under (m + 1)ϕMΣ +m of MΣ +fb(s), +(ii) (m + 1)ϕMΣ +m�MΣ +fb(s) ∩ {x = 0} +� = MΣ +fb,m(s) ∩ (m + 1)S2, +(iii) ϕMΣ +m commutes with R{z=0}, and +(iv) MΣ +m(s) = (m + 1)−1Am+1MΣ +fb,m(s) is a surface with smooth boundary. +Moreover, for every s > 0 and α ∈ ]0, 1[ +(v) +� +(m + 1) ◦ ϕMΣ +m +�∗ gΣ +m +C1,α(MΣ +fb(s),gMΣ) +−−−−−−−−−−−→ +m→∞ +gMΣ and +(vi) +� +(m + 1) ◦ ϕMΣ +m +�∗ AΣ +m +C0,α(MΣ +fb(s),gMΣ) +−−−−−−−−−−−→ +m→∞ +AMΣ. +All the above statements have analogues for Ξ−K0∪K0 +n +in place of Σ−K0∪B2∪K0 +m +, mutatis mutandis. +39 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +MΞ +n +KΞ +n +BΣm +KΣ +m +MΣ +m +Figure 5: Decomposition of Ξ−K0∪K0 +n +(left) and Σ−K0∪B2∪K0 +m +(right, cutaway view). +x +MΞ +fb,n +x +MΣ +fb,m +Figure 6: The dilated truncations MΞ +fb,n (left) and MΣ +fb,m (right). +40 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +The first four claims are immediate from the definitions, while the convergence assertions are ensured, +in the case of Σ−K0∪B2∪K0 +m +, by the following estimates from [6], the case of Ξ−K0∪K0 +n +being completely +analogous. Namely, the estimate [6, (5.20)] provides C2,α bounds for the defining function of +Σ−K0∪B2∪K0 +m +as a graph over the corresponding initial surface, so controlling the projection map ϖΣ +m +from Σ−K0∪B2∪K0 +m +to the initial surface. The same estimate [6, (5.20)] also bounds the parameter +value for the initial surface from the one-parameter family that is selected to produce the final +one. On the other hand, [6, Proposition 3.18] provides estimates on the initial surface, in terms +of the datum g as well as the value of the continuous parameter. (As an aid to extracting the +required information, we point out that the map ϖMm,ξ in [6, (3.43)] is essentially (that is: up to +some quotienting and the exact extent of the domains) the inverse of the map ϖΣ +m−1 ◦ ϕMΣ +m−1 of the +present article.) +Let us consider the other portions of our surfaces. By construction ϖΞ +n(KΞ +n ) and ϖΣ +m(KΣ +m) (subsets +of the initial surfaces) are graphs (under the Euclidean metric gR3) over subsets of KϵΞ(n) and +KϵΣ(m), and ϖΣ +m(BΣ +m) a graph over B2. Thus, by composition with a further projection, we obtain +injective maps ϖΞ +n(KΞ +n ) → KϵΞ(n), ϖΣ +m(KΣ +m) → KϵΣ(m), and BΣ +m → B2. Moreover, the image of each +of these three maps is O(2) invariant: the image of the third is a disc with radius tending to 1 as +m → ∞, the image of the second is a catenoidal annulus with upper boundary circle coinciding +with that of KϵΣ(m) and lower boundary circle tending to that of KϵΣ(m) as m → ∞; the image of +the first admits an analogous description. +In particular, by composing further with dilations of scale factor tending to 1, we obtain diffeomor- +phisms +ϕBΣ +m : B2 → BΣ +m; +similarly reparametrizing in the radial direction one also obtains diffeomorphisms +ϕKΞ +n : K0 → KΞ +n , +ϕKΣ +m : K0 → KΣ +m. +The inverses of these maps may be regarded as small perturbations (for n and m large) of nearest- +point projection onto R2 ⊂ B2 or onto the complete catenoid containing K0, as appropriate. +Somewhat more formally, by reference to [6] (specifically Proposition 3.18 and estimate (5.20) +therein), much as in the proof of Lemma 5.18, we confirm the following properties of KΞ +n , KΣ +m, and +BΣ +m. +Lemma 5.19 (Convergence of KΞ +n and KΣ +m). There exists m0 > 0 such that for each integer +m > m0 +(i) ϕKΣ +m is defined and a diffeomorphism from K0 onto KΣ +m, +(ii) ϕKΣ +m commutes with each element of Ym+1, and +(iii) ϕKΣ +m takes the upper boundary component of K0 to the upper boundary component of KΣ +m. +Moreover, for every α ∈ ]0, 1[ +(iv) (ϕKΣ +m)∗gΣ−K0∪B2∪K0 +m +��� +KΣ +m +C1,α(K0,gK0) +−−−−−−−−→ +m→∞ +gK0 and +(v) (ϕKΣ +m)∗AΣ−K0∪B2∪K0 +m +��� +KΣ +m +C0,α(K0,gK0) +−−−−−−−−→ +m→∞ +AK0. +41 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +All the above statements have analogues for Ξ−K0∪K0 +n +in place of Σ−K0∪B2∪K0 +m +, mutatis mutandis. +Lemma 5.20 (Convergence of BΣ +g ). There exists m0 > 0 such that for each integer m > m0 +(i) ϕBΣ +m is defined and a diffeomorphism from B2 onto BΣ +m and +(ii) ϕBΣ +m commutes with each element of Am+1. +Moreover, for each α ∈ ]0, 1[ +(iii) (ϕBΣ +m)∗gΣ−K0∪B2∪K0 +m +��� +BΣ +m +C1,α(B2,gB2) +−−−−−−−−→ +m→∞ +gB2 and +(iv) (ϕBΣ +m)∗AΣ−K0∪B2∪K0 +m +��� +BΣ +m +C0,α(B2,gB2) +−−−−−−−−→ +m→∞ +0. +Last we focus on the transition regions. Let us agree to write tΞ +n and tΣ +m for the distance functions on +nKϵΞ(n) and (m + 1)KϵΣ(m) from their respective lower boundary circles. By construction (assuming +s large enough in absolute terms) nϖΞ +n(n−1ΛΞ +n(s)) has two connected components, one a graph over +the catenoidal annular wedge +{s ≤ tΞ +n ≤ √n} ∩ W π/(2n) +−π/(2n) ⊂ nKϵΞ(n) +and the other the reflection of this last one through {z = 0}, while (m + 1)ϖΣ +m((m + 1)−1ΛΣ +n(s)) +has three connected components, one a graph over the planar annular wedge +{s ≤ (m + 1) − r ≤ +√ +m + 1} ∩ W π/(2(m+1)) +−π/(2(m+1)) ∩ (m + 1)B2, +another a graph over the catenoidal annular wedge +{s ≤ tΣ +m ≤ +√ +m + 1} ∩ W π/(2(m+1)) +−π/(2(m+1)) ⊂ (m + 1)KϵΣ(m), +and the third the reflection of this last one through {y = z = 0}. +Projecting onto these rotationally invariant sets and parametrizing them by arc length t in the +“radial” direction and ϑ := nθ or, respectively, ϑ := (m + 1)θ, in the angular direction (with θ +restricted to the appropriate interval containing 0), we obtain injective maps +ϕΛΞ +n(s),K : +� +s, √n +� +× +� +−π +2 , π +2 +� +→ ΛΞ +n, +ϕΛΣ +m(s),K, ϕΛΣ +m(s),B2 : +� +s, +√ +m + 1 +� +× +� +−π +2 , π +2 +� +→ ΛΣ +m +whose images are components of ΛΞ +n(s) and ΛΣ +m(s) that generate the latter regions under {z = 0} +and {y = z = 0} respectively. +Lemma 5.21 (Estimates on ΛΞ +n(s) and ΛΣ +m(s)). Let α ∈ ]0, 1[. There exists s0 > 0 such that for +each s > s0 there exists ms > 0 such that for every integer m > ms +(i) (ϕΛΣ +m(s),K)∗|AΣ +m|2 +gΣ +m(t, ϑ) = a1(t)m−2 + a2(t, ϑ)e−t/4 for some smooth functions a1, a2 having +C0,α(dt2 + dϑ2) norm bounded independently of m and s, +42 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +(ii) (ϕΛΣ +m(s),B2)∗|AΣ +m|2 +gΣ +m(t, ϑ) = a3(t, ϑ)e−t/4 for some smooth function a3 having C0,α(dt2 + dϑ2) +norm bounded independently of m and s, +(iii) (ϕΛΣ +m(s),K)∗gΣ +m = dt2 + (1 + m−1tf1(t))dϑ2 + f1 +uv(t, ϑ)e−t/4 du dv for some smooth functions +f1, f1 +uv having C1,α(dt2 + dϑ2) norm bounded independently of m and s, +(iv) (ϕΛΣ +m(s),B2)∗gΣ +m = dt2 + (1 + m−1tf2(t))dϑ2 + f2 +uv(t, ϑ)e−t/4 du dv for some smooth functions +f2, f2 +uv having C1,α(dt2 + dϑ2) norm bounded independently of m and s, +(v) ∆(ϕΛΣ +m(s),K)∗gΣ +m = ∂2 +t + m−1ct +1(t)∂t + (1 + m−1/2bϑϑ +1 (t))∂2 +ϑ + e−t/4(buv +2 (t, ϑ)∂u∂v + cu +2(t, ϑ)∂u) for +some smooth functions bϑϑ +1 , buv +2 , ct +1, cu +2 having C0,α(dt2 + dϑ2) norm bounded independently of +m and s, and +(vi) ∆(ϕΛΣ +m(s),B2)∗gΣ +m = ∂2 +t + m−1ct +3(t)∂t + (1 + m−1/2bϑϑ +3 (t))∂2 +ϑ + e−t/4(buv +4 (t, ϑ)∂u∂v + cu +4(t, ϑ)∂u) for +some smooth functions bϑϑ +3 , buv +4 , ct +3, ci +4 having C0,α(dt2 + dϑ2) norm bounded independently of +m and s. +It is understood that, in items (iii), (iv), (v), (vi) one takes u, v ∈ {t, ϑ}. +Furthermore, +(vii) lim +s→∞ lim +m→∞H 2(hΣ +m)(ΛΣ +m(s)) = 0. +The same claims hold for ΛΞ +n(s), mutatis mutandis. +Proof. Again the estimates are ultimately justified by reference to the construction [6], most +specifically (5.20) and Proposition 3.18 therein. That said, we also note how claim (v) follows +easily from (iii), as does claim (vi) from (iv); furthermore, it is clear that the justification of (ii) is +analogous to (in fact simpler than) (i), and (iv) is analogous to (iii). As a result, we briefly explain +the ideas behind the elementary computations required for the proof, in the case of ΛΣ +m(s), so with +regard to items (i) and (iii). +The projection of this region onto the blown-up initial surface (m+1)�Σ−K0∪B2∪K0 +m +is itself constructed +as a graph over (m+1)KϵΣ(m) or B2. Estimate [6, (5.20)] ensures that mϵΣ(m) is bounded uniformly +in m. The defining function of the above graph is obtained by “transferring” the defining functions +of the corresponding ends of MΣ over their asymptotic planes. These defining functions decay +exponentially in the distance along the planes. In turn ΛΣ +m(s) is a graph over this portion of the +initial surface with defining function that is also guaranteed (by [6, (5.20)]) to decay exponentially, +though a priori at a slower rate; we have chosen 1/4 somewhat arbitrarily. This accounts for all +exponential factors appearing in the estimates. +The m-dependent terms in the estimates for the metric (and Laplacian) arise simply from the choice +of (t, ϑ) coordinates on disc and catenoidal models. The m−2 term in the first item arises from +scaling the second fundamental form of the “asymptotic” catenoid to this component (while the +corresponding term for the disc vanishes). With the estimates for the second fundamental form in +place, the final item – the area estimate – follows (recalling the definitions (5.8)) from the bound +� π/2 +−π/2 +� √m+1 +s +�a1m−2 + a2e−t/4� dt dϑ ≤ C +� +m−3/2 + e−s/4� +, +and the analogous estimate concerning the disk-type component instead. +43 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +5.4 Regionwise spectral convergence +For each region S among MΞ +n , MΣ +m, KΞ +n , KΣ +n , and BΣ +m (depicted in Figure 5) we write QS +N for the +Jacobi form of S as a minimal surface in B3 with boundary, subject to the Robin condition (4.1) +where ∂S meets ∂B3 and subject to the Neumann condition elsewhere: recalling (2.17), we set +QS +N := +� +� +� +� +� +� +� +� +QΞ−K0∪K0 +n +�Nint +S +for S ⊂ Ξ−K0∪K0 +n +� +QΣ−K0∪B2∪K0 +m +�Nint +S +for S ⊂ Σ−K0∪B2∪K0 +m +(where on the right-hand side we slightly abuse notation in that in place of S we really mean +its interior). Similarly, for S either MΞ +fb,n or MΣ +fb,m we write QS +N for the Jacobi form of S as a +minimal surface in either nB3 or (m + 1)B3, subject to the Robin condition either du(η) = n−1u +or du(η) = (m + 1)−1u where ∂S meets either nS2 or (m + 1)S2, respectively, and subject to the +Neumann condition elsewhere. Keeping in mind the statement of Theorem 3.1, we stress that the +adjunction of Neumann conditions in the “interior” boundaries is motivated by our task of deriving +upper bounds on the Morse index of our examples. Recalling the notation � +MΞ +fb,n and � +MΣ +fb,n, we +remark that the bilinear forms QS +N and Q�S +N agree by definition for each S as above, but whenever +we refer to the eigenvalues, eigenfunctions, index, and nullity of the latter we shall always mean +those defined with respect to the hΞ +n or hΣ +m metric. +In the notation of (2.4) we have in particular (cf. Proposition 3.11) +Q +MΞ +fb,n +N += T +� +MΞ +fb,n, gΞ +n, qΞ +n = |AΞ +n|2 +gΞ +n, rΞ +n = n−1, +∂DMΞ +fb,n = ∅, ∂NMΞ +fb,n = ∂MΞ +fb,n \ nS2, ∂RMΞ +fb,n = ∂MΞ +fb,n \ ∂NMΞ +fb,n +� += T +� +MΞ +fb,n, hΞ +n, +� +ρΞ +n +�−2 +qΞ +n, +� +ρΞ +n +�−1 +n−1, ∅, ∂MΞ +fb,n \ nS2, ∂MΞ +fb,n \ ∂NMΞ +fb,n +� += Q +� +MΞ +fb,n +N +(5.9) +and similarly for Q +MΣ +fb,m +N += Q +� +MΣ +fb,m +N +. Observe further (cf. Lemma 3.5 and Proposition 3.11) +indPn(QMΞ +n +N ) = ind+ +{z=0} +� +Q +� +MΞ +fb,n +N +� +, +indYn(QMΞ +n +N ) = ind +� +Q +� +MΞ +fb,n +N +� +, +indAm+1(QMΣ +m +N +) = ind− +{y=z=0} +� +Q +� +MΣ +fb,m +N +� +, +indYm+1(QMΣ +m +N +) = ind +� +Q +� +MΣ +fb,m +N +� +, +and likewise for the corresponding nullities. +Lemma 5.22 (Equivariant index and nullity on KΞ +n , KΣ +m, and BΣ +m). There exist n0, m0 > 0 such +that we have the following indices and nullities for all integers n > n0 and m > m0: +S +G +indG(QS +N) +nulG(QS +N) +KΞ +n +Yn +1 +0 +KΣ +m +Ym+1 +1 +0 +BΣ +m +Am+1 +0 +0 +44 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +Additionally, still assuming m > m0 we have the upper bound +indYm+1 +� +QBΣ +m +N +� ++ nulYm+1 +� +QBΣ +m +N +� +≤ 1. +Proof. We use the convergence described in Lemma 5.19 and Lemma 5.20 along with Proposition 3.8 +to compare the low eigenvalues of the regions in question with those of their limiting models, as +recorded in Lemma 5.8 and Lemma 5.10. +While we have cut the surfaces Ξ−K0∪K0 +n +and Σ−K0∪B2∪K0 +m +in such a way that the resulting regions +KΞ +n and KΣ +m converge uniformly to K0 and likewise BΣ +m to B2, thereby securing the preceding +lemma in a straightforward fashion, the cases of MΞ +n and MΣ +m are more subtle. Our approach here +(especially the proof of eigenfunction bounds in Lemma 5.25 and their application to Lemma 5.26) +draws inspiration from the analysis Kapouleas makes of the invertibilty of the Jacobi operator on +“extended standard regions” in many gluing construction; for a specific example, concerning Scherk +towers glued to catenoids, we refer the reader to the proof of [17, Lemma 7.4]. +To proceed, recalling (2.17), for each s > 0 and each integer n (sufficiently large in terms of s) we +define +Q +� +MΞ +fb,n(s) +D +:= +� +Q +� +MΞ +fb,n +N +�Dint +� +MΞ +fb,n(s) +and +Q +� +MΞ +fb,n(s) +N +:= +� +Q +� +MΞ +fb,n +N +�Nint +� +MΞ +fb,n(s) +and analogously for � +MΣ +fb,m(s) in place of � +MΞ +fb,n(s). +Lemma 5.23 (Spectral convergence for � +MΞ +fb,n(s) and � +MΣ +fb,m(s)). With the above notation, we have +λ{z=0},± +i +� +Q� +MΞ +fb +� += lim +s→∞ lim +n→∞ λ{z=0},± +i +� +Q +� +MΞ +fb,n(s) +D +� += lim +s→∞ lim +n→∞ λ{z=0},± +i +� +Q +� +MΞ +fb,n(s) +N +� +for each integer i ≥ 1 and each common choice of sign ± on both sides of each equation. The +analogous statements hold, mutatis mutandis, for � +MΣ +fb,m in place of � +MΞ +fb,n. +Proof. Fix i. By Lemma 5.18 and Proposition 3.8 for each s > 0 we have +lim +n→∞ λ{z=0},+ +i +� +Q +� +MΞ +fb,n(s) +D +� += λ{z=0},+ +i +� +Q� +MΞ +fb(s) +D +� +, +lim +n→∞ λ{z=0},+ +i +� +Q +� +MΞ +fb,n(s) +N +� += λ{z=0},+ +i +� +Q� +MΞ +fb(s) +N +� +. +An application of Lemma 5.17 completes the proof in this case, and the proofs of the remaining +three cases are structurally identical to this one. +45 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +Lemma 5.24 (Eigenvalue upper bounds on � +MΞ +fb,n and � +MΣ +fb,m). With the above notation, we have +lim sup +n→∞ λ{z=0},± +i +� +Q +� +MΞ +fb,n +N +� +≤ λ{z=0},± +i +� +Q� +MΞ +fb +� +, +lim sup +m→∞ λ{y=z=0},± +i +� +Q +� +MΣ +fb,m +N +� +≤ λ{y=z=0},± +i +� +Q� +MΣ +fb +� +for each integer i ≥ 1 and each common choice of sign ± on both sides of each equation. +Proof. We give the proof for the + choice on both sides of the top equation, the proofs for the +remaining three cases being identical in structure to this one. Fix i ≥ 1. By (2.13), considering +extensions by zero of functions corresponding to the right-hand side below to obtain valid test +functions corresponding to the left, we get at once the inequality +λ{z=0},+ +i +� +Q +� +MΞ +fb,n +N +� +≤ λ{z=0},+ +i +� +Q +� +MΞ +fb,n(s) +D +� +for all s > 0 and all n sufficiently large in terms of s that � +MΞ +fb,n(s) is defined. We then finish by +applying Lemma 5.23. +Lemma 5.25 (Uniform bounds on eigenvalues and eigenfunctions of Q +� +MΞ +fb,n +N +and Q +� +MΣ +fb,m +N +). For each +integer i ≥ 1 there exist Ci, ki > 0 such that for each integer k > ki and whenever λ(k) +i +is the ith +eigenvalue of Q +� +MΞ +fb,k +N +or Q +� +MΣ +fb,k +N +and v(k) +i +is any corresponding eigenfunction of unit L2-norm (under +either hΞ +k or hΣ +k as appropriate), we have the bounds +max +� +|λ(k) +i +|, ∥v(k) +i +∥H1, ∥v(k) +i +∥C0 +� +≤ Ci +(where the H1 norm is defined via either hΞ +n or hΣ +m as applicable and we emphasize that Ci does not +depend on k). +Proof. We will give the proof for � +MΞ +fb,n that for � +MΣ +fb,m being identical in structure. Fix i ≥ 1 and +let λ(n) and v(n) be as in the statement for each integer n (suppressing the fixed index i); it is our +task to show that by assuming n large enough in terms of just i we can ensure the asserted bounds +on λ(n) and v(n). In particular our assumptions include the normalization ∥v(n)∥L2(MΞ +fb,n,hΞ +n) = 1. +Lemma 5.24 provides an upper bound on λ(n), independent of n. We deduce a lower bound on λ(n) +as follows. Keeping in mind the min-max characterization (2.13) we observe that in the ratio +�u, qnu +� +L2(MΞ +fb,n,hΞ +n) + +�u|∂RMΞ +fb,n, rnu|∂RMΞ +fb,n +� +L2(∂RMΞ +fb,n,hΞ +n) +∥u∥2 +L2(MΞ +fb,n,hΞ +n) +, +(5.10) +with +rn := +� +ρΞ +n +�−1 ��� +∂RMΞ +fb,n +n−1 = (1 + e−2n)−1/2n−1, +qn := +� +ρΞ +n +�−2 ���AΞ +n +��� +2 +gΞ +n +. +46 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +we have not only a uniform upper bound on rn, but also, by inspecting (5.8) and bearing in mind +the convergence described in Lemma 5.18 as well as the boundedness (with decay) of the second +fundamental form of MΞ, +sup +n +∥qn∥C0(MΞ +fb,n) < ∞. +In addition, the convergence in Lemma 5.18 further ensures that the constants appearing in (2.3), +with (Ω, g) = (MΞ +fb,n, hΞ +n) and ∂RΩ in place of ∂Ω can be chosen uniformly in n: thus, employing +such a trace inequality and exploiting the foregoing uniform bounds we secure the promised uniform +lower bound on λ(n). +In turn, from the definitions of eigenvalues and eigenfunctions and the normalization of v(n) we have +���∇hΞ +nv(n)��� +2 +L2(MΞ +fb,n,hΞ +n) = λ(n) + +� +v(n), qnv(n)� +L2(MΞ +fb,n,hΞ +n) ++ +� +v(n)|∂RMΞ +fb,n, rnv(n)|∂RMΞ +fb,n +� +L2(∂RMΞ +fb,n,hΞ +n). +The uniform bound on ∥v(n)∥H1(MΞ +fb,n,hΞ +n) now follows, in view of the above equality, from the upper +bound on λ(n) as well as again the above uniform bounds on qn and rn. +It remains to establish the uniform C0 bound. To start, by Lemma 3.5 and standard elliptic +regularity v(n) is smooth up to the boundary: indeed, it satisfies +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∆hΞ +n + +� +ρΞ +n +�−2 ���AΞ +n +��� +2 +gΞ +n ++ λ(n)� +v(n) = 0 +in MΞ +fb,n, +hΞ +n +� +ηΞ +n, ∇hΞ +nv(n)� += (1 + e−2n)−1/2n−1v(n) +on ∂RMΞ +fb,n, +hΞ +n +� +ηΞ +n, ∇hΞ +nv(n)� += 0 +on ∂NMΞ +fb,n, +(5.11) +with ηΞ +n the outward hΞ +n unit conormal to MΞ +fb,n. As established above, we have bounds independent +of n on |λ(n)| and the qn and rn functions. By Lemma 5.18 (and the uniform geometry of MΞ) we +also have uniform control over the geometry of (MΞ +fb,n(s), hΞ +n) for all s > 0 and all n sufficiently +large in terms of s. +That being said, it is convenient for us to stipulate that throughout this proof C denotes a strictly +positive constant whose value may change from instance to instance but can always be selected +independently of s and n. +Hence, first of all standard elliptic regularity therefore ensures that for every s > 0 there is ns > 0 +so that +���v(n)��� +MΞ +fb,n(s) +��� +C0(MΞ +fb,n(s),hΞ +n) ≤ C for every integer n > ns. +(5.12) +Since we do not have uniform control on the geometry of (MΞ +fb,n = MΞ +fb,n(√n), hΞ +n), we do not obtain +a global bound independent of n in the same fashion. Instead the proof will be completed by +securing a C0 bound for v(n), independent of n, on ΛΞ +n(s) for some s > 0 to be determined. +To proceed we multiply both sides of the PDE in (5.11) by +� +ρΞ +n +�2 to get +� +∆gΞ +n + +���AΞ +n +��� +2 +gΞ +n ++ λ(n) � +ρΞ +n +�2� +v(n) = 0, +(5.13) +47 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +and we aim to bound v(n) on ΛΞ +n(s) on the basis of this equation, with unknown but controlled (as +we explain momentarily) Dirichlet data on the portion of ∂ΛΞ +n(s) contained in the interior of MΞ +fb,n +and with homogeneous Neumann data on the rest of the boundary. By the symmetries it suffices to +establish the estimate on just the component of ΛΞ +n(s) that is a graph over a subset of nK0. (For +ΛΣ +m(s) one must also consider the component which is a graph over a subset of (m + 1)B2, but this +case does not differ in substance from the one we treat now.) +Recall the map +ϕΛΞ +n(s),K : +� +s, √n +� +× +� +−π +2 , π +2 +� +→ Λn(s) +introduced above Lemma 5.21 and continue to write (t, ϑ) for the standard coordinates on its +domain. For the remainder of this proof we abbreviate ϕΛΞ +n(s),K to ϕn,s and its domain to Rn,s. +Setting w(n) := ϕ∗ +n,sv(n), we pull back (5.13) to get +∆ϕ∗n,sgΞ +nw(n) = −w(n)ϕ∗ +n,s +����AΞ +n +��� +2 +gΞ +n ++ λ(n) � +ρΞ +n +�2� +. +From the uniform bound on λ(n), the expression for the conformal factor in (5.8), and item (i) of +Lemma 5.21 we in turn obtain +∆ϕ∗n,sgΞ +nw(n) = +�cn,se−t/4 + dn,sn−2�w(n) +(5.14) +for some smooth functions cn,s, dn,s having C0,α(dt2 + dϑ2) norms uniformly bounded in n and s, +with α ∈ ]0, 1[ now fixed for the rest of the proof. (Here and below when referring to items of +Lemma 5.21 we have in mind of course the corresponding statements for ΛΞ +n(s) in place of ΛΣ +m(s).) +Noting that we have (5.14) for all sufficiently large s, it now follows from the C0 bound (5.12) and +standard interior Schauder estimates (using also item (iii) of Lemma 5.21) that +���w(n)(s, ·) +��� +C2,α(dϑ2) ≤ C for every integer n > ns+1. +(5.15) +Since v(n) satisfies the homogeneous Neumann condition along ∂MΞ +fb,n, with the aid of item (iii) of +Lemma 5.21 we have +(∂tw(n))(√n, ϑ) = en,s e−√n/4(∂ϑw(n))(√n, ϑ), +(5.16) +(∂ϑw(n))(·, ±π/2) = 0 +(5.17) +for some smooth function en,s having C1,α(dt2 + dϑ2) norm bounded independently of n and s. (For +(5.17) we simply use the fact that ϕn,s has been constructed by composing and restricting maps +which commute with the symmetries of the construction, including the reflections through planes +corresponding to ϑ = ±π/2.) +Appealing again to standard Schauder estimates, now also up to the boundary, we can conclude +from (5.14), (5.15), (5.16), and (5.17) that +∥w(n)∥C2,α(dt2+dϑ2) ≤ C +�1 + ∥w(n)∥C0 +� +(5.18) +48 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +for n and s sufficiently large in terms of the bounds assumed on the functions cn,s, dn,s, and en,s, as +well as constants, which can be chosen uniformly, that appear in local Schauder estimates on Rn,s. +If we exploit (5.18) in (5.16) we get +∥(∂tw(n))(√n, ·)∥C1,α(dϑ2) ≤ Ce−√n/4�1 + ∥w(n)∥C0 +�, +(5.19) +once again for n and s assumed large enough in terms of absolute constants. +We next decompose w(n) into +w(n) +0 +:= 1 +π +� π/2 +−π/2 +w(n)(·, ϑ) dϑ, +w(n) +⊥ +:= w(n) − w(n) +0 . +From (5.14), (5.18), and item (v) of Lemma 5.21 we obtain +∂2 +t w(n) +0 += a0 +n,se−t/4 + b0 +n,sn−2 + c0 +n,sn−1∂tw(n) +0 , +with +∥a0 +n,s∥C0 + ∥b0 +n,s∥C0 +1 + ∥w(n)∥C0 ++ ∥c0 +n,s∥C0 ≤ C +(5.20) +and +∥∆dt2+dϑ2w(n) +⊥ ∥C0 ≤ C +� +e−s/4 + n−1/2��1 + ∥w(n)∥C0 +�. +(5.21) +For (5.20) we have in particular integrated (5.14) in ϑ, making use of the ϑ-invariance (see item +(v) of Lemma 5.21) of the coefficients of the n−1∂t and n−1/2∂2 +ϑ terms and observing that the +n−1/2∂2 +ϑ term integrates to zero because of (5.17); for (5.21) we have made use of the fact that +∥∆dt2+dϑ2w(n) +⊥ ∥C0 ≤ 2∥∆dt2+dϑ2w(n)∥C0 and then appealed to (5.14). +To complete the analysis we will need some basic estimates for ∆dt2+dϑ2 = ∂2 +t + ∂2 +ϑ on Rn,s. For +any bounded (real-valued) function f on Rn,s and for each non-negative integer κ let us define on +[s, √n] the Fourier coefficients fκ by +fκ(t) := +� +� +� +1 +π +� π/2 +−π/2 f(t, ϑ) dϑ +for κ = 0 +2 +π +� π/2 +−π/2 f(t, ϑ) cos κ(ϑ − π/2) dϑ +for κ > 0. +Then the Fourier coefficients of any u ∈ C2(Rn,s, dt2 + dϑ2) satisfying (∂ϑu) = 0 at ϑ = ±π/2 admit +the representations +u0(t) = u0(s) + (∂tu0)(√n) · (t − s) + +� t +s +� τ +√n +∂2 +t u0(σ) dσ dτ, += u0(s) + (∂tu0)(√n) · (t − s) + +� t +s +� τ +√n +(∆dt2+dϑ2u)0(σ) dσ dτ, +(5.22) +uκ̸=0(t) = +uκ(s) +cosh κ(√n − s) cosh κ(t − √n) + +(∂tuκ)(√n) +κ cosh κ(√n − s) sinh κ(t − s) +− cosh κ(t − √n) +κ cosh κ(√n − s) +� t +s +(∆dt2+dϑ2u)κ(τ) sinh κ(τ − s) dτ +(5.23) +− +sinh κ(t − s) +κ cosh κ(√n − s) +� √n +t +(∆dt2+dϑ2u)κ(τ) cosh κ(τ − √n) dτ. +49 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +In particular (5.23) implies, for any κ ≥ 1 the inequality +|uκ(t)| ≤ |uκ(s)| + 1 +κ|(∂tuκ)(√n)| + 1 +κ2 ∥(∆dt2+dϑ2u)κ∥C0. +(5.24) +Since u is C2 the Fourier series �∞ +κ=0 uκ(t) cos κ(ϑ − π/2) converges (at least) pointwise to u(t, ϑ); +furthermore (again appealing to the C2 assumption in order to control the first two terms of (5.24)) +we obtain the implication +� π/2 +−π/2 +u(·, ϑ) dϑ = 0 +⇓ +∥u∥C0 ≤ C +� +∥u(s, ·)∥C2(dϑ2) + ∥(∂tu)(√n, ·)∥C1(dϑ2) + ∥∆dt2+dϑ2u∥C0 +� +. +(5.25) +This last estimate in conjunction with (5.21), (5.15), and (5.19) yields +∥w(n) +⊥ ∥C0 ≤ C + C(e−√n/4 + e−s/4 + n−1/2)∥w(n)∥C0. +(5.26) +On the other hand, differentiating (5.22) with respect to t and applying (5.20) and (5.19) we find +∥∂tw(n) +0 ∥C0 ≤ C +�1 + ∥w(n)∥C0 +�� +e−√n/4 + e−s/4 + n−3/2� ++ Cn−1/2∥∂tw(n) +0 ∥C0 +and therefore, by absorption, +∥∂tw(n) +0 ∥C0 ≤ C +�1 + ∥w(n)∥C0 +�� +e−s/4 + n−3/2� +(5.27) +for n sufficiently large in terms of s and the constants appearing in the above estimate. Feeding +(5.27) into (5.20) and applying the result, along with (5.15) and (5.19), in (5.22), we get +∥w(n) +0 ∥C0 ≤ C + C(√ne−√n/4 + e−s/4 + n−1)∥w(n)∥C0. +(5.28) +Finally, since ∥w(n)∥C0 ≤ ∥w(n) +0 ∥C0 + ∥w(n) +⊥ ∥C0, estimates (5.28) and (5.26) jointly imply the desired +bound on the C0 norm of w(n) provided we first choose s and then, in turn, n sufficiently large, in +terms of the absolute constants appearing in the two estimates, to be able to absorb the ∥w(n)∥C0 +terms appearing on their right-hand sides. This ends the proof. +Lemma 5.26 (Eigenvalue lower bounds on � +MΞ +fb,n and � +MΣ +fb,m). For each integer i ≥ 1 +lim inf +n→∞ λ{z=0},± +i +� +Q +� +MΞ +fb,n +N +� +≥ λ{z=0},± +i +� +Q� +MΞ +fb +� +, +lim inf +m→∞ λ{y=z=0},± +i +� +Q +� +MΣ +fb,m +N +� +≥ λ{y=z=0},± +i +� +Q� +MΣ +fb +� +for each common choice of sign ± on both sides of each equation. +50 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +Proof. We give the proof for the + choice on both sides of the top equation, the argument for +the remaining three cases being identical in structure to this one. Fix i ≥ 1, and for each n let +{v(n) +j +}i +j=1 be an L2(MΞ +fb,n, hΞ +n) orthonormal set such that each v(n) +j +is a jth ({z = 0}, +)-invariant +eigenfunction of Q +� +MΞ +fb,n +N +. Fix C > 0, as afforded by Lemma 5.25, such that +sup +n +sup +1≤j≤i +� +∥v(n) +j +∥C0 + λ{z=0},+ +j +� +Q � +MΞ +fb,n +�� +≤ C. +Given any ϵ > 0 (fixed from now on) and taking s > 0 and correspondingly ns > 0 large enough, as +afforded by Lemma 5.21 and Lemma 5.23, we have +H 2(hΞ +n)(Λn(s)) < ϵ, +λ{z=0},+ +i +� +Q� +MΞ +fb +� +< λ{z=0},+ +i +� +Q +� +MΞ +fb,n(s) +N +� ++ ϵ. +(5.29) +Now, for n > ns and 1 ≤ j ≤ i we consider the restrictions +v(n,s) +j +:= v(n) +j +|MΞ +fb,n(s) +and estimate (using the notation δjk for the Kronecker delta) +���δjk − +� +v(n,s) +j +, v(n,s) +k +� +L2(MΞ +fb,n(s),hΞ +n) +��� ≤ C2ϵ, +���∇hΞ +nv(n,s) +j +��� +L2(MΞ +fb,n(s),hΞ +n) ≤ +���∇hΞ +nv(n) +j +��� +L2(MΞ +fb,n,hΞ +n), +� +v(n,s) +j +, +� +ρΞ +n +�−2���AΞ +n +��� +2 +gΞ +n +v(n,s) +j +� +L2(MΞ +fb,n(s),hΞ +n) ≥ +� +v(n) +j +, +� +ρΞ +n +�−2���AΞ +n +��� +2 +gΞ +n +v(n) +j +� +L2(MΞ +fb,n,hΞ +n) − 2C2ϵ, +where for the last inequality we have used the fact that on Λn(s) the potential function appearing +here is bounded above by 2, as is obvious from inspection of (5.8). +We conclude that for all n > ns the set {v(n,s) +j +}i +j=1 is linearly independent, and for all 1 ≤ j ≤ i +Q +� +MΞ +fb,n(s) +N +� +v(n,s) +j +, v(n,s) +j +� +��v(n,s) +j +��2 +L2(MΞ +fb,n(s),hΞ +n) +≤ +λ{z=0},+ +j +� +Q +� +MΞ +fb,n +N +� ++ 2C2ϵ +1 − C2ϵ +and so by virtue of the min-max characterization (2.13) of the eigenvalues +λ{z=0},+ +j +� +Q +� +MΞ +fb,n(s) +N +� +≤ +λ{z=0},+ +j +� +Q +� +MΞ +fb,n +N +� ++ 2C2ϵ +1 − C2ϵ +for all n > ns and 1 ≤ j ≤ i. Thus, using the second inequality in (5.29), we get in particular +λ{z=0},+ +i +� +Q� +MΞ +fb +� +≤ +λ{z=0},+ +i +� +Q +� +MΞ +fb,n +N +� ++ 2C2ϵ +1 − C2ϵ ++ ϵ +for all n > ns. The claim now follows, since this inequality holds for all ϵ > 0, with C independent +of ϵ and n. +51 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +By combining Lemma 5.24 with Lemma 5.26 we immediately derive the following conclusion. +Corollary 5.27 (Eigenvalues on � +MΞ +fb,n and � +MΣ +fb,m). For each integer i ≥ 1 +lim +n→∞ λ{z=0},± +i +� +Q +� +MΞ +fb,n +N +� += λ{z=0},± +i +� +Q� +MΞ +fb +� +, +lim +m→∞ λ{y=z=0},± +i +� +Q +� +MΣ +fb,m +N +� += λ{y=z=0},± +i +� +Q� +MΣ +fb +� +, +for each common choice of sign ± on both sides of each equation. +Corollary 5.28 (Equivariant index and nullity on MΞ +n and MΣ +m). There exist n0, m0 > 0 such that +we have the following indices and nullities for all integers n > n0 and m > m0. +S +G +indG(QS +N) +nulG(QS +N) +MΞ +n +Pn +1 +0 +MΣ +m +Am+1 +1 +0 +Additionally, still assuming m > m0 we have the upper bound +indYm+1 +� +QMΣ +m +N +� ++ nulYm+1 +� +QMΣ +m +N +� +≤ 3. +Proof. All claims follow from the conjunction of Lemma 3.5 (to reduce to the appropriately even +and odd indices and nullities on n−1MΞ +fb,n and (m + 1)−1MΣ +fb,m with Neumann boundary data), +Proposition 3.11 (to dispense with the above scale factors n, m + 1 and, more substantially, to +pass from the natural metric to hΞ +n or hΣ +m), Lemma 5.27 (to reduce to the appropriate indices and +nullities of � +MΞ +fb and � +MΣ +fb ), and finally Lemma 5.16 (which provides these last quantities). +5.5 Proofs of Theorem 1.2 and 1.1 +The following statement collects, from the broader analysis conducted in the previous section, those +conclusions we shall need to prove the two main results stated in the introduction. +Corollary 5.29 (Equivariant index and nullity upper bounds for Σ−K0∪B2∪K0 +m +and Ξ−K0∪K0 +n +). There +exists m0, n0 > 0 such that for all integers m > m0 and n > n0 we have the bounds +indAm+1(Σ−K0∪B2∪K0 +m +) + nulAm+1(Σ−K0∪B2∪K0 +m +) ≤ 2, +indYm+1(Σ−K0∪B2∪K0 +m +) + nulYm+1(Σ−K0∪B2∪K0 +m +) ≤ 5, +indPn(Ξ−K0∪K0 +n +) ++ nulPn(Ξ−K0∪K0 +n +) +≤ 2. +Proof. We apply item (ii) of Proposition 3.1, for the partition “into building blocks” defined in +Section 5.3 (cf. Figure 5), in conjunction with Lemma 5.22 and Corollary 5.28 for the ancillary +estimates for the index and nullity of the various blocks. +52 + +References +A. Carlotto, M. B. Schulz, D. Wiygul +So, we are in position to fully determine the (maximally) equivariant index and nullity for the two +families of free boundary minimal surfaces we constructed in [6]. +Proof of Theorem 1.2. We combine the upper bounds of the preceding corollary with the lower +bounds from our earlier paper [6], specifically with the content of Proposition 7.1 (cf. Remark 7.5) +therein for what pertains the index. At that stage, the fact that both nullities are zero then follows +from the first and third inequality in Corollary 5.29. +Finally, we can obtain the absolute estimates on the Morse index of the same families. +Proof of Theorem 1.1. The lower bounds have already been established: specifically, for Σ−K0∪B2∪K0 +m +this is just part of Proposition 5.4, while for Ξ−K0∪K0 +n +it follows from just combining Proposition 5.4 +with Proposition 5.5. For the upper bound we can apply the Montiel–Ros argument making use +of the equivariant upper bounds above, as we are about to explain. In the case of Ξ−K0∪K0 +n +, the +Pn-equivariant upper bound on the Morse index (and nullity) is equivalent to an upper bound on +the index an nullity on each domain Ωn +i = Ξ−K0∪K0 +n +∩ Wi where W1, . . . , W4n are the open domains +defined, in B3, by the horizontal plane {z = 0} together with the 2n vertical planes passing through +the origin and having equations θ = π/(2n)+iπ/n, i = 0, 1, . . . , 2n−1 (in the cylindrical coordinates +defined at the beginning of Section 4), subject to Neumann conditions in the interior boundary +as prescribed by Lemma 3.5. Thus the conclusion comes straight by appealing to Corollary 3.2. +Similarly, for Σ−K0∪B2∪K0 +m +we can interpret the second inequality in the statement of Corollary 5.29 +as a statement on the index and nullity of the portions of surfaces that are contained in any of the +2(m + 1) sets obtained by intersecting the horizontal plane {z = 0} with the m + 1 vertical planes +passing through the origin and having equations θ = π/(2(m + 1)) + 2iπ/(m + 1), i = 0, 1, . . . , m, +again subject to Neumann conditions. This completes the proof. +References +[1] L. Ambrozio, A. Carlotto, and B. Sharp, Comparing the Morse index and the first Betti number of minimal +hypersurfaces, J. Differential Geom. 108 (2018), no. 3, 379–410. +[2] +, Index estimates for free boundary minimal hypersurfaces, Math. Ann. 370 (2018), no. 3-4, 1063–1078. +[3] N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities +of second order, J. Math. 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Scherk, Bemerkungen über die kleinste Fläche innherhalb gegebener Grenzen, J. Reine Angew. Math. 13 +(1835), 185–208. +[36] G. Smith and D. Zhou, The Morse index of the critical catenoid, Geom. Dedicata 201 (2019), 13–19. +54 + +References +A. Carlotto, M. B. Schulz, D. Wiygul +[37] M. E. Taylor, Partial Differential Equations I, Applied Mathematical Sciences, vol. 115, Springer-Verlag, New +York, 1996. Basic theory. +[38] H. Tran, Index characterization for free boundary minimal surfaces, Comm. Anal. Geom. 28 (2020), no. 1, +189–222. +10 January 2023 +Alessandro Carlotto +ETH D-Math, Rämistrasse 101, 8092 Zürich, Switzerland +E-mail address: alessandro.carlotto@math.ethz.ch +Università di Trento, Dipartimento di Matematica, via Sommarive 14, 38123 Povo di Trento, Italy +E-mail address: alessandro.carlotto@unitn.it +Mario B. Schulz +University of Münster, Mathematisches Institut, Einsteinstrasse 62, 48149 Münster, Germany +E-mail address: mario.schulz@uni-muenster.de +David Wiygul +ETH D-Math, Rämistrasse 101, 8092 Zürich, Switzerland +E-mail address: david.wiygul@math.ethz.ch +55 + diff --git a/1NE1T4oBgHgl3EQfRgP7/content/tmp_files/load_file.txt b/1NE1T4oBgHgl3EQfRgP7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..77430cf2d59405750d72817d58cb65c46d669521 --- /dev/null +++ b/1NE1T4oBgHgl3EQfRgP7/content/tmp_files/load_file.txt @@ -0,0 +1,1794 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf,len=1793 +page_content='Spectral estimates for free boundary minimal surfaces via Montiel–Ros partitioning methods Alessandro Carlotto, Mario B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, David Wiygul Abstract We adapt and extend the Montiel–Ros methodology to compact manifolds with boundary, allowing for mixed (including oblique) boundary conditions and also accounting for the action of a finite group G together with an additional twisting homomorphism σ: G → O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We then apply this machinery in order to obtain quantitative lower and upper bounds on the growth rate of the Morse index of free boundary minimal surfaces with respect to the topological data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' the genus and the number of boundary components) of the surfaces in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In particular, we compute the exact values of the equivariant Morse index and nullity for two infinite families of examples, with respect to their maximal symmetry groups, and thereby derive explicit two-sided linear bounds when the equivariance constraint is lifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 1 Introduction Despite a profusion of constructions of free boundary minimal surfaces in the Euclidean unit ball B3 over the course of the past decade ([14–16,24,31] via optimization of the first Steklov eigenvalue, [4,25,26] via min-max methods for the area functional, and [6,11,18–20,22] via gluing methods), many basic questions about the space of such surfaces remain open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The reader is referred to [12, 13, 27] for recent overviews of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In particular, so far it is only for the rotationally symmetric examples, planar discs through the origin and critical catenoids, that the exact value of the Morse index is actually known (see [8,36,38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The present manuscript is the first in a series of works aimed at shedding new light on this fundamental invariant, which (also due to its variational content, and thus to its natural connection with min-max theory, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' [28–30] and references therein) has acquired great importance within geometric analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Partly motivated by the corresponding conjectures concerning closed minimal hypersurfaces in manifolds of positive Ricci curvature (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' [1, 33]), five years ago the first-named author proved with Ambrozio and Sharp a universal lower bound for the index of any free boundary minimal surface in any mean-convex subdomain Ω of R3 in terms of the topological data of the surface under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Specifically, it was shown in [2] that the following estimate holds: index(Σ) ≥ 1 3(2g + b − 1) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1) where Σ is any free boundary minimal surface in Ω, and g, b denote respectively its genus and the number of its boundary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' It is then a natural, and by now well-known question, whether such a lower bound can be complemented by an affine upper bound or whether – instead – it is 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='03055v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='DG] 8 Jan 2023 1 Introduction A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul conceivable to have a superlinear growth rate of the index with respect to g and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In this article we show that there are in fact infinite families of free boundary minimal surfaces in B3 whose index is bounded from above (and below) by explicit affine functions of the topological data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' More broadly, we embed such a result in a network of index estimates that in turn build on a generalization of the fundamental Montiel–Ros methodology – as first presented in [32] – that is of independent interest and wider applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In general terms, we shall be concerned here with proving effective estimates for (part of) the spectrum of Schrödinger-type operators on bounded Lipschitz domains of Riemannian manifolds, combined with mixed boundary conditions, that will be – on disjoint portions of the boundary in question – of Dirichlet or Robin (oblique) type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Summarizing and oversimplifying things to the extreme, the number of eigenvalues of any such operator below a given threshold can be estimated by suitably partitioning the domain into finitely many subdomains, provided one adjoins Dirichlet boundary conditions in the interior boundaries when aiming for lower bounds, and Neumann boundary conditions in the interior boundaries for upper bounds instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We refer the reader to Section 2 for the setup of our problem together with our standing assumptions, and to the first part of Section 3 (specifically to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1, and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='2) for precise statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In fact, often times (yet not always) the partitions mentioned above naturally relate to the underlying symmetries of the problem in question, which is in particular the case for some of the classes of free boundary minimal surfaces in B3 that have so far been constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' With this remark in mind, a peculiar (and, a posteriori, fundamental) feature of our work is the development of the Montiel–Ros methodology in the presence of the action of a group G together with an additional twisting homomorphism σ: G → O(1), in the terms explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' This allows, for instance, to explicitly and transparently study how the Morse index of a given free boundary minimal surface depends on the symmetries one imposes, namely to look at the “functor” (G, σ) → indσ G(T), where T denotes the index (Jacobi) form of the surface in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' As apparent even from the simplest examples we shall discuss, this perspective turns out to be very natural and effective in tackling the geometric problems we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' With this approach, lower bounds are sometimes relatively cheap to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' One way they can derived is from ambient Killing vector fields, once it is shown that the associated (scalar-valued) Jacobi field on the surface under consideration vanishes along the (interior) boundary of any domain of the chosen partition, which in practice amounts to suitably designing the partition and picking the Killing field given the geometry of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We present one simple yet paradigmatic such result in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='2, which concerns free boundary minimal surfaces with pyramidal or prismatic symmetry in B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Instead, upper bounds are often a lot harder to obtain and shall typically rely on finer information than the sole symmetries of the scene one deals with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Said otherwise, one needs to know how (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' by which method) the surface under study has been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We will develop here a detailed analysis of the Morse index of the two families of free boundary minimal surfaces we constructed in our recent, previous work [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Very briefly, using gluing methods of essentially PDE–theoretic character, we obtained there a sequence Σ−K0∪B2∪K0 m of surfaces having genus m, three boundary components and antiprismatic symmetry group Am+1, and a sequence Ξ−K0∪K0 n of surfaces having genus zero, n + 2 boundary components and prismatic symmetry group Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' As we described at length in Section 7 therein, with data (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Table 2 and Table 3) and heuristics, numerical simulations for the Morse index of the surfaces in the former sequence display a seemingly “erratic” behaviour, as such values do not align on the graph of any affine function, nor seem to exhibit any obvious periodic pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' This is a rather unexpected behaviour (by comparison 2 1 Introduction A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' with other families of examples, say in the round three-dimensional sphere, see [21]), which obviously calls for a careful study that we carry through in Section 5 of the present article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In particular, we establish the following statement: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1 (Index estimates for Σ−K0∪B2∪K0 m and Ξ−K0∪K0 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' There exist m0, n0 > 0 such that for all integers m > m0 and n > n0 the Morse index and nullity of the free boundary minimal surfaces Σ−K0∪B2∪K0 m , Ξ−K0∪K0 n ⊂ B3 satisfy the bounds 2m + 1 ≤ ind(Σ−K0∪B2∪K0 m ), ind(Σ−K0∪B2∪K0 m ) + nul(Σ−K0∪B2∪K0 m ) ≤ 10m + 10, 2n + 2 ≤ ind(Ξ−K0∪K0 n ), ind(Ξ−K0∪K0 n ) + nul(Ξ−K0∪K0 n ) ≤ 8n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' To the best of our knowledge, this is the very first upper bound obtained for the Morse index of a sequence of free boundary minimal surfaces in the Euclidean unit ball B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In fact, the upper bound in this “absolute estimate” follows quite easily by combining the “relative estimate”, associated to the equivariant Morse index of these surfaces (with respect to their respective maximal symmetry groups) with the aforementioned Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The next statement thus pertains to such equivariant bounds, for which we do obtain equality, thus settling part of Conjecture 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='7 (iv) and Conjecture 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='9 (iv) of [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We stress that neither family is constructed variationally, and thus there is actually no cheap index bound one can extract from the design methodology itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' on the contrary, this statement indicates a posteriori that the families of surfaces in question may in principle be constructed (even in a non-asymptotic regime) by means of min-max schemes generated by 2-parameter sweepouts, modulo the well-known problem of fully controlling the topology in the process (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='2 (Equivariant index and nullity of Σ−K0∪B2∪K0 m and Ξ−K0∪K0 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' There exist m0, n0 > 0 such that for all integers m > m0 and n > n0 the equivariant Morse index and nullity of the free boundary minimal surfaces Σ−K0∪B2∪K0 m , Ξ−K0∪K0 n ⊂ B3 satisfy indAm+1(Σ−K0∪B2∪K0 m ) = 2, nulAm+1(Σ−K0∪B2∪K0 m ) = 0, indPn(Ξ−K0∪K0 n ) = 2, nulPn(Ξ−K0∪K0 n ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The main idea behind the proof of these results, or – more precisely – for the upper bounds can only be explained by recalling, in a few words, how the surfaces in question have been constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Following the general methodology of [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' one first considers a singular configuration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' that is a formal union of minimal surfaces in B3 (not necessarily free boundary),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' then its regularization – which needs the use of (wrapped) periodic minimal surfaces in R3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' to desingularize near the divisors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' and controlled interpolation processes between the building blocks in play – and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' thirdly and finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' the perturbation of such configurations to exact minimality (at least for some values of the parameters),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' while also ensuring proper embeddedness and accommodating the free boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Here we first get a complete understanding of the index and nullities of the building blocks, for the concrete cases under consideration in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In somewhat more detail, the analysis of the Karcher–Scherk towers (the periodic building blocks employed in either construction) exploits, in a substantial fashion, the use of the Gauss map, which allows one to rephrase the initial geometric question into as one for the spectrum of simple elliptic operators of the form ∆gS2 + 2 on suitable (typically singular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' spherical triangles, wedges or lunes) subdomains of round S2, 3 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul with mixed boundary conditions, and possibly subject to additional symmetry requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The analysis of the other building blocks – disks and asymmetric catenoidal annuli – is more direct, although, in the latter case, trickier than it may first look (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Once that preliminary analysis is done, we then prove that, corresponding to the (local) geometric convergence results (that are implied by the very gluing methodology) there are robust spectral convergence results that serve our scopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' However, a general challenge in the process is that gluing constructions typically have transition regions where different scales interact with each another: in our constructions of the sequences Σ−K0∪B2∪K0 m and Ξ−K0∪K0 n such regions occur between the catenoidal annuli K0 (as well as the disk B2 in the former case) and the wrapped Karcher– Scherk towers, roughly at distances between m−1 and m−1/2 (respectively n−1 and n−1/2) from the equatorial S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' As a result, we need to deal with delicate scale-picking arguments, an ad hoc study of the geometry of such regions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='21) and – most importantly – prove the corresponding uniform bounds for eigenvalues and eigenfunctions (collected in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='25), which allow to rule out pathologic concentration phenomena, thereby leading to the desired conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The authors wish to express their sincere gratitude to Giada Franz for a number of conversations on themes related to those object of the present manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 947923).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The research of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics– Geometry–Structure, and the Collaborative Research Centre CRC 1442, Geometry: Deformations and Rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Part of this article was finalized while A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' was visiting the ETH-FIM, whose support and excellent working conditions are gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 2 Notation and standing assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1 Boundary value problems for Schrödinger operators on Lipschitz domains Let Ω be a Lipschitz domain of a smooth, compact d-dimensional manifold M with (possibly empty) boundary ∂M, by which we mean here a nonempty, open subset of M whose boundary is everywhere locally representable as the graph of a Lipschitz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We do not require – at least in general – Ω to be connected, and we admit the case Ω = M (where Ω denotes the closure of Ω in M), when of course ∂Ω = ∂M, the boundary of the ambient manifold in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Throughout this article we will in fact assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We are going to study the spectrum of a given Schrödinger operator on Ω subject to boundary conditions and, sometimes, symmetry constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Such symmetry constraints will be encoded in terms of equivariance with respect to a certain group action, which we shall specify at due place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The Schrödinger operator ∆g + q is determined by the data of a given smooth Riemannian metric g on Ω and a given smooth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' C∞) function q: Ω → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' To avoid ambiguities, we remark here that a function (or tensor field) on 4 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul Ω smooth if it is the restriction of a smooth tensor field on M or – equivalently – on a relatively open set containing Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The boundary conditions are specified by another smooth function r: Ω → R and a decomposition ∂Ω = ∂DΩ ∪ ∂NΩ ∪ ∂RΩ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1) where the sets on the right-hand side are the closures of pairwise disjoint open subsets ∂DΩ, ∂NΩ, and ∂RΩ of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Somewhat more specifically, we will consider the spectrum of the operator ∆g + q subject to the Dirichlet, Neumann, and Robin conditions � � � � � � � u = 0 on ∂DΩ, du(ηΩ g ) = 0 on ∂NΩ, du(ηΩ g ) = ru on ∂RΩ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='2) where ηΩ g is the almost-everywhere defined outward unit normal induced by g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' It is obviously the case that the Neumann boundary conditions can be regarded as a special case of their inhomogenous counterpart, however it is convenient – somewhat artificially – to distinguish them in view of the later applications we have in mind, to the study of the Morse index of free boundary minimal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='2 Sobolev spaces and traces To pose the problem precisely we introduce the Sobolev space H1(Ω, g) consisting of all real-valued functions in L2(Ω, g) which have a weak g-gradient whose pointwise g-norm is also in L2(Ω, g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' then H1(Ω, g) is a Hilbert space equipped with the inner product ⟨u, v⟩H1(Ω,g) := � Ω �uv + g(∇gu, ∇gv) � dH d(g), integrating with respect to the d-dimensional Hausdorff measure induced by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (We say a function u ∈ L1 loc(Ω, g) has a weak g-gradient ∇gu if ∇gu is a measurable vector field on Ω with pointwise g norm in L1 loc(Ω, g) and � Ω g(X, ∇gu) dH d(g) = − � Ω u divg X dH d(g) for every smooth vector field X on Ω of relatively compact support, where divg X is the g divergence of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' ∇gu is uniquely defined whenever it exists, modulo vector fields vanishing almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=') Under our assumptions on ∂Ω we have a bounded trace map H1(Ω, g) → L2(∂Ω, g), extending the restriction map C1(Ω) → C0(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (The Hilbert space L2(∂Ω, g) is defined using either the (d−1)-dimensional Hausdorff measure H d−1(g) induced by g or, equivalently, the almost-everywhere defined volume density induced by g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=') In fact, we have not only boundedness of this map but also the stronger inequality ∥u|∂Ω∥L2(∂Ω,g) ≤ C(Ω, g) � ϵ∥u∥H1(Ω,g) + C(ϵ)∥u∥L2(Ω,g) � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='3) for all u ∈ H1(Ω, g), all ϵ > 0, some C(Ω, g) independent of u and ϵ, and some C(ϵ) independent of u and (Ω, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (This can be deduced, for example, by inspecting the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='6 in [9]: 5 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul specifically, we can apply the Cauchy–Schwarz inequality (weighting with ϵ, as standard) to the inequality immediately above the line labeled (⋆ ⋆ ⋆) on page 158 of the preceding reference, whose treatment of Lipschitz domains in Euclidean space is readily adapted to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=') For each C ∈ {D, N, R}, indicating one of the boundary conditions we wish to impose, by composing the preceding trace map with the restriction L2(∂Ω, g) → L2(∂CΩ, g) , since ∂CΩ is open in ∂Ω, we also get a trace map ·|∂C : H1(Ω, g) → L2(∂CΩ, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In practice we will consider traces on just ∂DΩ and ∂RΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Considering the condition on ∂DΩ we will then define H1 ∂DΩ(Ω, g) := {u ∈ H1(Ω, g) : u|∂DΩ = 0}, that is obviously to be understood in the sense of traces, in the terms we just described, and we remark that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='3) also clearly holds with ∂Ω on the left-hand side replaced by ∂RΩ (or by ∂DΩ or ∂NΩ, but we have no need of the inequality in these cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='3 Bilinear forms and their eigenvalues and eigenspaces Corresponding to the above data we define the bilinear form T = T[Ω, g, q, r, ∂DΩ, ∂NΩ, ∂RΩ] by T : H1 ∂DΩ(Ω, g) × H1 ∂DΩ(Ω, g) → R (u, v) �→ � Ω � g(∇gu, ∇gv) − quv � dH d(g) − � ∂RΩ ruv dH d−1(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='4) Then T is symmetric, bounded, and coercive as encoded in the following three equations respectively: ∀u, v ∈ H1 ∂DΩ(Ω, g) T(u, v) = T(v, u), ∀u ∈ H1 ∂DΩ(Ω, g) T(u, u) ≤ �1 + C(Ω, g, q, r) �∥u∥2 H1(Ω,g), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='5) ∀u ∈ H1 ∂DΩ(Ω, g) T(u, u) ≥ 1 2∥u∥2 H1(Ω,g) − C(Ω, g, q, r)∥u∥2 L2(Ω,g), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='6) where, for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='6), one can take C(Ω, g, q, r) = ∥q∥C0(Ω) + C(Ω, g)∥r∥C0(∂RΩ), thanks to the trace inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' From these three properties and the Riesz representation theorem for Hilbert spaces it follows that for some constant Λ = Λ(Ω, g, q, r) > 0 there exists a linear map R: L2(Ω, g) → H1 ∂DΩ(Ω, g) such that T(Rf, v) + Λ⟨Rf, ιv⟩L2(Ω,g) = ⟨f, ιv⟩L2(Ω,g) for all functions f ∈ L2(Ω, g) and v ∈ H1 ∂DΩ(Ω, g), where we have introduced the inclusion map ι: H1 ∂DΩ(Ω, g) → L2(Ω, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (Of course, if f is smooth then standard elliptic interior regularity results ensures that u is as well smooth on Ω and there satisfies the equation −(∆g + q − Λ)u = f in a classical pointwise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=') Since the inclusion H1(Ω, g) �→ L2(Ω, g) is compact (see for example Section 7 of Chapter 4 of [37]) and of course the inclusion of the closed subspace H1 ∂DΩ(Ω, g) �→ H1(Ω, g) is bounded, the aforementioned maps ι: H1 ∂DΩ(Ω, g) → L2(Ω, g) and the composite ιR: L2(Ω, g) → L2(Ω, g) are also both compact operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Furthermore, to confirm that ιR is symmetric we simply note that (by appealing to the equation defining the operator R, with Rf1 and Rf2 in place of v) ⟨f2, ιRf1⟩L2(Ω,g) = T(Rf2, Rf1) + Λ⟨ιRf2, ιRf1⟩L2(Ω,g) = T(Rf1, Rf2) + Λ⟨ιRf1, ιRf2⟩L2(Ω,g) = ⟨f1, ιRf2⟩L2(Ω,g) 6 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul for all f1, f2 ∈ L2(Ω, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' That being clarified, to improve readability we will from now on refrain from explicitly indicating the inclusion map ι in our equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' With slight abuse of language, in the setting above we call λ ∈ R an eigenvalue of T if there exists a nonzero u ∈ H1 ∂DΩ(Ω, g) such that ∀v ∈ H1 ∂DΩ(Ω, g) T(u, v) = λ⟨u, v⟩L2(Ω,g), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='7) and we call any such u an eigenfunction of T with eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (We caution that the notions of eigenfunctions and eigenvalues depend not only on T but also on the underlying metric g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' for the sake of convenience we choose to suppress the latter dependence from our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=') Hence, as a consequence of the key facts we presented before this definition, one can prove by well- known arguments the existence of a discrete spectrum for the “shifted” elliptic operator (∆g +q)−Λ subject to the very same boundary conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' As a straightforward corollary, by accounting for the shift, we obtain the following conclusions for T: the set of eigenvalues of T is discrete in R and bounded below, for each eigenvalue of T the corresponding eigenspace has finite dimension, there exists an Hilbertian basis {ej}∞ j=1 for L2(Ω, g) consisting of eigenfunctions of T, and {ej}∞ j=1 has dense span in H1 ∂DΩ(Ω, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (To avoid ambiguities, we remark that the phrase Hilbertian basis refers to a countable, complete orthonormal system for the Hilbert space in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=') For each integer i ≥ 1 we write λi (T) for the ith eigenvalue of T (listed with repetitions in nondecreasing order, in the usual fashion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' There holds the usual min-max characterization λi (T) = min � max � T(w, w) ∥w∥2 L2(Ω,g) : 0 ̸= w ∈ W � : W ⊂ subspace H1 ∂DΩ(Ω, g), dim W = i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='8) Next, for any t ∈ R we let E=t(T) denote the (possibly trivial) linear span, in H1 ∂DΩ(Ω, g), of the eigenfunctions of T with eigenvalue t, and, more generally, for any t ∈ R and any binary relation ∼ on R (in practice <, ≤, >, ≥, or =) we set E∼t(T) := ClosureL2(Ω,g) � Span �� s∼t E=s(T) �� and we denote the corresponding orthogonal projection by π∼t T : L2(Ω, g) → E∼t(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' That is, the space E∼t(T) has been defined to be the closure in L2(Ω, g) of the span of all eigenfunctions of T having eigenvalue λ such that λ ∼ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Of course E∼t(T) is a subspace of H1 ∂DΩ(Ω, g) – in particular – whenever the former has finite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Taking ∼ to be equality clearly reproduces the originally defined space E=t(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' For future use observe that the above spectral theorem for T implies (E∼t(T))⊥L2(Ω,g) = E̸∼t(T), E, ≥, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='9) 7 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul and ∀u ∈ H1 ∂DΩ(Ω, g) ∩ � E≤t(T) ∪ E≥t(T) � T(u, u) = t∥u∥2 L2(Ω,g) ⇒ u ∈ E=t(T), throughout which t is any real number (not necessarily an eigenvalue of T) and where in the first equality of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='9) ∼ is any relation on R and ̸∼ its negation (so that {s ̸∼ t} = R \\ {s ∼ t} for any t ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Index and nullity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In the setting above, and under the corresponding standing assumption, we shall define the non-negative integers ind(T) := dim E<0(T) and nul(T) := dim E=0(T), called, respectively, the index and nullity of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Such invariants will be of primary interest in our applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='4 Group actions Let G be a finite group of smooth diffeomorphisms of M, each restricting to a surjective isometry of (Ω, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Then, as for any group of diffeomorphism of Ω, we have the standard (left) action of G on functions on Ω via pullback: (φ, u) �→ u ◦ φ−1 = φ−1∗u for all φ ∈ G, u: Ω → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We say that a function u is G-invariant if it is invariant under this action: equivalently u ◦ φ = u for all φ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We can also twist this action by orthogonal transformations on the fiber R: given in addition to G a group homomorphism σ: G → O(1) = {−1, 1}, we define the action (φ, u) �→ σ(φ)(u ◦ φ−1) = σ(φ)φ−1∗u for all φ ∈ G, u: Ω → R, and we call a function (G, σ)-invariant if it is invariant under this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Obviously the above standard action (φ, u) �→ u ◦ φ−1 is recovered by taking the trivial homomorphism σ ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We also comment that one could of course replace R by C and correspondingly O(1) by U(1) (and in the preceding sections instead work with Sobolev spaces over C) though we restrict attention to real-valued functions in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Since, by virtue of our initial requirement, G is a group of isometries of (Ω, g), the above twisted action yields a unitary representation of G in L2(Ω, g), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' a group homomorphism �σ: G → O �L2(Ω, g) � φ �→ σ(φ)φ−1∗ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='10) whose target are the global isometries of L2(Ω, g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' we note that the same conclusions hold true with H1(Ω, g) in place of L2(Ω, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The corresponding subspaces of (G, σ)-invariant functions, in L2(Ω, g) 8 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul or H1(Ω, g), are readily checked to be closed, and thus Hilbert spaces themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' That said, we define the orthogonal projection πG,σ : L2(Ω, g) → L2(Ω, g) u �→ 1 |G| � φ∈G �σ(φ)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='11) Here |G| is the order of G, which – we recall – is assumed throughout to be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The image of L2(Ω, g) under πG,σ thus consists of (G, σ)-invariant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' One could lift the finiteness assumption, say by allowing G to be a compact Lie group, requiring σ to be continuous, and replacing the finite average in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='11) with the average over G with respect to its Haar measure (which reduces to the former for finite G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' However, with a view towards our later applications, in this article we will content ourselves with the finiteness assumption, which allows for a lighter exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Henceforth we make the additional assumptions that G globally (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' as sets) preserves each of ∂DΩ, ∂NΩ, and ∂RΩ, and that q and r are both G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Each element of �σ(G) then preserves also H1 ∂DΩ(Ω, g) and the bilinear form T, and the projection πG,σ commutes with the projection π∼t T , for any t ∈ R and binary relation ∼ on R (as above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In particular πG,σ preserves each eigenspace E=t(T) of T, and more generally the space E∼t G,σ(T) := πG,σ(E∼t(T)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='12) is a subspace of E∼t(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' For each integer i ≥ 1 we can then define λG,σ i (T), the ith (G, σ)-eigenvalue of T, to be the ith eigenvalue of T having a (G, σ)-invariant eigenfunction (by definition nonzero), counting with multiplicity as before;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' equivalently one can work with spaces of (G, σ)-invariant functions and derive the analogous conclusions as in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='3 directly in that setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We explicitly note, for the sake of completeness, that under no additional assumptions on the group G and the homomorphism σ it is possible that the space of (G, σ)-invariant functions be finite dimensional (possibly even of dimension zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' This type of phenomenon happens, for instance, when every point of the manifold M is a fixed point of a σ-odd isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In this case, all conclusions listed above still hold true, but need to be understood with a bit of care: the corresponding sequence of eigenvalues λG,σ 1 (T) ≤ λG,σ 2 (T) ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' will in fact just be a finite sequence, consisting say of I(G, σ) elements, counted with multiplicity as usual;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' we shall formally convene that λG,σ i (T) = +∞ for i > I(G, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' That being said, we also remark that this phenomenon patently does not occur for the Jacobi form of the two sequences of free boundary minimal surfaces we examine in Sections 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In this equivariant framework we still have the corresponding min-max characterization λG,σ i (T) = min � max � T(w, w) ∥w∥2 L2(Ω,g) : 0 ̸= w ∈ W � : W ⊂ subspace πG,σ �H1 ∂DΩ(Ω, g) �, dim W = i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='13) 9 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul We also define the (G, σ)-index and (G, σ)-nullity indσ G(T) := dim E<0 G,σ(T) and nulσ G(T) := dim E=0 G,σ(T) of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Obviously we can recover E∼t(T), λi (T), and the standard index and nullity by taking G to be the trivial group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' As mentioned in the introduction, we reiterate that it is one of the goals of the present article to study, for fixed g and T, how these numbers (index indσ G(T) and nullity nulσ G(T)) depend on G and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' For the sake of brevity, we shall employ the phrase admissible data to denote any tuple (Ω, g, q, r, ∂DΩ, ∂NΩ, ∂RΩ, G, σ) satisfying all the standing assumptions presented up to now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We digress briefly to highlight two important special cases, which warrant additional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='3 (Actions of order-2 groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' When |G| = 2, there are precisely two homomorphisms G → O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Considering such homomorphisms, and the corresponding (G, σ)-invariant functions, we may define G-even or G-odd functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Hence, we may call ind+ G and ind− G the G-even and G-odd index, and likewise for the nullity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Clearly, we always have � � � ind(T) = ind+ G(T) + ind− G(T), nul(T) = nul+ G(T) + nul− G(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='14) Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='4 (Actions of self-congruences of two-sided hypersurfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Suppose, momentarily, that (M, g) is isometrically embedded (as a codimension-one submanifold) in a Riemannian manifold (N, h), that the set Ω be connected and assume further that the normal bundle of M over Ω is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Then we can pick a unit normal ν on Ω and thereby identify – as usual – sections of the normal bundle of M|Ω with functions on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' With this interpretation of functions on Ω in mind and G now a finite group of diffeomorphisms of N that map Ω onto itself (as a set), and everywhere on Ω preserve the ambient metric h meaning that φ∗h = h for any φ ∈ G, we have a natural action given by (φ, u) �→ sgnν(φ)(u ◦ φ−1) for all φ ∈ G, u: Ω → R, where sgnν(φ) := h(φ∗ν, ν) is a constant in O(1) = {1, −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We shall further assume that the action of G on Ω is faithful, meaning that only the identity element fixes Ω pointly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' this assumption is always satisfied in our applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In this context we continue to say that a function u: Ω → R is G-invariant if u = u ◦ φ for all φ ∈ G, and we say rather that u is G-equivariant if u = sgnν(φ)u ◦ φ for all φ ∈ G (that is, noting the identity sgnν(φ) = sgnν(φ−1), provided u is invariant under the sgnν-twisted G action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Similarly, in this context, we set indG(T) := indsgnν G (T) and nulG T := nulsgnν G (T), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='15) which we may refer to as simply the G-equivariant index and G-equivariant nullity of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We point out that we are abusing notation in the above definitions in that, on the right-hand side of each, in place of G we mean really the group, isomorphic to G by virtue of the faithfulness assumption, obtained by restricting each element of G to Ω, and in place of sgnν we mean really the corresponding homomorphism, well-defined by the faithfulness assumption, on this last group of isometries of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We now return to the more general assumptions on G preceding this paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 10 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='5 Subdomains Suppose that Ω1 ⊂ Ω is another Lipschitz domain of M (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We shall define ∂intΩ1 := ∂Ω1 ∩ Ω, ∂extΩ1 := ∂Ω1 \\ ∂intΩ1, ∂Dint D Ω1 := (∂extΩ1 ∩ ∂DΩ) ∪ ∂intΩ1, ∂Nint D Ω1 := ∂extΩ1 ∩ ∂DΩ, ∂Dint N Ω1 := ∂extΩ1 ∩ ∂NΩ, ∂Nint N Ω1 := (∂extΩ1 ∩ ∂NΩ) ∪ ∂intΩ1, ∂Dint R Ω1 := ∂extΩ1 ∩ ∂RΩ, ∂Nint R Ω1 := ∂extΩ1 ∩ ∂RΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='16) In this way we prepare to pose two different sets of boundary conditions on Ω1, whereby, roughly speaking, in both cases ∂Ω1 inherits whatever boundary condition is in effect on ∂Ω wherever the two meet (corresponding to ∂extΩ1) and the two sets of conditions are distinguished by placing either the Dirichlet or the Neumann condition on the remainder of the boundary (corresponding to ∂intΩ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Naturally associated to these two sets of conditions are the bilinear forms T Dint Ω1 := T[Ω1, g, q, r, ∂Dint D Ω1, ∂Dint N Ω1, ∂Dint R Ω1], T Nint Ω1 := T[Ω1, g, q, r, ∂Nint D Ω1, ∂Nint N Ω1, ∂Nint R Ω1], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='17) defined, respectively, on the Sobolev spaces H1 ∂Dint D Ω1(Ω1, g) and H1 ∂Nint D Ω1(Ω1, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Recalling (G, σ) from above, with the tacit understanding that (Ω, g, q, r, ∂DΩ, ∂NΩ, ∂RΩ, G, σ) is admissible, we further assume that each element of G maps Ω1 onto itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' since G preserves Ω and respects the decomposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1), it follows that it also respects the decompositions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Somewhat abusively, we shall write �σ and πG,σ not only for the maps (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='11) but also for their counterparts with Ω replaced by Ω1, which are well-defined under our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The spaces E∼t G,σ(T Dint Ω1 ) and E∼t G,σ(T Nint Ω1 ) as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='12), are then also well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' ∂RΩ ∂DΩ ∂DΩ ∂NΩ ∂NΩ Ω1 ∂extΩ1 ∩ ∂NΩ ∂extΩ1 ∩ ∂DΩ ∂extΩ1 ∩ ∂RΩ ∂intΩ1 Figure 1: Example of a Lipschitz domain Ω with subdomain Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 11 3 Fundamental tools A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul 3 Fundamental tools 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1 Index and nullity bounds in the style of Montiel and Ros Recalling the notation and assumptions of Section 2, suppose now that we have not only Ω1 ⊂ Ω as above, but also (open) Lipschitz subdomains Ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' , Ωn ⊂ Ω which are pairwise disjoint, each of which satisfies the same assumptions as Ω1 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='5, and whose closures cover Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In particular, we assume that each element of the group G maps each subdomain Ωi onto itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We assume further that G acts transitively on the connected components of Ω and note that this last condition is always satisfied in the important special case that Ω is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1 (Montiel–Ros bounds on the number of eigenvalues below a threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' With assumptions as in the preceding paragraph and notation as in Section 2, the following inequalities hold for any t ∈ R (i) dim E 0.05, age range: 88-100 +months). All participants were righthanded native +Spanish speakers with normal or corrected-to-normal +vision; none had a hearing impairment. All participants +in the dyslexic group had been formally diagnosed with +dyslexia at school. All participants in the skilled reader +group were free from reading and writing difficulties and +had not been formally diagnosed with dyslexia. The +participants’ +legal +guardians +expressed +their +understanding of the study, gave their written consent, +and were present throughout the experiment. +All participants experienced an auditory stimulus in 15- +minute sessions. The stimulus, which was modulated at + +Rodríguez-Rodríguez et al. + +4 +4.8 Hz (prosodic-syllabic frequency) in 2.5-minute +segments, was band-limited white noise. This type of +stimulus was chosen to identify what synchronicity +patterns the low-level auditory processing would induce +and on the basis of the expert knowledge of linguistic +psychologists +concerning +the +main +frequency +components representing words in the human voice. The +participants’ EEG signals were recorded with a +BrainVision actiCHamp Plus with 32 active electrodes +(actiCAP, Brain Products GmbH, Germany) at a 500 Hz +sampling rate. The 10–20 standardized system was used +to place the 32 electrodes. +3.2. Preprocessing +The preprocessing involved removing all eye-blinking +and movement/impedance variation artifacts from the +EEG signals. The former were eliminated via +independent component analysis (ICA) 35 based on the +eye movements observed in the EOG channel, while for +the latter the relevant EEG segments were excluded. The +channels were then referenced to the Cz channel. +Then, a band-pass filter was applied to the EEG channels +to collect information for the five EEG frequency bands +(Delta, 1.5–4 Hz; Theta, 4–8 Hz, Alpha, 8–13 Hz; Beta, +13–30 Hz; and Gamma, 30–80 Hz). We used finite +impulse response (FIR) filters because these ensure a +constant phase lag that can later be corrected. To be +specific, each signal was sent forward and backward +through the two-way zero-phase lag band-pass FIR least- +squares filter, producing a zero-lag phase in the overall +filtering process that addressed the issue of phase lag 36. +As low-pass filtering with an 80 Hz threshold was +employed, we added a 50 Hz notch filter during +preprocessing to eliminate this frequency component. +3.3. Hilbert Transform +A Hilbert transform (HT) transforms real signals into +analytic signals, i.e. complex-valued time series without +negative frequency components, allowing the time- +varying amplitude, phase and frequency, i.e., the +instantaneous amplitude, phase and frequency, to be +calculated from the analytic signal. + +We define HT for a signal x(t) as: + +ℋ[𝑥(𝑡)] = 1 +𝜋 ∫ +𝑥(𝑡) +𝑡 − 𝜏 𝑑𝜏 ++∞ +−∞ + +(1) + +and we obtain the analytic signal zi(t) for signal x(t) as: + +𝑧𝑖(𝑡) = 𝑥𝑖(𝑡) + 𝑗ℋ{𝑥𝑖(𝑡)} = 𝑎(𝑡)𝑒𝑗𝜙(𝑡) +(2) + +From zi(t), computing the instantaneous amplitude is +straightforward: + +𝑎(𝑡) = √𝑟𝑒(𝑧𝑖(𝑡))2 + 𝑖𝑚(𝑧𝑖(𝑡))2 +(3) + +with the instantaneous, unwrapped phase as: + +𝜙(𝑡) = 𝑡𝑎𝑛−1 𝑖𝑚(𝑧𝑖(𝑡)) +𝑟𝑒(𝑧𝑖(𝑡)) +(4) + +The above technique gives the phase value for each time +point, allowing the inter-channel synchronization to be +estimated based on the phase variation. +3.4. Granger Causality test +Developed for the field of econometrics by Clive +Granger, +Granger +causality +37 +describes +causal +interactions occurring between continuous-valued time +series. As a statistical hypothesis test, it essentially states +that “the past and present may cause the future, but the +future cannot cause the past”; hence, knowing a cause +will be more helpful in predicting future effects than an +auto-regression will. Specifically, variable x will +Granger-cause y if the auto-regression for y that uses past +values of x and y is significantly more accurate than one +using only past values of y. We may exemplify this by +taking two stationary time-series sequences, xt and yt, +whereby xt−k and yt−k are, respectively, the past k values +of xt and yt. We then use two regressions to perform +Granger causality: + +𝑦𝑡̂ 1 = ∑ 𝑎𝑘 +𝑙 +𝑘=1 +𝑦𝑡−𝑘 + 𝜀𝑡 +(5) + +𝑦𝑡̂ 2 = ∑ 𝑎𝑘 +𝑙 +𝑘=1 +𝑦𝑡−𝑘 + ∑ 𝑏𝑘 +𝑤 +𝑘=1 +𝑥𝑡−𝑘 + 𝜂𝑡 +(6) + +where 𝑦𝑡̂ 1 and 𝑦𝑡̂ 2 are, respectively, the fitting values of +the first and second regressions; l and w are the maximum +numbers of the lagged observations of xt and yt; ak; bk ∈ +R are the regression coefficient vectors estimated using +least squares; and εt and ηt are white noise (prediction +errors). Note that even though w can be infinite, due to +the finite nature of our data, we consider w finite and give +it a length well below the time series length, estimated +using model selection, such as the Akaike information +criterion (AIC) 38. Next, an F-test is applied to give a p- +value indicating whether the regression model produced + + +Neural source/sink phase connectivity in Developmental Dyslexia + +5 +by Eq. (5) is statistically better than that of Eq. (6). If it +is, then x Granger-causes y. +We perform Granger causality testing for each +participant and evaluate the channels’ interactions, +producing an n x n square matrix of p-values (n = number +of channels). +Using Granger causality to analyze the neural network’s +directed functional connectivity intuitively demonstrates +the directionality with which information is transmitted +between neurons or brain regions. Previous studies have +already applied this technique to EEG analysis with great +success 39 40. +3.5. Connectivity vectors +The field of neuroscience tends to consider the brain as a +network using functional information +41 +42 +43, +culminating in the so-called connectome. This refers to +the complete mapping of all connections between brain +regions as an adjacency matrix, and often includes the +covariance, as well as other metrics, between fMRI +signals measured for different regions. Several studies +have also examined the temporal covariance between +EEG electrodes. +Once we had assembled the Granger causality matrices +for each participant subject, we established a threshold +value that evidenced a causal relationship between the +channels. Then, we formulated the three scenarios used +to produce each participant’s feature set: +• Sources: Array of n x 1 elements; each element +relates each channel with the number of channels that +it influences. +• Sinks: Array of n x 1 elements; each element relates +each channel with the number of channels that it is +influenced by. +• Total activity: Array of n x 1 elements; the sum of +the two previous scenarios, acting as a reference for +each channel’s global activity. +By organizing the information thus, we receive the same +number of features as there are channels for each +participant, each with a number that indicates its activity +as a source, as a sink, or the total. A summary of this +process is presented in Figure 1. + + +Fig. 1. Assembling the source and sink connectivity arrays for +a participant, given the relevant Granger matrix. [P(k)] is an +Iverson bracket function. +3.6. Ensemble feature selection +If the model includes many features, it will be more +complex, potentially leading to data overfitting. +Moreover, some of the features may be noise and could +adversely affect the model. Thus, we removed such +features to ensure the better generalization of the model. +We hereby selected the variables based on majority +voting through the application of several techniques. If a +variable was chosen by an algorithm, it received one +vote. The votes were then summed for each variable, and +those with the most votes were selected. (Fig. 2). This +method has been found to be suitable for datasets that are +high-dimensional yet have few instances 44. The voting +strategy used a variety of feature selection methods 45, as +outlined in the following: +Information value (IV) using weight of evidence +(WOE): This indicates the predictive power of an +independent variable concerning the dependent variable +46. It allows a continuous independent variable to be +transformed into a set of groups or bins based on the +similarity of the dependent variable distribution (i.e. +numbers of events and non-events). Using WOE allows +outliers and missing values to be addressed and +eliminates the need for dummy variables 47: + +𝑊𝑂𝐸 = ln ( +𝐸𝑣𝑒𝑛𝑡% +𝑁𝑜𝑛 𝐸𝑣𝑒𝑛𝑡%) +(7) + +𝐼𝑉 = Σ[(𝐸𝑣𝑒𝑛𝑡% − 𝑁𝑜𝑛 𝐸𝑣𝑒𝑛𝑡%) ∗ 𝑊𝑂𝐸] +(8) + +An IV statistic above 0.3 is held to indicate a strong +relationship between the predictor and the event/non- +event odds ratio 48. + +FP1 +FP2 +F7 +PO10 +Sinks = +FP1 +1 +pvalue<0.01 +GrM[FP1,k], +[P(R) +FP2 +1 +, GrM[F7,k]. +F7 +pvalue<0.01 +1 +pvalue<0.01 +P(a) +... +PO10 +pvalue<0.01 +1 + GrM[P010, k] +P(k) +, GrM[k, FP2] +Sources = +GrM[k, FP1], +[P(k) +l0otherwiseRodríguez-Rodríguez et al. + +6 +Variable importance using random forest/extra trees +classifier: Calculated using a tree-based estimator, this +can be used to eliminate irrelevant features. Variable +importance is conventionally computed using the mean +decrease in impurity (i.e., gini importance +49) +mechanism, wherein the improvement in the split +criterion for each split of each tree is the importance +measure assigned to the splitting variable. For each +variable, this is separately accumulated over all the trees +in the forest. This measure is similar to the R2 in the +training set regression. +Recursive Feature Elimination: This can be used to +select features by recursively considering feature sets +with diminishing size based on an external estimator (a +linear regression model) that assigns weights to the +features 50. The estimator is trained on the first feature +set, noting each feature’s importance based on a given +attribute. The least important features are subsequently +removed from the current set. The process is performed +recursively on the pruned set until the desired number of +features is achieved. +Chi-square best variables: This uses a chi-square (χ2) +test to assess the correlations among a dataset’s features +and identify multicollinearity. The aim is revealing any +relationships between the dependent variable and any of +the independent variables 51. In the chi-square test, H₀ +(null hypothesis) assumes that two features are +independent, while H₁ (alternative hypothesis) predicts +that they are related. We set a α=0.05 and a p-value of +0.05 or greater is considered critical, anything less means +the deviations are significant hence the hypothesis must +be rejected. +L1-based feature selection: Some features can be +eliminated using a linear model with an L1 penalty. This +method involves regularization, wherein a penalty is +added to various parameters of a machine learning model +to reduce the model’s freedom and prevent overfitting. +When regularizing linear models, the penalty is applied +in addition to the coefficients multiplying the predictors +52. Unlike other forms of regularization, L1 can reduce +some coefficients to zero, meaning the feature is +removed. +Once the best variables had been chosen by voting, we +performed a multicollinearity check on them. + + +Fig. 2. The feature selection procedure for the ‘Sources’ +scenario using a vote-based approach. +3.7. Classification process +In an ensemble method, multiple models are first +generated and then integrated to produce higher-quality +results. The respective predictions are hereby combined +using weighted majority voting to make the final +prediction. At each boosting iteration, the data are +modified by applying w1, w2 , …, wn to each training +sample. As the weights are initially wi=1/N, a weak +learner is trained in the first step using the raw data. At +each successive iteration, the sample weights are +modified individually, and the algorithm is then applied +to the reweighted data. Training examples that are +incorrectly predicted relative to the previous step’s +boosted model are given increased weights; correctly +predicted examples are given decreased weights. As a +result, the examples that were difficult to predict become +increasingly influential as the number of iterations +increases, and the weak learners that follow are forced to +focus on the examples previously missed. +Ensemble methods deliver more accurate results than +single models, and are particularly suitable for improving +binary prediction on small data sets. We use the Gradient +Boosting classifier, as well as an Ada Boost for results +verification. This latter classifier 53 is a meta-estimator +that initially fits to the data, with further copies then being +fit to the same data, while incorrectly classified +instances’ weights are modified to force subsequent +classifiers to focus on them. The Gradient Boosting +classifier 54 creates an additive model based on a forward +stage-wise construction, allowing the optimization of the +arbitrary differentiable loss function. At each stage, n +regression trees are fit to the multinomial or deviance +binomial loss function’s negative gradient, with a single +regression tree being used for the special case of binary +classification. To identify the best parameter set, we +cross-validate with 20 folds and a parameter grid, as +shown in Table 1. + +Method:IVusingWOE-→ +Feature subset: f1, f2, f3,.., fn +All features for Scenario Sources: +Voting +FP1. FP2,F7... PO10 +Method:Var.Imp.using RF→ +Feature subset:f'1, f'2,f3, ...,f'n +M +Method:Var.Imp.using Trees → +Feature subset: f"1, f"2, f"3,., f"n +Method:RFE→Feature subset +Reducedranked feature +subsetbasedonvotes +→fr1,f2,f3,.,frn +Method:Chi Squared→Feature subset +Neural source/sink phase connectivity in Developmental Dyslexia + +7 +Table 1. Parameter grid of machine learning classifiers. +Algorithm +Parameter +Range +Gradient +n_estimators +1 to 12 +Boosting +Loss +deviance, exponential + +Learning rate +0.05 to 1.5 + +Criterion +friedm_mse, sq_error, mse, mae + +Min_samples_split +0.01 to 3 + +Min_samples_leaf +0.01 to 3 + + +Max_depth +1 to 4 +Ada Boost +n_estimators +1 to 25 + +Learning rate +1 to 3.5 + +Boosting algorithm +SAMME, SAMME.R +4. Results +Plotting each learner’s array of sources and sinks permits +the visual extraction of the respective patterns of the +dyslexic and control groups. To this end, we examined +the channel distributions for both groups by calculating +the means and dispersions and producing a box-and- +whisker plot. We also constructed a topoplot as this can +illustrate the results with greater clarity. For example, +Fig. 3 shows the Theta band connectivity of the control +and dyslexic groups specifically for total activity. Please +note that Fig. 3 and Fig. 4 do not directly represent the +electrical activity of the cerebral cortex, but rather show +the levels of the cause-effect relationships between the +channels, i.e. in one direction or in the other direction or +in total. It immediately becomes clear that despite the +similarity of the patterns, the dyslexic group has a +significantly higher activity level in the Theta band. + + + + + +Fig. 3. Boxplot of the total activity in the Alpha band. +Fig 4. The equivalent graphical representation of Fig. 3 in a +topoplot. + + + +Fig. 5. Source/sink activity in the Theta, Beta and Gamma bands in the control and dyslexic groups. Numbers represent how many +channels are affected by each channel as a source, or how many channels are affecting each channel as a sink. + +ActivityofsourcesinThetaband +Control +22.5 +Dyslexic +20.0 +nels +7.5 +5.0 +2.5 - +Fp1 Fp2 F7 F3 FZ F4 F8 FC5 FC1 FC2FC6 T7 C3 C4 T8 TP9CP5 CP1CP2CP6IP1O P7 P3 PZ P4 P8 PO9 O1 OZ O2PO10 +ChannelActivityofsources inThetaband +Control group +Dyslexic group +19 +19 +Fp1 +Fp2 +F1 +2 +F8 +IFG +13 +FC6 +FQ1 +FG2 +FG6 +. +13 +TZ +T8 +G3 +18 +TP9CR5 +CP1 +CPEFP.i +CP5 +CBI +Cp2 +CP6 +P3 +RZ +P4 +P8 +P7 +PZ +F4 +P8 +kod +01 +02 +Pig +60 +01 +02 +Poip +QzActivity of sources in Theta band +19 +Control group +Dyslexlc group +19 +13 +13Activity of sources in Beta band +19 +Control group +Dyslexlc group +19 +F8 +13 +13Activityof sources in Gamma band +19 +Control group +Dyslexlc group +19 +13 +13Activity of sinks in Theta band +19 +Control group +Dyslexic group +19 +Fp2 +13 +13Activity of sinks in Beta band +19 +Control group +Dyslexic group +19 +13 +6 +13Activity of sinks in Gamma band +19 +Controlgroup +Dyslexicgroup +19 +Fp. +F4 +13 +T651 +FE6 +13 +4 +9P5 +EPFTotal activity per channel in Theta band +Control group +Dyslexic group +38 +38 +27 +27 +15 +15Total activity per channel in Beta band +Control group +Dyslexic group +38 +38 +27 +27 +CP +CPI +15 +TE +15Total activity per channel in Gamma band +Control group +Dyslexic group +38 +38 +H +27 +Fe6 +27 +4 +15 +15Rodríguez-Rodríguez et al. + +8 +Fig. 5 compares the channel activity in the Theta, Beta +and Gamma bands, and can be viewed separately as +sources, sinks, or total activity for both the control and +dyslexic groups. Please note that the range of +visualization is the same in all sinks/sources topoplots, +while different in the total activity ones, for better +representation. Once more, it is immediately clear that +while the patterns are broadly similar, the activity level is +higher in the dyslexic group, primarily observed in the +sink activity (less in the source activity). Thus, although +the sources, broadly speaking, behave similarly between +the groups, the dyslexic group has significantly more +concentrated sinks and more activity. Consequently, the +overall activity level is also affected. + + +Fig. 6. Feature importance in Theta, Beta and Gamma bands considering sources, sinks and total activity. +With as many arrays as subjects, and with each array +having as many components as channels, we performed +feature selection to identify channels that can help +differentiate between the control and dyslexic groups. +The feature selection procedure outlined above was thus +applied for the cases of sources, sinks and total activity, +according to the band. Fig. 6 presents the results for the +Theta, Beta and Gamma bands, whereby the importance +values are normalized to permit fair and simple +comparison. Channels showing a higher significance are +those with more dissimilarity between the control and + +Feature importance in Theta band +1.0 +Activity of sources +Activity of sinks +Total activity per channel +0.8 +0.2 - +0.0- +Fp1 +Fp2 +F7 +F3 +Fz +F4 +F8 +FC5 +FC1 +FC2 +FC6 +T7 +C3 +C4 +T8 +CP5 +CP1 +CP2 +CP6 TP10 +P3 +Pz +P4 +P8 +PO9 +Q1 +Qz +ZO +PO10 +ChannelFeature importance in Beta band +1.0 +Activity of sources +I Activity of sinks +Total activity per channel +0.8- +Fea +0.2 +0.0- +Fp1 +Fp2 +F7 +F3 +Fz +F4 +F8 +FC5 +FC1 +FC2 +FC6 +T7 +C3 +C4 +T8 +CP5 +CP1 +CP2 +CP6TP10 +P7 +P3 +Zd +P4 +P8 +PO9 +Q1 +Qz +02 +PO10 +ChannelFeatureimportanceinGammaband +1.0 +Activity of sources +Activity of sinks +Total activity per channel +0.8 +0.2 +0.0- +Fp1 +Fp2 +F7 +F3 +Fz +F4 +F8 +FC5 +FC1 +FC2 +FC6 +C3 +C4 +T8 +CP5 +CP1 +CP2 +CP6TP10 +3 +P4 +P8 +PO9 +Q1 +Qz +02PO10 +Channel +Neural source/sink phase connectivity in Developmental Dyslexia + +9 +dyslexic groups, directing us to where we can find +different patterns of functioning. +After performing the feature selection for each band, for +each case (sources, sinks and total activity), we optimize +the Gradient Boosting classifier to obtain the best +performance. The results are summarized in Table 2, with +performances achieving at least 80% marked bold. +According to the results, the greatest differences between +the control and dyslexic groups (i.e., the best classifier +results) emerge in the Theta and Gamma bands when +accounting for the activity sink role of the different +channels, achieving accuracies of 84% and 88%, +respectively. We also wish to highlight the results for the +Beta band for the activity sources regarding the Area +Under the Curve (AUC), in addition to accuracy. +Table 2. Results of the Gradient Boosting machine +learning classifier. +Band +Features set +Accuracy +AUC +Delta +Sources +0.77 ± 0.14 +0.65 ± 0.31 + +Sinks +0.79 ± 0.20 +0.70 ± 0.29 + +Total activity +0.74 ± 0.19 +0.76 ± 0.25 +Theta +Sources +0.77 ± 0.17 +0.77 ± 0.30 + +Sinks +0.84 ± 0.15 +0.87 ± 0.18 + +Total activity +0.74 ± 0.17 +0.72 ± 0.28 +Alpha +Sources +0.79 ± 0.19 +0.74 ± 0.25 + +Sinks +0.76 ± 0.21 +0.71 ± 0.29 + +Total activity +0.79 ± 0.17 +0.77 ± 0.21 +Beta +Sources +0.80 ± 0.17 +0.86 ± 0.18 + +Sinks +0.79 ± 0.24 +0.81 ± 0.27 + +Total activity +0.76 ± 0.23 +0.75 ± 0.32 +Gamma +Sources +0.81 ± 0.18 +0.83 ± 0.22 + +Sinks +0.88 ± 0.14 +0.93 ± 0.16 + +Total activity +0.82 ± 0.12 +0.87 ± 0.18 + +The Receiver Operating Curve (ROC) space is a valuable +data interpretation tool that can be used to assess the +performance of a binary classifier, wherein it indicates +the cutoff point at which sensitivity is traded for +specificity. Hence, it can be used to evaluate the +classifier’s performance in distinguishing positive and +negative samples. Related to this, AUC is the probability +that the classifier will assign a random positive instance +a more extreme value than a random negative instance. +Fig. 7 presents the ROC curves for the Theta, Beta and +Gamma bands, to identify those with the best +performance. Notably, the Gamma band with the +channels’ sinks activity as the features presents a 93% +under the curve. +The obtained results were verified by repeating the +classification process using the Ada Boost algorithm. +Table 3 presents the results for the Gamma band while +Fig. 8 shows the ROC curve. While the performance is +slightly diminished, it remains consistent across all bands +and cases (sources, sinks and total activity) with the +results from the Gradient Boosting. + + + +Fig. 7. ROC curves for the Theta, Beta and Gamma bands with +the Gradient Boosting classifier. + + +Thetaband +1.0 +: Rate (Positive label: +0.8 +0.6 +Positive +0.4 +True +0.2 +Chance +SourcesMeanROC(AUC =0.77±0.31) +Sinks Mean ROC (AUC = 0.87±0.18) +0.0 +Total activityMeanROC (AUC = 0.72 ± 0.28) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate (Positive label: 1)Betaband +1.0 +: Rate (Positive label: +0.8 +0.6 +Positive +0.4 +True +0.2 +Chance +SourcesMeanROC(AUC=0.86±0.18) +Sinks Mean ROC (AUC = 0.81 ± 0.27) +0.0 +Total activityMeanROC (AUC = 0.75 ± 0.33) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate (Positive label:1)Gammaband +1.0 + Rate (Positive label: 1) +0.8 +0.6 +Positive +0.4 +True +0.2 +Chance +SourcesMeanROC(AUC=0.83±0.23) +SinksMeanROC(AUC=0.93± 0.17) +0.0 +Total activityMean ROC (AUC = 0.87 ± 0.18) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate (Positive label: 1)Rodríguez-Rodríguez et al. + +10 +Table 3. Results for the Ada Boost classifier for +the Gamma band. +Band +Feature set +Accuracy +AUC +Gamma +Sources +0.83 ± 0.17 +0.82 ± 0.27 + +Sinks +0.88 ± 0.11 +0.86 ± 0.21 + +Total activity +0.77 ± 0.19 +0.76 ± 0.31 + + +Fig. 8. ROC curves for the Gamma band with the Ada Boost +classifier. +As is often the case in biomedical studies, statistical tests +are required to check that the number of samples has not +introduced bias in the classification stage (e.g., through +overfitting). Moreover, there is a need to check the +probability of these results having been obtained by +chance. For large datasets, such tests need not be as +stringent, but real-world studies demand special attention +due to the small sample sizes and unbalanced classes. +Specifically, in experimental studies the prevalence of +the disorder among the population being treated must be +taken into account. For DD, this is around 5-12%, as +mentioned above. +To this end, a null distribution is generated by estimating +the classifier’s accuracy for 1000 permutations of the +labels. This indicates the distribution for the null +hypothesis that the features are not dependent on the +labels, and enables the estimation of the probability that +the classification results will be reproduced with shuffled +labels. The result is an empirical p-value determined by: + +𝑝 − 𝑣𝑎𝑙𝑢𝑒 = #𝑝𝑒𝑟𝑚 𝑤𝑖𝑡ℎ 𝑎𝑐𝑐. ℎ𝑖𝑔ℎ𝑒𝑟 𝑡ℎ𝑎𝑛 𝑏𝑎𝑠𝑒𝑙𝑖𝑛𝑒 +#𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛𝑠 + +(9) + +Fig. 9 gives the permutation test results for the Theta, +Beta, and Gamma bands for sources, sinks and total +activity. +The +null +distribution +from +the +label +permutations, as outlined above, is in blue, while the +vertical red line represents the accuracy obtained for the +non-permuted case. At each permutation iteration, a 20- +fold stratified cross-validation is performed, and based on +the average of the results obtained at these 20 folds, the +corresponding permutation iteration is determined. +Hence, Fig. 9 presents the classification’s probability +density. According to the permutation tests, the results +have low p-values and are significant. +5. Discussion +The participants were subjected to white noise at 4.8 Hz, +i.e. between the syllabic and prosodic frequencies, as the +sole stimulus. DD has been shown to link to impairments +in syllabic and prosodic perception 55, suggesting general +difficulties in identifying the different modulation +frequencies. This influences the slower temporal rates of +speech processing in particular, as well as the tracking of +the amplitude envelope of speech, diminishing learners’ +syllabic segmentation efficiency. +Multi-time resolution models of speech processing 16 +have evidenced that phonetic segment identification +associates with faster temporal modulations (Gamma +rate, 30–80 Hz), syllable identification is linked to slower +modulations (Theta rate, 4–10 Hz), and syllable stress +and prosodic patterning information correlates with very +slow modulations (Delta rate, 1.5–4 Hz). Nonetheless, +anomalies can emerge in various frequency ranges due to +inter-band entrainment. +As it offers adequate time resolution, examining the +patterns occurring in EEG channels at different bands can +unveil the speech encoding linked to problems with +speech prosody and sensorimotor synchronization. +Exemplifying this, previous research 18 used speech- +based stimuli and time-frequency descriptors to reveal +the link between speech features and neural dynamics. +We find that the classifier performs better in the Theta +and Gamma bands. The results for the Theta band are +expected as the TSF suggests that the phonological +deficit of DD – regardless of language – may be partially +attributed to functionally atypical or impaired phonology +entrainment mechanisms in the auditory cortex, +especially as oscillations at slower temporal rates, i.e. +Theta and Delta, relate to syllabic and prosodic +processing 56. + + + +Gammaband +1.0 +: Rate (Positive label: +0.8 +0.6 +Positive +0.4 +True +0.2 +Chance +SourcesMeanROC(AUC=0.78±0.27) +Sinks MeanROC (AUC=0.86±0.21) +0.0 +Total activity Mean ROC(AUC=0.76± 0.31) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate (Positive label:1) +Neural source/sink phase connectivity in Developmental Dyslexia + +11 + + + +Fig. 9. Permutation tests for Gradient Boosting classifier in Theta, Beta and Gamma bands. +As per the TSF, group differences are expected in +neuronal oscillatory entrainment at slower rates (approx. +4 Hz, in line with the stimulus used) 57. Higher causality +relationships emerged in the frontal area in all scenarios +for the Theta band. In addition, the number of channels +that g-causes causality is higher in the dyslexic domain, +which was the case for the sources, sinks and total +activity. This higher activity in terms of overall causality +relations was evident across all bands. However, in the +participants with DD there was significantly less +entrainment in the auditory networks of the right +hemisphere in the Theta band. As Fig. 6 (feature +selection) shows, the C4 channel in the upper part, i.e. the +Theta band, is predominantly influential for the causality +regarding the sources, as well as the sinks and total +activity. It has already been established that the right- +lateralized Theta sampling network tends to involve +slower temporal rates and codes the speech signal’s lower +modulation frequencies 57, facilitating syllable-scale +temporal integration. In other words, spoken sentences +are tracked and distinguished by the Theta band phase +pattern, allowing the incoming speech signal to be broken +into syllable-sized packets and speech dynamics to be +tracked through resetting and sliding, such as with +varying rates of speech 58. Fig. 5 (topoplots) clearly +demonstrates that the C4 channel is the most interesting +as it has the most Granger causality (causing and being +caused) for all scenarios for the dyslexic group. For the +sources, the frontal area contains other noteworthy +channels (FP2, F7, F3 and Fz) that show differences +between the control and dyslexic groups in terms of +activity. The most influential channels in the sinks are F3 +and F4 (frontal area) and P3. +Hence, it seems pertinent to suggest that the main +differences in the causality relationships of the Theta +band lie in the so-called dorsal and ventral pathways. In +particular, the right area seems critical, as evidenced in + +Thetaband-Sources +8 +7 +Score on original data: o.77 +6 +(p-value:0.002) +5 +I +- +4 - +- +FE +2 +1 +0- +0.40 +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +Accuracy scoreTheta band - Sinks +- +8 +score +on original data: 0.84 +6 - +(p-value:0.001) +*. +- +- +- +0 +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +Accuracy scoreTheta band -Total activity +- +8 - +- +Score on original data: o.74 +(p-value:0.001) +6 - +- +robability + 4 +- +2 - +0 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +Accuracy scoreBeta band-Sources +10 - +8 +Score onoriginal data: o.80 +6 +(p-value:0.001) +4 +2 - +0 +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +Accuracy scoreBeta band -Sinks +8 - +！！ +7 - +Score on original data: o.79 +6 - +(p-value:0.001) +. +4. +E +2 +1 +0 +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +Accuracy scoreBeta band -Total activity +7 +Score on original data: o.76 +6 +D- +value:0.002 +5 +3 - +2 - +1 +0 +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +Accuracy scoreGammaband-Sources +10 +8 +Score on original data: 0.81 +(p-value: 0.001) +4 +2 . +0 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +Accuracy scoreGammaband-Sinks +8 +Score 0n original data: 0.88 +6 +(p-value:0.001) +2 +0 +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +00.75 +0.80 +0.85 +Accuracy scoreGammaband-Totalactivity +-8 +7 +Scoreor +n original data: 0.82 +6 · +(p-value:0.001) +3 +2 +1 - +0 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +Accuracy scoreRodríguez-Rodríguez et al. + +12 +the prior research and especially demonstrated here with +the sinks scenario. +Another interesting result worth discussing is that for the +Beta band. Here, more activity was observed for all three +scenarios in the dyslexic group; this agrees with the +results for the Theta band as well as those from previous +studies 59. For the sources, differences in the causal +relationships were mainly identified in the C3 and C4 +channels, pointing to areas responsible for motor +processing 11. It is becoming increasingly clear that +speech perception is at least partially located in the motor +areas, especially under less-than-optimal listening +conditions. This cruciality of the C4 channel was +similarly seen in the Theta band and is in line with prior +research evidencing the important role played by the +lower frequency bands in general and Beta band coupling +in particular 60. Hence, inefficient phase locking in the +auditory cortex may affect visual and motor processing +development, which may in turn cause some of the +visual, motor and attentional difficulties seen in DD 61. +It should be noted, however, that the C3-C4 interaction is +mostly relevant for the sources and is not important for +either the C3 for the sinks or, as a result, for the total +activity. Meanwhile, the causal activity in the Beta band +is different in the occipital area in the sinks scenario, and +it is remarkably different in the frontal area, especially in +FP1 for all three scenarios and in the F3 channel for the +sinks scenario. +In the Gamma band the activity is higher than in the Theta +band for maximum values, although the occipital area +shows more concentrated activity among the causality +relations, as Fig. 5 shows. Nevertheless, the effect is +different between the control and dyslexic groups, +whereby the participants with DD show higher activity +for the sinks, which increases their total activity. +For the sources, the channels with the most explicit +differences are FC1 and, more generally, TP9 in the left +temporal area. In the case of sinks, this is also an +important channel, although O1 and, as highlighted +above, C3 also play a role. +Meanwhile, +in +the +Gamma +band, +despite +the +discrepancies between the dorsal and ventral pathways, +the latter offers the main difference for the classification +of TP9 for both sources and sinks. FC1 is linked to +sources and C3 to sinks, suggesting a significant cause- +effect relationship, albeit with potentially less activity in +the dyslexic group, facilitating classification. +We can confirm that the classifier performs better in the +Theta and Gamma bands, which can evidence atypical +oscillatory differences based on both speech and non- +speech stimuli 56. According to Leong’s models 62, the +slower rates (Delta and Theta) temporally constrain +entrainment at the faster rates, such as Gamma. +Lehongre et al. 65 contended that the oscillatory nesting +seen between the Theta/Delta phase and the Gamma +power 63 64 offers a way to integrate information at the +phonemic (Gamma) rate into the syllabic rate. +Meanwhile, the integration of the various acoustic +features that contribute to the same phoneme being +perceived may be hindered by impairments in the phase +locking by Theta generators. Otherwise, flaws in certain +Theta mechanisms could influence the development of +the phonological system, which thus tends to code +information bilaterally with the Gamma oscillations +independently and then link them perceptually with the +Theta oscillator output. In this case, the impaired phase +locking of the right hemisphere Theta oscillatory +networks causes difficulties with lower frequency +modulations 17 66. +In addition, the spontaneous oscillatory neural activity +identified in the auditory cortex in both the Theta and +Gamma bands is known to associate with spontaneous +activity in the visual and premotor areas 66. +A bilateral Gamma sampling network codes the signal’s +higher frequency modulations, thereby facilitating +temporal integration at the phonetic (i.e., phoneme) scale. +If we apply this model to DD, it is indicated that impaired +processing at the syllable level (i.e., less efficient Theta +phase locking) occurs alongside unimpaired Gamma +sampling, meaning more weight is assigned to phonetic +feature information during phonological development. +Hence, as is the case in typical infant development, +children with DD may have sensitivity to all phonetic +contrasts of human languages 67. +Leong and Goswami 62 found that learners with DD show +a preference for different phase alignment between +amplitude modulations (AMs) when these respectively +convey syllable and phoneme information (Theta and +Gamma-AMs). A different phase locking angle suggests +a discrepancy in the integration of speech information +that arrives at a temporal rate different to that of the final +perception of the speech 14. Our results concerning the +interaction between the Theta and Gamma bands support +this. +Finally, our results also seem to confirm that the dyslexic +brain is less efficient at encoding the amplitude +modulation hierarchy’s highest levels, i.e. those bearing +information on the prosodic-syllabic structure, leading to +cascade effects that impact the encoding of the +phonological structure’s levels nested within the Delta +band, such as the syllable-level (Theta band) and +phoneme-level (Gamma band) AM information. +Importantly, our results have been validated using a +demanding permutation test, with the aim of ensuring +that the results are not coincidental, despite the medium +sample size. + + +Neural source/sink phase connectivity in Developmental Dyslexia + +13 +6. Conclusion and future works +Our results support the main assumption of the TSF that +DD involves a specific deficit in the low-frequency phase +locking mechanisms in the auditory cortex, thereby +potentially affecting phonological development 56. +In confirmation of this, we find an anomaly that emerges +primarily in the causal relationships of channels that +function as sinks, which is significantly more pronounced +than when only the total activity is considered. Hence, it +is reasonable to consider a division into Granger-causing +or Granger-caused relationships. This, in turn, suggests +that the main differences contributing to DD emerge +when certain brain areas must function as receptors in the +interactions between channels. +Furthermore, our results are in line with previous +research, which has already detected an anomaly in the +right-lateralized Theta band. We have clearly identified +this here across all three scenarios (sources, sinks, total +activity). +We also find confirmation for the higher brain activity in +learners with DD, although differences are more +significant for the sinks in the Theta and Gamma bands, +in turn leading to more total activity. The highest +classifier performance (accuracy and AUC) is hereby +found in the sink scenario. For the Beta band, the +difference in activity is more consistent across all three +scenarios. The classifier also performs well for the Beta +band in all three scenarios, with few differences +observed, thereby confirming the important role played +by this band in the sensorimotor coding of speech. +The results reflect the causal activity generated in the +brain subjected to prosodic-syllabic stimulus at 4.8 Hz. +Consequently, future work could consider the Granger +causality relationships in the phases across channels and +bands using higher frequency stimuli to stimulate +syllabic-phonetic and phonetic activity. +Acknowledgements +This work was supported by projects PGC2018-098813- +B-C32 (Spanish “Ministerio de Ciencia, Innovación y +Universidades”), UMA20-FEDERJA-086 (Consejería de +econnomía y conocimiento,Junta de Andalucía) and by +European Regional Development Funds (ERDF). + +References +1. Peterson RL, Pennington BF. Developmental dyslexia. +The lancet. 2012;379:1997–2007. +2. Thompson PA, Hulme C, Nash HM, Gooch D, Hayiou- +Thomas E, Snowling MJ. 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Nature reviews neuroscience. 2004;5:831–843. + + + diff --git a/AdAyT4oBgHgl3EQfq_lh/content/tmp_files/load_file.txt b/AdAyT4oBgHgl3EQfq_lh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..99f93cb0e3a05e71e05c7da71c44b3f6c3b5c7fe --- /dev/null +++ b/AdAyT4oBgHgl3EQfq_lh/content/tmp_files/load_file.txt @@ -0,0 +1,1165 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf,len=1164 +page_content='1 NEURAL SOURCE/SINK PHASE CONNECTIVITY IN DEVELOPMENTAL DYSLEXIA BY MEANS OF INTERCHANNEL CAUSALITY I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' RODRÍGUEZ-RODRÍGUEZ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' ORTIZ, N.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='uk While the brain connectivity network can inform the understanding and diagnosis of developmental dyslexia, its cause-effect relationships have not yet enough been examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Employing electroencephalography signals and band- limited white noise stimulus at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8 Hz (prosodic-syllabic frequency), we measure the phase Granger causalities among channels to identify differences between dyslexic learners and controls, thereby proposing a method to calculate directional connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' As causal relationships run in both directions, we explore three scenarios, namely channels’ activity as sources, as sinks, and in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Our proposed method can be used for both classification and exploratory analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' In all scenarios, we find confirmation of the established right-lateralized Theta sampling network anomaly, in line with the temporal sampling framework’s assumption of oscillatory differences in the Theta and Gamma bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Further, we show that this anomaly primarily occurs in the causal relationships of channels acting as sinks, where it is significantly more pronounced than when only total activity is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' In the sink scenario, our classifier obtains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='84 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='88 accuracy and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='87 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='93 AUC for the Theta and Gamma bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Keywords: Developmental Dyslexia, EEG, Granger causality, functional connectivity, anomaly detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Introduction Developmental dyslexia (DD) is a learning difficulty that typically causes various reading difficulties, including letter migration and frequent spelling errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' In any given population, between 5% and 12% of learners are likely to have DD, depending on the test battery used 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' DD is traditionally diagnosed using behavioral tests of reading and writing skills, but these are vulnerable to exogenous factors, such as attitude or disposition, leading to diagnoses that may be fundamentally unsound 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' It is, therefore, imperative to develop more objective metrics that can offer a more accurate diagnosis among young learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' A stimulus system that remains uninfluenced by the learner’s behavior, actions and context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=', native language or learning level) would be extremely valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' If the stimulus further involves the simulation of prosody, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' the white noise at the usual frequency of the language envelope, it may also inform our understanding of the brain areas active in auditory processing, indicating the differences between learners with and without dyslexia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' While various neuroscience methods for gathering functional brain data exist, including functional magnetic resonance imaging (fMRI) 3, magnetoencephalography (MEG) and functional near-infrared spectroscopy (fNIRS), electroencephalography (EEG) continues to be the most widely used and least costly method to assess cortical brain activity with enhanced temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' An EEG measures several frequency bands, namely the Delta, Theta, Alpha, Beta, and Gamma bands, which do not experience stimulation equally, and it is thus generally held that the stimulation of one band can transfer to the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Using EEG to separately investigate the patterns emerging in these bands may offer valuable insights for the research on DD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' EEG is well-established in DD studies exploring the functional network connectivity and organization of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Functional connectivity means the level of coordination between the activities in different areas of the brain while the learner is engaging in a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Prior research has produced various techniques that employ EEG to assess functional connectivity to, for example, determine what patterns are characteristic of neurological conditions, including Parkinson’s disease 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Studies in cognitive neuroscience have also used brain connectivity to identify brain areas crucial to language and learning 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Rodríguez-Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 2 Connectivity analysis has allowed neuroscience to provide even deeper insights 6 by analyzing the parameters linking two signals gathered through two distinct channels, such as their correlation, causality and covariance 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Employing connectivity analysis to brain signals measured in different regions allows us to explore the neural network, in line with the notion that the brain is hyper-connected 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Notably, brain connectivity is not solely limited to the interactions between areas, with regions potentially influencing each other through, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=', phase-phase or phase-amplitude modulations among bands 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We here consider areas that primarily exert an influence as sources, while those more likely to be subject to this influence are sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' This novel consideration of connectivity in terms of sources and sinks can not only serve classification but also facilitate exploratory analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We propose extracting the frequency components of each band from the EEG signals acquired under prosodic auditory stimuli, subsequently using them to generate a connectivity model based on inter-channel Granger causality 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' By modelling connectivity as sources and sinks, we seek to clarify the existence of abnormalities between learners with and without DD, aiming to offer an enhanced understanding of the mechanisms underlying DD, ultimately allowing early diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The remainder of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Section 2 describes the most relevant extant work in this field, and Section 3 outlines the data and methodology, including the preprocessing, Granger causality matrices and connectivity matrices construction, and classification algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Section 4 presents our main results, leading to the discussion in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Finally, Section 6 presents the main conclusions and contributions of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Related works Previous studies have indicated that the phonological deficit that causes DD may be due to an impairment in the neural encoding of low-frequency speech envelopes, relating to speech prosody 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' There is evidence of significant difficulties among that learners with DD in tasks relying on prosodic awareness, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' identifying syllable stress, compared to controls at an earlier reading level 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' This indicates the presence of atypical oscillatory functioning in low-frequency brain rhythms in DD 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' There has been substantial research on the important role played by the ability to perceive prosodic frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' After directly measuring the neural encoding of children’s speech using EEG, Power et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 11 reconstructed the participants’ speech stimulus envelopes using the emergent patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The EEG recordings were done while the participants were performing a word report task using noise-vocoded speech, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' still with a low-frequency envelope yet with a degraded temporal fine structure (TFS) of speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Due to this degradation, the participants necessarily derived the spoken words and sentences from the information given by the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' If the learners could accurately perceive the words and sentences, it was possible to evaluate the functioning of their neural encoding of the low-frequency envelopes in speech, which is likely impaired in learners with DD according to temporal sampling theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Brain activity, and thus the connectivity network, occurs across various frequency bands, as demonstrated via the temporal sampling framework (TSF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Temporal coding is thought to be partially attributed to synchronous auditory cortex activity, wherein the network neurons synchronize endogenous oscillations at different preferred rates while matching the temporal information of the acoustic speech signal 14 15 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The auditory and visual parts of speech unfold across different timescales, and thus, when the neurons in auditory and visual cortices oscillate, they are believed to phase-align their activity to match the input’s modulation rates 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' TSF proposes that atypical oscillatory sampling at various temporal rates may be the cause of the phonological impairment in DD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Furthermore, a potential biological mechanism for DD has recently been suggested, highlighting the presence of atypical dominant neural entrainment 18 for the slow rhythmic prosodic (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='5–1 Hz), syllabic (4–8 Hz) and phoneme (12– 40Hz) rhythm categories 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Following this line of thought, we might consider learners with DD to have atypical oscillatory sampling for at least one temporal rate, leading to difficulties in phonologically capturing linguistic units such as syllables or phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' However, this phenomenon is not likely to be experienced equally across all frequency bands (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Delta, Theta, Alpha, Beta, and Gamma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Thus, it seems pertinent to examine these bands’ connectivity patterns separately using EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Prior research has indeed used EEG or MEG to investigate the fundamental mechanisms underlying DD, implementing speech-based stimuli under the premise that DD is essentially derived from a lesser awareness of individual speech units 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Using visual and auditory stimulus, Power et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 21, for example, identified differences between learners with DD and a control group in the preferred entrainment phase of the Delta and Theta bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Based on changes in the frequency, phase, and power spectrum, it thus becomes feasible to derive measures of spectral connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' In line with this, there are techniques showing the statistical Neural source/sink phase connectivity in Developmental Dyslexia 3 relationship between electrodes on the same frequency band 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The prior research has also explored the inference from connectivity patterns during reading tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' For example, Žarić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 23 used visual word and false font processing tasks to investigate disruptions in the connectivity between the visual and language processing networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' They hereby calculated the connectivity patterns based on how statistically significant the differences in the power spectral density (PSD) were for each EEG band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Language-based reading or writing-related tasks have also been used in previous studies identifying discriminant patterns in EEG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' For instance, using graph theory, González et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 24 compared the EEG measurements of participants performing audiovisual tasks or at rest to determine differences in the connectivity patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Meanwhile, Stam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 25 used a phase lag index to compute multiple weighted connectivity matrices for multiple frequency bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Assessing the connectivity of two channels requires a separate analysis of their respective phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' A signal’s phase, φ(t), changes over time when being captured with an electrode, and thus it must be measured for each channel i, referring to the instantaneous phase, captured using a Hilbert transform and computed via band-pass filtered signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Consequently, the phase value can be pinpointed at each time point, allowing inter-channel correlation and causality to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Using this method to track changes in the phase synchronization of epileptic patients, Mormann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 26 showed that epileptic episodes are often preceded by characteristic changes in synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Following this, we can estimate the inter-channel connectivity based on the cause-effect relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The Granger causality test can hereby show whether one of the factors is a time series, allowing the characteristics of additional time series to be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' First employed in the 1980s in the economics field, Granger causality is a statistical hypothesis test that has been used to produce good results in a wide range of other fields 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Neuroscience research has applied it to EEG measurements, producing findings on brain activity in emotion recognition 27, Vagus nerve stimulation 28, and pain perception 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Connectivity based on causality implies cause-effect relationships between various areas of the brain, but these are not necessarily bidirectional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Thus, some brain areas will be very active because they are influencing others, and other areas may be very active because they are being influenced by remote areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Likewise, it could be the case that high activity may be due to both situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' While this concept of sources/sinks is not new, it has been subject to a variety of different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' For example, Rimehaug et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 30 integrated it into their model of the visual cortex’s local field potential, while Sotero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 31 used it to explain the laminar distribution of phase- amplitude coupling of spontaneous current sources and sinks in rat brains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' However, neither of those studies based their modeling of sources and sinks on causality relationships, instead using the electrical activity in the cerebral cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The concepts of Granger causality and source/sink relationships have been used to address the clinical issue of surgical resection planning by capturing high- frequency ictal and preictal oscillations on an intracranial EEG 32, although no connectivity maps were constructed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' furthermore, the study did not use machine learning to examine whether this approach could be applied in the differential diagnosis of impairments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Building on the work outlined above, we apply machine learning classification algorithms to assess the potential of diagnosing DD via a learner’s sources, sinks and total activity under stimulus, identified using Granger causality matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Due to the impenetrable nature of EEG signal classification and the complexity of the problem being addressed, machine learning is highly suitable 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Briefly, we seek to demonstrate that different connectivity patterns are induced in certain brain networks by low-level auditory processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' To this end, we delineate this connectivity by establishing the source and sink relationships through the application of Granger causality to the phase synchronization among EEG channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Materials and methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Data acquisition The dataset comprised EEG data from the University of Málaga’s Leeduca Study Group 34, gathered from 48 age- matched child participants (32 skilled readers and 16 dyslexic readers) (t(1) = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='4, p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='05, age range: 88-100 months).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' All participants were righthanded native Spanish speakers with normal or corrected-to-normal vision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' none had a hearing impairment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' All participants in the dyslexic group had been formally diagnosed with dyslexia at school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' All participants in the skilled reader group were free from reading and writing difficulties and had not been formally diagnosed with dyslexia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The participants’ legal guardians expressed their understanding of the study, gave their written consent, and were present throughout the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' All participants experienced an auditory stimulus in 15- minute sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The stimulus, which was modulated at Rodríguez-Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8 Hz (prosodic-syllabic frequency) in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='5-minute segments, was band-limited white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' This type of stimulus was chosen to identify what synchronicity patterns the low-level auditory processing would induce and on the basis of the expert knowledge of linguistic psychologists concerning the main frequency components representing words in the human voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The participants’ EEG signals were recorded with a BrainVision actiCHamp Plus with 32 active electrodes (actiCAP, Brain Products GmbH, Germany) at a 500 Hz sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The 10–20 standardized system was used to place the 32 electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Preprocessing The preprocessing involved removing all eye-blinking and movement/impedance variation artifacts from the EEG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The former were eliminated via independent component analysis (ICA) 35 based on the eye movements observed in the EOG channel, while for the latter the relevant EEG segments were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The channels were then referenced to the Cz channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Then, a band-pass filter was applied to the EEG channels to collect information for the five EEG frequency bands (Delta, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='5–4 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Theta, 4–8 Hz, Alpha, 8–13 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Beta, 13–30 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' and Gamma, 30–80 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We used finite impulse response (FIR) filters because these ensure a constant phase lag that can later be corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' To be specific, each signal was sent forward and backward through the two-way zero-phase lag band-pass FIR least- squares filter, producing a zero-lag phase in the overall filtering process that addressed the issue of phase lag 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' As low-pass filtering with an 80 Hz threshold was employed, we added a 50 Hz notch filter during preprocessing to eliminate this frequency component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Hilbert Transform A Hilbert transform (HT) transforms real signals into analytic signals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' complex-valued time series without negative frequency components, allowing the time- varying amplitude, phase and frequency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=', the instantaneous amplitude, phase and frequency, to be calculated from the analytic signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We define HT for a signal x(t) as: ℋ[𝑥(𝑡)] = 1 𝜋 ∫ 𝑥(𝑡) 𝑡 − 𝜏 𝑑𝜏 +∞ −∞ (1) and we obtain the analytic signal zi(t) for signal x(t) as: 𝑧𝑖(𝑡) = 𝑥𝑖(𝑡) + 𝑗ℋ{𝑥𝑖(𝑡)} = 𝑎(𝑡)𝑒𝑗𝜙(𝑡) (2) From zi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' computing the instantaneous amplitude is straightforward: 𝑎(𝑡) = √𝑟𝑒(𝑧𝑖(𝑡))2 + 𝑖𝑚(𝑧𝑖(𝑡))2 (3) with the instantaneous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' unwrapped phase as: 𝜙(𝑡) = 𝑡𝑎𝑛−1 𝑖𝑚(𝑧𝑖(𝑡)) 𝑟𝑒(𝑧𝑖(𝑡)) (4) The above technique gives the phase value for each time point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' allowing the inter-channel synchronization to be estimated based on the phase variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Granger Causality test Developed for the field of econometrics by Clive Granger, Granger causality 37 describes causal interactions occurring between continuous-valued time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' As a statistical hypothesis test, it essentially states that “the past and present may cause the future, but the future cannot cause the past”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' hence, knowing a cause will be more helpful in predicting future effects than an auto-regression will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Specifically, variable x will Granger-cause y if the auto-regression for y that uses past values of x and y is significantly more accurate than one using only past values of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We may exemplify this by taking two stationary time-series sequences, xt and yt, whereby xt−k and yt−k are, respectively, the past k values of xt and yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We then use two regressions to perform Granger causality: 𝑦𝑡̂ 1 = ∑ 𝑎𝑘 𝑙 𝑘=1 𝑦𝑡−𝑘 + 𝜀𝑡 (5) 𝑦𝑡̂ 2 = ∑ 𝑎𝑘 𝑙 𝑘=1 𝑦𝑡−𝑘 + ∑ 𝑏𝑘 𝑤 𝑘=1 𝑥𝑡−𝑘 + 𝜂𝑡 (6) where 𝑦𝑡̂ 1 and 𝑦𝑡̂ 2 are, respectively, the fitting values of the first and second regressions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' l and w are the maximum numbers of the lagged observations of xt and yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' ak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' bk ∈ R are the regression coefficient vectors estimated using least squares;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' and εt and ηt are white noise (prediction errors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Note that even though w can be infinite, due to the finite nature of our data, we consider w finite and give it a length well below the time series length, estimated using model selection, such as the Akaike information criterion (AIC) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Next, an F-test is applied to give a p- value indicating whether the regression model produced Neural source/sink phase connectivity in Developmental Dyslexia 5 by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' (5) is statistically better than that of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' If it is, then x Granger-causes y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We perform Granger causality testing for each participant and evaluate the channels’ interactions, producing an n x n square matrix of p-values (n = number of channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Using Granger causality to analyze the neural network’s directed functional connectivity intuitively demonstrates the directionality with which information is transmitted between neurons or brain regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Previous studies have already applied this technique to EEG analysis with great success 39 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Connectivity vectors The field of neuroscience tends to consider the brain as a network using functional information 41 42 43, culminating in the so-called connectome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' This refers to the complete mapping of all connections between brain regions as an adjacency matrix, and often includes the covariance, as well as other metrics, between fMRI signals measured for different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Several studies have also examined the temporal covariance between EEG electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Once we had assembled the Granger causality matrices for each participant subject, we established a threshold value that evidenced a causal relationship between the channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Then, we formulated the three scenarios used to produce each participant’s feature set: Sources: Array of n x 1 elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' each element relates each channel with the number of channels that it influences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Sinks: Array of n x 1 elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' each element relates each channel with the number of channels that it is influenced by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Total activity: Array of n x 1 elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' the sum of the two previous scenarios, acting as a reference for each channel’s global activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' By organizing the information thus, we receive the same number of features as there are channels for each participant, each with a number that indicates its activity as a source, as a sink, or the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' A summary of this process is presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Assembling the source and sink connectivity arrays for a participant, given the relevant Granger matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' [P(k)] is an Iverson bracket function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Ensemble feature selection If the model includes many features, it will be more complex, potentially leading to data overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Moreover, some of the features may be noise and could adversely affect the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Thus, we removed such features to ensure the better generalization of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We hereby selected the variables based on majority voting through the application of several techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' If a variable was chosen by an algorithm, it received one vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The votes were then summed for each variable, and those with the most votes were selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' This method has been found to be suitable for datasets that are high-dimensional yet have few instances 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The voting strategy used a variety of feature selection methods 45, as outlined in the following: Information value (IV) using weight of evidence (WOE): This indicates the predictive power of an independent variable concerning the dependent variable 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' It allows a continuous independent variable to be transformed into a set of groups or bins based on the similarity of the dependent variable distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' numbers of events and non-events).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Using WOE allows outliers and missing values to be addressed and eliminates the need for dummy variables 47: 𝑊𝑂𝐸 = ln ( 𝐸𝑣𝑒𝑛𝑡% 𝑁𝑜𝑛 𝐸𝑣𝑒𝑛𝑡%) (7) 𝐼𝑉 = Σ[(𝐸𝑣𝑒𝑛𝑡% − 𝑁𝑜𝑛 𝐸𝑣𝑒𝑛𝑡%) ∗ 𝑊𝑂𝐸] (8) An IV statistic above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='3 is held to indicate a strong relationship between the predictor and the event/non- event odds ratio 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' FP1 FP2 F7 PO10 Sinks = FP1 1 pvalue<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='01 GrM[FP1,k], [P(R) FP2 1 , GrM[F7,k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' F7 pvalue<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='01 1 pvalue<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='01 P(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' PO10 pvalue<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='01 1 GrM[P010, k] P(k) , GrM[k, FP2] Sources = GrM[k, FP1], [P(k) l0otherwiseRodríguez-Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 6 Variable importance using random forest/extra trees classifier: Calculated using a tree-based estimator, this can be used to eliminate irrelevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Variable importance is conventionally computed using the mean decrease in impurity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=', gini importance 49) mechanism, wherein the improvement in the split criterion for each split of each tree is the importance measure assigned to the splitting variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' For each variable, this is separately accumulated over all the trees in the forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' This measure is similar to the R2 in the training set regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Recursive Feature Elimination: This can be used to select features by recursively considering feature sets with diminishing size based on an external estimator (a linear regression model) that assigns weights to the features 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The estimator is trained on the first feature set, noting each feature’s importance based on a given attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The least important features are subsequently removed from the current set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The process is performed recursively on the pruned set until the desired number of features is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Chi-square best variables: This uses a chi-square (χ2) test to assess the correlations among a dataset’s features and identify multicollinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The aim is revealing any relationships between the dependent variable and any of the independent variables 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' In the chi-square test, H₀ (null hypothesis) assumes that two features are independent, while H₁ (alternative hypothesis) predicts that they are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We set a α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='05 and a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='05 or greater is considered critical, anything less means the deviations are significant hence the hypothesis must be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' L1-based feature selection: Some features can be eliminated using a linear model with an L1 penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' This method involves regularization, wherein a penalty is added to various parameters of a machine learning model to reduce the model’s freedom and prevent overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' When regularizing linear models, the penalty is applied in addition to the coefficients multiplying the predictors 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Unlike other forms of regularization, L1 can reduce some coefficients to zero, meaning the feature is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Once the best variables had been chosen by voting, we performed a multicollinearity check on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The feature selection procedure for the ‘Sources’ scenario using a vote-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Classification process In an ensemble method, multiple models are first generated and then integrated to produce higher-quality results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The respective predictions are hereby combined using weighted majority voting to make the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' At each boosting iteration, the data are modified by applying w1, w2 , …, wn to each training sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' As the weights are initially wi=1/N, a weak learner is trained in the first step using the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' At each successive iteration, the sample weights are modified individually, and the algorithm is then applied to the reweighted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Training examples that are incorrectly predicted relative to the previous step’s boosted model are given increased weights;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' correctly predicted examples are given decreased weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' As a result, the examples that were difficult to predict become increasingly influential as the number of iterations increases, and the weak learners that follow are forced to focus on the examples previously missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Ensemble methods deliver more accurate results than single models, and are particularly suitable for improving binary prediction on small data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We use the Gradient Boosting classifier, as well as an Ada Boost for results verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' This latter classifier 53 is a meta-estimator that initially fits to the data, with further copies then being fit to the same data, while incorrectly classified instances’ weights are modified to force subsequent classifiers to focus on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The Gradient Boosting classifier 54 creates an additive model based on a forward stage-wise construction, allowing the optimization of the arbitrary differentiable loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' At each stage, n regression trees are fit to the multinomial or deviance binomial loss function’s negative gradient, with a single regression tree being used for the special case of binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' To identify the best parameter set, we cross-validate with 20 folds and a parameter grid, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Method:IVusingWOE-→ Feature subset: f1, f2, f3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='., fn All features for Scenario Sources: Voting FP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' FP2,F7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' PO10 Method:Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Imp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content="using RF→ Feature subset:f'1, f'2,f3, ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=",f'n M Method:Var." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Imp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='using Trees → Feature subset: f"1, f"2, f"3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=', f"n Method:RFE→Feature subset Reducedranked feature subsetbasedonvotes →fr1,f2,f3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=',frn Method:Chi Squared→Feature subset Neural source/sink phase connectivity in Developmental Dyslexia 7 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Parameter grid of machine learning classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Algorithm Parameter Range Gradient n_estimators 1 to 12 Boosting Loss deviance, exponential Learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='05 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='5 Criterion friedm_mse, sq_error, mse, mae Min_samples_split 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='01 to 3 Min_samples_leaf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='01 to 3 Max_depth 1 to 4 Ada Boost n_estimators 1 to 25 Learning rate 1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='5 Boosting algorithm SAMME, SAMME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='R 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Results Plotting each learner’s array of sources and sinks permits the visual extraction of the respective patterns of the dyslexic and control groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' To this end, we examined the channel distributions for both groups by calculating the means and dispersions and producing a box-and- whisker plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We also constructed a topoplot as this can illustrate the results with greater clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' For example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 3 shows the Theta band connectivity of the control and dyslexic groups specifically for total activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Please note that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 4 do not directly represent the electrical activity of the cerebral cortex, but rather show the levels of the cause-effect relationships between the channels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' in one direction or in the other direction or in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' It immediately becomes clear that despite the similarity of the patterns, the dyslexic group has a significantly higher activity level in the Theta band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Boxplot of the total activity in the Alpha band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The equivalent graphical representation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 3 in a topoplot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Source/sink activity in the Theta, Beta and Gamma bands in the control and dyslexic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Numbers represent how many channels are affected by each channel as a source, or how many channels are affecting each channel as a sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' ActivityofsourcesinThetaband Control 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='5 Dyslexic 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 nels 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='5 - Fp1 Fp2 F7 F3 FZ F4 F8 FC5 FC1 FC2FC6 T7 C3 C4 T8 TP9CP5 CP1CP2CP6IP1O P7 P3 PZ P4 P8 PO9 O1 OZ O2PO10 ChannelActivityofsources inThetaband Control group Dyslexic group 19 19 Fp1 Fp2 F1 2 F8 IFG 13 FC6 FQ1 FG2 FG6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 13 TZ T8 G3 18 TP9CR5 CP1 CPEFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='CP5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='CBI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Cp2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='CP6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='P3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='RZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='P4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='P8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='P7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='PZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='F4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='P8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='kod ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='01 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='02 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Pig ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='01 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='02 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Poip ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='QzActivity of sources in Theta band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Control group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Dyslexlc group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='13Activity of sources in Beta band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Control group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Dyslexlc group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='F8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='13Activityof sources in Gamma band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Control group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Dyslexlc group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='13Activity of sinks in Theta band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Control group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Dyslexic group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Fp2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='13Activity of sinks in Beta band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Control group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Dyslexic group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='13Activity of sinks in Gamma band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Controlgroup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Dyslexicgroup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' F4 13 T651 FE6 13 4 9P5 EPFTotal activity per channel in Theta band Control group Dyslexic group 38 38 27 27 15 15Total activity per channel in Beta band Control group Dyslexic group 38 38 27 27 CP CPI 15 TE 15Total activity per channel in Gamma band Control group Dyslexic group 38 38 H 27 Fe6 27 4 15 15Rodríguez-Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 5 compares the channel activity in the Theta, Beta and Gamma bands, and can be viewed separately as sources, sinks, or total activity for both the control and dyslexic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Please note that the range of visualization is the same in all sinks/sources topoplots, while different in the total activity ones, for better representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Once more, it is immediately clear that while the patterns are broadly similar, the activity level is higher in the dyslexic group, primarily observed in the sink activity (less in the source activity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Thus, although the sources, broadly speaking, behave similarly between the groups, the dyslexic group has significantly more concentrated sinks and more activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Consequently, the overall activity level is also affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Feature importance in Theta, Beta and Gamma bands considering sources, sinks and total activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' With as many arrays as subjects, and with each array having as many components as channels, we performed feature selection to identify channels that can help differentiate between the control and dyslexic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The feature selection procedure outlined above was thus applied for the cases of sources, sinks and total activity, according to the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 6 presents the results for the Theta, Beta and Gamma bands, whereby the importance values are normalized to permit fair and simple comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Channels showing a higher significance are those with more dissimilarity between the control and Feature importance in Theta band 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 Activity of sources Activity of sinks Total activity per channel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0- Fp1 Fp2 F7 F3 Fz F4 F8 FC5 FC1 FC2 FC6 T7 C3 C4 T8 CP5 CP1 CP2 CP6 TP10 P3 Pz P4 P8 PO9 Q1 Qz ZO PO10 ChannelFeature importance in Beta band 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 Activity of sources I Activity of sinks Total activity per channel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8- Fea 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0- Fp1 Fp2 F7 F3 Fz F4 F8 FC5 FC1 FC2 FC6 T7 C3 C4 T8 CP5 CP1 CP2 CP6TP10 P7 P3 Zd P4 P8 PO9 Q1 Qz 02 PO10 ChannelFeatureimportanceinGammaband 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 Activity of sources Activity of sinks Total activity per channel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0- Fp1 Fp2 F7 F3 Fz F4 F8 FC5 FC1 FC2 FC6 C3 C4 T8 CP5 CP1 CP2 CP6TP10 3 P4 P8 PO9 Q1 Qz 02PO10 Channel Neural source/sink phase connectivity in Developmental Dyslexia 9 dyslexic groups, directing us to where we can find different patterns of functioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' After performing the feature selection for each band, for each case (sources, sinks and total activity), we optimize the Gradient Boosting classifier to obtain the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The results are summarized in Table 2, with performances achieving at least 80% marked bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' According to the results, the greatest differences between the control and dyslexic groups (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=', the best classifier results) emerge in the Theta and Gamma bands when accounting for the activity sink role of the different channels, achieving accuracies of 84% and 88%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We also wish to highlight the results for the Beta band for the activity sources regarding the Area Under the Curve (AUC), in addition to accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Results of the Gradient Boosting machine learning classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Band Features set Accuracy AUC Delta Sources 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='31 Sinks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='29 Total activity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='25 Theta Sources 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='30 Sinks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='18 Total activity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='28 Alpha Sources 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='25 Sinks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='29 Total activity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='21 Beta Sources 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='18 Sinks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='27 Total activity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='32 Gamma Sources 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='22 Sinks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='16 Total activity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='18 The Receiver Operating Curve (ROC) space is a valuable data interpretation tool that can be used to assess the performance of a binary classifier, wherein it indicates the cutoff point at which sensitivity is traded for specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Hence, it can be used to evaluate the classifier’s performance in distinguishing positive and negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Related to this, AUC is the probability that the classifier will assign a random positive instance a more extreme value than a random negative instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 7 presents the ROC curves for the Theta, Beta and Gamma bands, to identify those with the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Notably, the Gamma band with the channels’ sinks activity as the features presents a 93% under the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The obtained results were verified by repeating the classification process using the Ada Boost algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Table 3 presents the results for the Gamma band while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 8 shows the ROC curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' While the performance is slightly diminished, it remains consistent across all bands and cases (sources, sinks and total activity) with the results from the Gradient Boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' ROC curves for the Theta, Beta and Gamma bands with the Gradient Boosting classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Thetaband 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 : Rate (Positive label: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='6 Positive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='4 True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='2 Chance SourcesMeanROC(AUC =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='31) Sinks Mean ROC (AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='18) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 Total activityMeanROC (AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='28) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 False Positive Rate (Positive label: 1)Betaband 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 : Rate (Positive label: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='6 Positive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='4 True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='2 Chance SourcesMeanROC(AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='86±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='18) Sinks Mean ROC (AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='27) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 Total activityMeanROC (AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='33) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 False Positive Rate (Positive label:1)Gammaband 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 Rate (Positive label: 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='6 Positive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='4 True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='2 Chance SourcesMeanROC(AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='23) SinksMeanROC(AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='93± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='17) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 Total activityMean ROC (AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='18) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 False Positive Rate (Positive label: 1)Rodríguez-Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 10 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Results for the Ada Boost classifier for the Gamma band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Band Feature set Accuracy AUC Gamma Sources 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='27 Sinks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='21 Total activity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='31 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' ROC curves for the Gamma band with the Ada Boost classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' As is often the case in biomedical studies, statistical tests are required to check that the number of samples has not introduced bias in the classification stage (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=', through overfitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Moreover, there is a need to check the probability of these results having been obtained by chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' For large datasets, such tests need not be as stringent, but real-world studies demand special attention due to the small sample sizes and unbalanced classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Specifically, in experimental studies the prevalence of the disorder among the population being treated must be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' For DD, this is around 5-12%, as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' To this end, a null distribution is generated by estimating the classifier’s accuracy for 1000 permutations of the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' This indicates the distribution for the null hypothesis that the features are not dependent on the labels, and enables the estimation of the probability that the classification results will be reproduced with shuffled labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The result is an empirical p-value determined by: 𝑝 − 𝑣𝑎𝑙𝑢𝑒 = #𝑝𝑒𝑟𝑚 𝑤𝑖𝑡ℎ 𝑎𝑐𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' ℎ𝑖𝑔ℎ𝑒𝑟 𝑡ℎ𝑎𝑛 𝑏𝑎𝑠𝑒𝑙𝑖𝑛𝑒 #𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛𝑠 (9) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 9 gives the permutation test results for the Theta, Beta, and Gamma bands for sources, sinks and total activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The null distribution from the label permutations, as outlined above, is in blue, while the vertical red line represents the accuracy obtained for the non-permuted case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' At each permutation iteration, a 20- fold stratified cross-validation is performed, and based on the average of the results obtained at these 20 folds, the corresponding permutation iteration is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Hence, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 9 presents the classification’s probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' According to the permutation tests, the results have low p-values and are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Discussion The participants were subjected to white noise at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8 Hz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' between the syllabic and prosodic frequencies, as the sole stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' DD has been shown to link to impairments in syllabic and prosodic perception 55, suggesting general difficulties in identifying the different modulation frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' This influences the slower temporal rates of speech processing in particular, as well as the tracking of the amplitude envelope of speech, diminishing learners’ syllabic segmentation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Multi-time resolution models of speech processing 16 have evidenced that phonetic segment identification associates with faster temporal modulations (Gamma rate, 30–80 Hz), syllable identification is linked to slower modulations (Theta rate, 4–10 Hz), and syllable stress and prosodic patterning information correlates with very slow modulations (Delta rate, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='5–4 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Nonetheless, anomalies can emerge in various frequency ranges due to inter-band entrainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' As it offers adequate time resolution, examining the patterns occurring in EEG channels at different bands can unveil the speech encoding linked to problems with speech prosody and sensorimotor synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Exemplifying this, previous research 18 used speech- based stimuli and time-frequency descriptors to reveal the link between speech features and neural dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We find that the classifier performs better in the Theta and Gamma bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The results for the Theta band are expected as the TSF suggests that the phonological deficit of DD – regardless of language – may be partially attributed to functionally atypical or impaired phonology entrainment mechanisms in the auditory cortex, especially as oscillations at slower temporal rates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Theta and Delta, relate to syllabic and prosodic processing 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Gammaband 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 : Rate (Positive label: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='6 Positive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='4 True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='2 Chance SourcesMeanROC(AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='27) Sinks MeanROC (AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='86±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='21) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 Total activity Mean ROC(AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='76± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='31) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='0 False Positive Rate (Positive label:1) Neural source/sink phase connectivity in Developmental Dyslexia 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Permutation tests for Gradient Boosting classifier in Theta, Beta and Gamma bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' As per the TSF, group differences are expected in neuronal oscillatory entrainment at slower rates (approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 4 Hz, in line with the stimulus used) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Higher causality relationships emerged in the frontal area in all scenarios for the Theta band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' In addition, the number of channels that g-causes causality is higher in the dyslexic domain, which was the case for the sources, sinks and total activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' This higher activity in terms of overall causality relations was evident across all bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' However, in the participants with DD there was significantly less entrainment in the auditory networks of the right hemisphere in the Theta band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 6 (feature selection) shows, the C4 channel in the upper part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' the Theta band, is predominantly influential for the causality regarding the sources, as well as the sinks and total activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' It has already been established that the right- lateralized Theta sampling network tends to involve slower temporal rates and codes the speech signal’s lower modulation frequencies 57, facilitating syllable-scale temporal integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' In other words, spoken sentences are tracked and distinguished by the Theta band phase pattern, allowing the incoming speech signal to be broken into syllable-sized packets and speech dynamics to be tracked through resetting and sliding, such as with varying rates of speech 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 5 (topoplots) clearly demonstrates that the C4 channel is the most interesting as it has the most Granger causality (causing and being caused) for all scenarios for the dyslexic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' For the sources, the frontal area contains other noteworthy channels (FP2, F7, F3 and Fz) that show differences between the control and dyslexic groups in terms of activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The most influential channels in the sinks are F3 and F4 (frontal area) and P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Hence, it seems pertinent to suggest that the main differences in the causality relationships of the Theta band lie in the so-called dorsal and ventral pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' In particular, the right area seems critical, as evidenced in Thetaband-Sources 8 7 Score on original data: o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='77 6 (p-value:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='002) 5 I 4 - FE 2 1 0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='45 0.' metadata={'source': 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original data: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='84 6 - (p-value:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} 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original data: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='81 (p-value: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='001) 4 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='85 Accuracy scoreGammaband-Totalactivity 8 7 Scoreor n original data: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='82 6 · (p-value:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='001) 3 2 1 - 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='80 Accuracy scoreRodríguez-Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 12 the prior research and especially demonstrated here with the sinks scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Another interesting result worth discussing is that for the Beta band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Here, more activity was observed for all three scenarios in the dyslexic group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' this agrees with the results for the Theta band as well as those from previous studies 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' For the sources, differences in the causal relationships were mainly identified in the C3 and C4 channels, pointing to areas responsible for motor processing 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' It is becoming increasingly clear that speech perception is at least partially located in the motor areas, especially under less-than-optimal listening conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' This cruciality of the C4 channel was similarly seen in the Theta band and is in line with prior research evidencing the important role played by the lower frequency bands in general and Beta band coupling in particular 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Hence, inefficient phase locking in the auditory cortex may affect visual and motor processing development, which may in turn cause some of the visual, motor and attentional difficulties seen in DD 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' It should be noted, however, that the C3-C4 interaction is mostly relevant for the sources and is not important for either the C3 for the sinks or, as a result, for the total activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Meanwhile, the causal activity in the Beta band is different in the occipital area in the sinks scenario, and it is remarkably different in the frontal area, especially in FP1 for all three scenarios and in the F3 channel for the sinks scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' In the Gamma band the activity is higher than in the Theta band for maximum values, although the occipital area shows more concentrated activity among the causality relations, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 5 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Nevertheless, the effect is different between the control and dyslexic groups, whereby the participants with DD show higher activity for the sinks, which increases their total activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' For the sources, the channels with the most explicit differences are FC1 and, more generally, TP9 in the left temporal area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' In the case of sinks, this is also an important channel, although O1 and, as highlighted above, C3 also play a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Meanwhile, in the Gamma band, despite the discrepancies between the dorsal and ventral pathways, the latter offers the main difference for the classification of TP9 for both sources and sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' FC1 is linked to sources and C3 to sinks, suggesting a significant cause- effect relationship, albeit with potentially less activity in the dyslexic group, facilitating classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We can confirm that the classifier performs better in the Theta and Gamma bands, which can evidence atypical oscillatory differences based on both speech and non- speech stimuli 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' According to Leong’s models 62, the slower rates (Delta and Theta) temporally constrain entrainment at the faster rates, such as Gamma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Lehongre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 65 contended that the oscillatory nesting seen between the Theta/Delta phase and the Gamma power 63 64 offers a way to integrate information at the phonemic (Gamma) rate into the syllabic rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Meanwhile, the integration of the various acoustic features that contribute to the same phoneme being perceived may be hindered by impairments in the phase locking by Theta generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Otherwise, flaws in certain Theta mechanisms could influence the development of the phonological system, which thus tends to code information bilaterally with the Gamma oscillations independently and then link them perceptually with the Theta oscillator output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' In this case, the impaired phase locking of the right hemisphere Theta oscillatory networks causes difficulties with lower frequency modulations 17 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' In addition, the spontaneous oscillatory neural activity identified in the auditory cortex in both the Theta and Gamma bands is known to associate with spontaneous activity in the visual and premotor areas 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' A bilateral Gamma sampling network codes the signal’s higher frequency modulations, thereby facilitating temporal integration at the phonetic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=', phoneme) scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' If we apply this model to DD, it is indicated that impaired processing at the syllable level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=', less efficient Theta phase locking) occurs alongside unimpaired Gamma sampling, meaning more weight is assigned to phonetic feature information during phonological development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Hence, as is the case in typical infant development, children with DD may have sensitivity to all phonetic contrasts of human languages 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Leong and Goswami 62 found that learners with DD show a preference for different phase alignment between amplitude modulations (AMs) when these respectively convey syllable and phoneme information (Theta and Gamma-AMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' A different phase locking angle suggests a discrepancy in the integration of speech information that arrives at a temporal rate different to that of the final perception of the speech 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Our results concerning the interaction between the Theta and Gamma bands support this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Finally, our results also seem to confirm that the dyslexic brain is less efficient at encoding the amplitude modulation hierarchy’s highest levels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' those bearing information on the prosodic-syllabic structure, leading to cascade effects that impact the encoding of the phonological structure’s levels nested within the Delta band, such as the syllable-level (Theta band) and phoneme-level (Gamma band) AM information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Importantly, our results have been validated using a demanding permutation test, with the aim of ensuring that the results are not coincidental, despite the medium sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Neural source/sink phase connectivity in Developmental Dyslexia 13 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Conclusion and future works Our results support the main assumption of the TSF that DD involves a specific deficit in the low-frequency phase locking mechanisms in the auditory cortex, thereby potentially affecting phonological development 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' In confirmation of this, we find an anomaly that emerges primarily in the causal relationships of channels that function as sinks, which is significantly more pronounced than when only the total activity is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Hence, it is reasonable to consider a division into Granger-causing or Granger-caused relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' This, in turn, suggests that the main differences contributing to DD emerge when certain brain areas must function as receptors in the interactions between channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Furthermore, our results are in line with previous research, which has already detected an anomaly in the right-lateralized Theta band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We have clearly identified this here across all three scenarios (sources, sinks, total activity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' We also find confirmation for the higher brain activity in learners with DD, although differences are more significant for the sinks in the Theta and Gamma bands, in turn leading to more total activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The highest classifier performance (accuracy and AUC) is hereby found in the sink scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' For the Beta band, the difference in activity is more consistent across all three scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The classifier also performs well for the Beta band in all three scenarios, with few differences observed, thereby confirming the important role played by this band in the sensorimotor coding of speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' The results reflect the causal activity generated in the brain subjected to prosodic-syllabic stimulus at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='8 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Consequently, future work could consider the Granger causality relationships in the phases across channels and bands using higher frequency stimuli to stimulate syllabic-phonetic and phonetic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Acknowledgements This work was supported by projects PGC2018-098813- B-C32 (Spanish “Ministerio de Ciencia, Innovación y Universidades”), UMA20-FEDERJA-086 (Consejería de econnomía y conocimiento,Junta de Andalucía) and by European Regional Development Funds (ERDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' References 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Kuhl PK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Early language acquisition: cracking the speech code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' Nature reviews neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} +page_content='5:831–843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfq_lh/content/2301.00552v1.pdf'} diff --git a/BdFJT4oBgHgl3EQfsS2B/content/tmp_files/2301.11612v1.pdf.txt b/BdFJT4oBgHgl3EQfsS2B/content/tmp_files/2301.11612v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d991a73fd89d3518a3a22bbcdfada5d85bacbb33 --- /dev/null +++ b/BdFJT4oBgHgl3EQfsS2B/content/tmp_files/2301.11612v1.pdf.txt @@ -0,0 +1,1807 @@ +A neural network potential with self-trained atomic fingerprints: +a test with the mW water potential +Francesco Guidarelli Mattioli, Francesco Sciortino, and John Russo∗ +Sapienza University of Rome, Piazzale Aldo Moro 2, 00185 Rome, Italy +(Dated: January 30, 2023) +We present a neural network (NN) potential based on a new set of atomic fingerprints built upon +two- and three-body contributions that probe distances and local orientational order respectively. +Compared to existing NN potentials, the atomic fingerprints depend on a small set of tuneable +parameters which are trained together with the neural network weights. To tackle the simultaneous +training of the atomic fingerprint parameters and neural network weights we adopt an annealing +protocol that progressively cycles the learning rate, significantly improving the accuracy of the NN +potential. We test the performance of the network potential against the mW model of water, which +is a classical three-body potential that well captures the anomalies of the liquid phase. Trained on +just three state points, the NN potential is able to reproduce the mW model in a very wide range of +densities and temperatures, from negative pressures to several GPa, capturing the transition from +an open random tetrahedral network to a dense interpenetrated network. The NN potential also +reproduces very well properties for which it was not explicitly trained, such as dynamical properties +and the structure of the stable crystalline phases of mW. +I. +INTRODUCTION +Machine learning (ML) potentials represent one of the +emerging trends in condensed matter physics and are +revolutionising the landscape of computational research. +Nowadays, different methods to derive ML potentials +have been proposed, providing a powerful methodology +to model liquids and solid phases in a large variety of +molecular systems [1–16, 16, 17]. Among these methods, +probably the most successful representation of a ML po- +tential so far is given by Neural Network (NN) potentials, +where the potential energy surface is the output of a feed- +forward neural network [18–35]. +In short, the idea underlying NN potentials construc- +tion is to train a neural network to represent the po- +tential energy surface of a target system. +The model +is initially trained on a set of configurations generated +ad-hoc, for which total energies and forces are known, +by minimizing a suitable defined loss-function based on +the error in the energy and force predictions. +If the +training set is sufficiently broad and representative, the +model can then be used to evaluate the total energy and +forces of any related atomic configuration with an accu- +racy comparable to the original potential. Typically the +original potential will include additional degrees of free- +dom, such as the electron density for DFT calculations, +or solvent atoms in protein simulations, which make the +full computation very expensive. +By training the net- +work only on a subset of the original degrees of freedom +one obtaines a coarse-grained representation that can be +simulated at a much reduced computational cost. NN +potentials thus combine the best of two worlds, retain- +ing the accuracy of the underlying potential model, at +the much lower cost of coarse-grained classical molecu- +∗Corresponding author: john.russo@uniroma1.it +lar dynamics simulations. The accuracy of the NN po- +tential depends crucially on how local atomic positions +are encoded in the input of the neural network, which +needs to retain the symmetries of the underlying Hamil- +tonian, i.e. rotational, translational, and index permu- +tation invariance. Several methods have been proposed +in the literature [12, 36], such as the approaches based +on the Behler-Parrinello (BP) symmetry functions [18], +the Smooth Overlap of Atomic Positions (SOAP) [37], +N-body iterative contraction of equivariants (NICE) [38] +and polynomial symmetry functions [39], or frameworks +like the DeepMD [23], SchNet [22] and RuNNer [18]. In +all cases, atomic positions are transformed into atomic +fingerprints (AFs). The choice of the AFs is particularly +relevant, as it greatly affects the accuracy and generality +of the resulting NN potential. +We develop here a fully learnable NN potential in +which the AFs, while retaining the simplicity of typi- +cal local fingerprints, do not need to be fixed beforehand +but instead are learned during the training procedure. +The coupled training of the atomic fingerprint param- +eters and of the network weights makes the NN train- +ing process more efficient since the NN representation is +spontaneously built on a variable atomic fingerprint rep- +resentation. To tackle the combined minimization of the +AF parameters and of the network weights we adopt an +efficient annealing procedure, that periodically cycles the +learning rate, i.e. the step size of the minimization algo- +rithm, resulting in a fast and accurate training process. +We validate the NN potential on the mW model of +water [40], which is a one-site classical potential that +has found widespread adoption to study water’s anoma- +lies [41, 42] and crystallization phenomena [43, 44]. Since +the first pioneering MD simulations [45, 46], water is of- +ten chosen as a prototypical case study, as the large num- +ber of distinct local structures that are compatible with +its tetrahedral coordination make it the molecule with +the most complex thermodynamic behavior [47], for ex- +arXiv:2301.11612v1 [cond-mat.soft] 27 Jan 2023 + +2 +ample displaying a liquid-liquid critical point at super- +cooled conditions [48–52]. NN potentials for water have +been developed starting from density functional calcu- +lations, with different levels of accuracy [53–60]. +NN +potentials have also been proposed to parametrise accu- +rate classical models for water with the aim of speeding +up the calculations when multi-body interactions are in- +cluded [61], as in the MBpol model [62–64] or for testing +the relevance of the long range interactions, as for the +SPC/E model [65]. We choose the mW potential as our +benchmark system because its explicit three-body poten- +tial term offers a challenge to the NN representation that +is not found in molecular models built from pair-wise +interactions. We stress that we train the NN-potential +against data which can be generated easily and for which +structural and dynamic properties are well known (or +can be evaluated with small numerical errors) in a wide +range of temperatures and densities. In this way, we can +perform a quantitative accurate comparison between the +original mW model and the hereby proposed NN model. +Our results show that training the NN potential at +even just one density-temperature state point provides +an accurate description of the mW model in a surround- +ing phase space region that is approximately a hundred +kelvins wide. A training based on three different state +points extends the convergence window extensively, ac- +curately reproducing state points at extreme conditions, +i.e. +large negative and (crushingly) positive pressures. +We will show that the NN reproduces thermodynamic, +structural and dynamical properties of the mW liquid +state, as well as structural properties of all the stable +crystalline phases of mW water. +The paper is organized as follows. In Section II we de- +scribe the new atomic fingerprints and the details about +the Neural Network potential implementation, including +the warm restart procedure used to train the weights +and the fingerprints at the same time. +In Section III +we present the results, which include the accuracy of the +models built from training sets that include one or three +state points, and a comparison of the thermodynamic, +structural and dynamic properties with those of the orig- +inal mW model. We conclude in Section IV. +II. +THE NEURAL NETWORK MODEL +The most important step in the design of a feed- +forward neural network potential is the choice on how to +define the first and the last layers of the network, respec- +tively named the input and output layers. We start with +the output layer, as it determines the NN potential ar- +chitecture to be constructed. Here we follow the Behler +Parrinello NN potential architecture [18], in which the +total energy of the system is decomposed as the sum of +local fields (Ei), each one representing the contribution of +a local environment centered around atom i. Being this +a many-body contribution, it is important to note that +Ei is not the energy of the single atom i, but of all its +environment (see also the Appendix A). With this choice, +the total energy of the system is simply the sum over all +atoms, E = � Ei, and the force ⃗fi acting on atom i is +the negative gradient of the total energy with respect to +the coordinates ν of atom i, e.g. fiν = ∂E/∂xiν. We +have to point out that a NN potential is differentiable +and hence it is possible to evaluate the gradient of the +energy analytically. This allows to compute forces of the +NN potential in the same way of other force fields, e.g. +by the negative gradient of the total potential energy. +The input layer is built from two-body (distances) and +three-body (angles) descriptors of the local environment, +⃗D(i) and T (i) respectively, ensuring translational and ro- +tational invariance. The first layer of the neural network +is the Atomic Fingerprint Constructor (AFC), as shown +in Fig. 1, which applies an exponential weighting on the +atomic descriptors, restoring the invariance under per- +mutations of atomic indexes. The outputs of this first +layer are the atomic fingerprints (AFs) and in turn these +are given to the first hidden layer. We will show how +this organization of the AFC layer allows for the inter- +nal parameters of the exponential weighting to be trained +together with the weights in the hidden layers of the net- +work. In the following we describe in detail the construc- +tion of the inputs and the calculation flow in the first +layers. +A. +The atomic fingerprints +The choice of input layer presents considerably more +freedom, and it is here that we deviate from previous NN +potentials. The data in this layer should retain all the in- +formation needed to properly evaluate forces and energies +of the particles in the system, possibly exploiting the in- +ternal symmetries of the Hamiltonian (which in isotropic +fluids are the rotational, translational and permutational +invariance) to reduce the number of degenerate inputs. +Given that the output was chosen as Ei, the energy of the +atomic environment surrounding atom i, the input uses +an atom-centered representation of the local environment +of atom i. +In the input layer, we define an atom-centered repre- +sentation of the local environment of atom i, consider- +ing both the distances rij with the nearest neighbours j +within a spatial cut-off Rc, and the angles θjik between +atom i and the pair of neighbours jk that are within a +cut-off Rc′. More precisely, for each atom j within Rc +from i we calculate the following descriptors +D(i) +j (rij; Rc) = +� +1 +2 +� +1 + cos +� +π rij +Rc +�� +rij ≤ Rc +0 +rij > Rc +(1) +and, for each triplet j − i − k within Rc′ from i, +T (i) +jk (rij, rik, θjik) = +(2) +1 +2 [1 + cos (θjik)] D(i) +j (rij; Rc +′) D(i) +k (rik; Rc +′) + +3 +C +Ei +α +γ +δ +β +Compression +⃗ +D(i) +⃗ +D(i) +T(i) +θjik +rij +≡ ( +⃗ +D(i), T(i)) +A +B +Atomic Fingerprint +Constructor +Hidden + layers +FIG. 1: Schematic representation of the Neural Network Potential flow. (A) Starting from the relative distances and the triplets +angles between neighbouring atoms, the input layer evaluates the atomic descriptors ⃗D(i) = {D(i) +j } (Eq. 1) and T (i) = {T (i) +jk } +(Eq. 2). (B) The first layer is the Atomic Fingerprint Constructor (AFC) and it combines the atomic descriptors into atomic +fingerprints, weighting them with an exponential function. The red nodes perform the calculation of Eq. 5, where from the two- +body descriptors a weighting vector ⃗D(i) +w (α) = {eαD(i) +j } is calculated (square with α) and then the scalar product ⃗D(i) · ⃗D(i) +w (α) +is computed (square with point) and finally a logarithm is applied (circle). The blue nodes perform the calculation of Eq. 7, +where two weighting vectors are calculated from the two-body descriptors namely ⃗D(i) +w (γ) and ⃗D(i) +w (δ) and one weighting +matrix from the three-body descriptors T (i) +w (β) = {eβT (i) +jk /2}. Finally in the compression unit (Eq 6) values are combined as +0.5[ ⃗D(i) ◦ ⃗D(i) +w (γ)]T [T (i) ◦T (i) +w (β)][ ⃗D(i) ◦ ⃗D(i) +w (δ)] where we use the circle symbol for the element-wise multiplication. The output +value of the compression unit is given to the logarithm function (circle). The complete network (D) is made of ten AFC units +and two hidden layers with 25 nodes per layer and here is depicted 2.5 times smaller. +Here i indicates the label of i-th particle while in- +dex j and k run over all other particles in the system. +In Eq. 1, D(i) +j (rij; Rc) is a function that goes continu- +ously to zero at the cut-off (including its derivatives). +The choice of this functional form guarantees that D(i) +j +is able to express contributions even from neighbours +close to the cut-off. +Other choices, based on polyno- +mials or other non-linear functions, have been tested +in the past [31]. +For example, we tested a parabolic +cutoff function which produced considerably worse re- +sults than the cutoff function in Eq. 1. +The function +T (i) +jk (rij, rik, θjik) is also continuous at the triplet cutoff +R′ +c. +The angular function +1 +2 [1 + cos (θjik)] guarantees +that 0 ≤ T (i) +jk (rij, rik, θjik) ≤ 1. We note that the use of +relative distances and angles in Eq. 1-2 guarantees trans- +lational and rotational invariance. +The pairs and triplets descriptors are then fed to the +AFC layer to compute the atomic fingerprints, AFs. +These are computed by projecting the D(i) +j +and T (i) +jk de- +scriptors on a exponential set of functions defined by + +4 +D +(i)(α) = ln +� +�� +j̸=i +D(i) +j eαD(i) +j ++ ϵ +� +� − Zα +(3) +T +(i)(β, γ, δ) = ln +� +� � +j̸=k̸=i +T (i) +jk eβT (i) +jk eγD(i) +j eδD(i) +k +2 ++ ϵ +� +�(4) +−Zβγδ +These AFs are built summing over all pairs and all +triplets involving particle i, making them invariant un- +der permutations, and multiplying each descriptor by an +exponential filter whose parameters are called α for dis- +tance AFs, and β, γ, δ for the triplet AFs. These param- +eters play the role of feature selectors, i.e. by choosing +an appropriate list of α, β, γ, δ the AFs can extract the +necessary information from the atomic descriptors. The +best choice of α, β, γ, δ will emerge automatically dur- +ing the training stage. In Eqs. 3-4, the number ϵ is set to +10−3 and fixes the value of energy in the rare event that +no neighbors are found inside the cutoff. Parameters Zα +and Zβγδ are optimized during the training process, shift- +ing the AFs towards positive or negative values, and act +as normalization factors that improve the representation +of the NN. +The definitions in equations 3-4 can be reformulated in +terms of product between vectors and matrices in the fol- +lowing way. The descriptors in equations 1-2 for particle i +can be represented as a vector ⃗D(i) = {D(i) +j } and a matrix +T (i) = {T (i) +jk } respectively. Given a choice of α, β, γ and +δ, three weighting vector ⃗D(i) +w (α) = {eαD(i) +j }, ⃗D(i) +w (γ) = +{eγD(i) +j } and ⃗D(i) +w (δ) = {eδD(i) +j } and one weighting ma- +trix T (i) +w (β) = {eβT (i) +jk /2} are calculated from ⃗D(i) and +T (i). +The 2-body atomic fingerprint (Eq. 3) is finally +computed as +D +(i)(α) = ln +� +⃗D(i) · ⃗D(i) +w (α) + ϵ +� +− Zα +(5) +The 3-body atomic fingerprint (Eq. 4) is computed first +by what we call compression step in Fig. 1 as +T c +(i) = [ ⃗D(i) ◦ ⃗D(i) +w (γ)]T [T (i) ◦ T (i) +w (β)][ ⃗D(i) ◦ ⃗D(i) +w (δ)] +2 +(6) +and finally by +T +(i)(β, γ, δ) = ln +� +T c +(i)(β, γ, δ) + ϵ +� +− Zβγδ +(7) +where we use the circle symbol for the element-wise mul- +tiplication. The NN potential flow is depicted in Figure +1 following the vectorial representation. +In summary, our AFs select the local descriptors use- +ful for the reconstruction of the potential by weight- +ing them with an exponential factor tuned with expo- +nents α, β, γ, δ. A similar weighting procedure has been +showed to be extremely powerful in the selection of com- +plex patterns and is widely applied in the so-called atten- +tion layer first introduced by Google Brain [66]. However +the AFC layer imposes additionally physically motivated +constraints on the neural network representation. +We note that the expression for the system energy is a +sum over the fields Ei, but the local fields Ei are not addi- +tive energies, involving all the pair distances and triplets +angles within the cut-off sphere centered on particle i. +This non-additive feature favours the NN ability to cap- +ture higher order correlations (multi-body contribution +to the energy), and has been shown to outperform ad- +ditive models in complex datasets [67]. +The NN non- +additivity requires the derivative of the whole energy E +(as opposed to Ei) to estimate the force on a particle i. In +this way, contributions to the force on particle i come not +only from the descriptors of i but also from the descrip- +tors of all particles who have i as a neighbour, de facto +enlarging the effective region in space where interaction +between particles are included. This allows the network +to include contributions from length-scales larger than +the cutoffs that define the atomic descriptors. The Ap- +pendix A provides further information on this point. +B. +Hidden layers +We employ a standard feed-forward fully-connected +neural network composed of two hidden layers with 25 +nodes per layer and using the hyperbolic tangent (tanh) +as the activation function. The nodes of the first hidden +layer are fully connected to the ones in the second layer, +and these connections have associated weights W which +are optimized during the training stage. +The input of the first hidden layer is given by the AFC +layer where we used five nodes for the two-body AFs +(Eq. 3) and five nodes for the three-body AFs (Eq. 4) +for a total of 10 AFs for each atom. +We explore the +performance of some combinations for the number of two- +body and three-body AF in Appendix D and we find that +the choice of five and five is the more efficient. +The output is the local field Ei, for each atomic envi- +ronment i, whose sum E = �N +i=1 Ei represents the NN +estimate of the potential energy E of the whole system. +C. +Loss function and training strategy +To train the NN-potential we minimize a loss function +computed over nf frames, i.e. the number of independent +configurations extracted from an equilibrium simulation +of the liquid phase of the target potential (in our case +the mW potential). The loss function is the sum of two +contributions. +The first contribution, H[{∆ϵk, ∆f k +iν}], expresses the +difference in each frame k between the NN estimates and +the target values for both the total potential energy (nor- +malized by total number of atoms) ϵk and the atomic + +5 +forces f k +iν acting in direction ν on atom i. The nf energy +ϵk values and 3Nnf force f k +iν values are combined in the +following expression +H[{∆ϵk, ∆f k +iν}] = pe +nf +nf +� +k=1 +hHuber(∆ϵk) + +pf +3Nnf +nf +� +k=1 +N +� +i=1 +3 +� +ν=1 +hHuber(∆f k +iν) +(8) +where pe = 0.1 and pf = 1 control the relative contri- +bution of the energy and the forces to the loss function, +and hHuber(x) is the so-called Huber function +hHuber(x) = +� +0.5x2 if |x| ≤ 1 +0.5 + (|x| − 1) if |x| > 1 +(9) +pe and pf are hyper-parameters of the model, and we se- +lected them with some preliminary tests that found those +values to be near the optimal ones. +The Huber func- +tion [68] is an optimal choice whenever the exploration +of the loss function goes through large errors caused by +outliers, i.e. data points that differ significantly from pre- +vious inputs. Indeed when a large deviation between the +model and data occur, a mean square error minimization +may gives rise to an anomalous trajectory in parameters +space, largely affecting the stability of the training pro- +cedure. This may happen especially in the first part of +the training procedure when the parameter optimization, +relaxing both on the energy and forces error surfaces may +experience some instabilities. +The second contribution to the loss function is a reg- +ularization function, R[{αl, βm, γm, δm}], that serves to +limit the range of positive values of αl and of the triplets +βm, γm, δm (where the indexes l and m run over the five +different values of α and five different triplets of values +for β, γ and δ) in the window −∞ to 5. To this aim we +select the commonly used relu function +rrelu(x) = +� +x − 5 +if x > 5 +0 +if x ≤ 5 +(10) +(11) +and write +R[{αl, βm, γm, δm}] = +5 +� +l=1 +rrelu(αl) + +5 +� +m=1 +[rrelu(βm) + rrelu(γm) + rrelu(δm)] +(12) +Thus, the R function is activated whenever one param- +eters of the AFC layer becomes, during the minimization, +larger than 5. +To summarize, the global loss function L used in the +training of the NN is +L[ϵ, f] = H[{∆ϵk, ∆f k +iν}] + pbR[{αl, βm, γm, δm}] (13) +where pb = 1 weights the relative contribution of R com- +pared to H. +Compared to a standard NN-potential, we train not +only the network weights W but also the AFs param- +eters Σ ≡ {αl, βm, γm, δm} at the same time. The si- +multaneous optimization of the weights W and AFs Σ +prevents possible bottleneck in the optimisation of W at +fixed representation of Σ. Other NN potential approaches +implement a separate initial procedure to optimise the Σ +parameters followed by the optimisation of W at fixed +Σ [69]. The two-step procedure not only requires a spe- +cific methodological choice for optimising Σ, but also may +not result in the optimal values, compared to a search in +the full parameter space (i.e. both Σ and W). Since the +complexity of the loss function has increased, we have +investigated in some detail some efficient strategies that +lead to a fast and accurate training. Firstly, we initial- +ize the parameters W via the Xavier algorithm, in which +the weights are extracted from a random uniform distri- +bution [70]. +To initialize the Σ parameters we used a +uniform distribution in interval [−5, 5]. We then mini- +mize the loss function using the warm restart procedure +proposed in reference [71]. In this procedure, the learn- +ing rate η is reinitialized at every cycle l and inside each +cycle it decays as a function of the number of training +steps t following +η(l)(t) = Al +�(1 − ξf) +2 +� +1 + cos +�πt +Tl +�� ++ ξf +� +(14) +0 ≤ t ≤ Tl +where ξf = 10−7, Al = η0ξl +0 is the initial learning rate of +the l-th cycle with η0 = 0.01 and ξ0 = 0.9, Tl = bτ l is +the period of the l-th cycle with τ = 1.4 and b = 40. The +absolute number of training steps n during cycle l can be +calculated summing over the length of all previous cycles +as n = τ + �l−1 +m=0 Tm. +We also select to evaluate the loss function for groups +of four frames (mini-batch) and we randomly select 200 +frames nf = 200 for a system of 1000 atoms and hence +we split this dataset in 160 frames (%80) for the training +set and the 40 frames (%20) for the test set. +In Fig. 2(A) we represent the typical decay of the learn- +ing rate of the warm restart procedure, which will be +compared to the standard exponential decay protocol in +the Results section. +D. +The Target Model +To test the quality of the proposed novel NN we train +the NN with data produced with the mW [40] model + +6 +100 +101 +102 +103 +104 +105 +n +0.000 +0.005 +0.010 +A +0 +1000 +2000 +3000 +ne +10 +1 +100 +B +Validation Loss +Training Loss +0 +1000 +2000 +3000 +ne +10 +2 +10 +1 +100 +101 + (kcal mol +1) +C +0 +1000 +2000 +3000 +ne +101 +f (kcal mol +1 nm +1) +D +FIG. 2: +Model convergence properties: +(A) Learning rate +schedule (Eq. 14) as a function of the absolute training step n +(one step is defined as an update of the network parameters). +(B) The training and validation loss (see L[ϵ, f] in Eq. 13) evo- +lution during the training procedure, reported as a function +of the number of epoch ne (an epoch is defined as a complete +evaluation of the training dataset). Root mean square (RMS) +error of the total potential energy per particle (C) and of the +force cartesian components (D) during the training evaluated +in the test dataset. Data in panels B-C-D refers to the NN3 +model and the green point shows the best model location. +of water. +This potential, a re-parametrization of the +Stillinger-Weber model for silicon [72], uses a combina- +tion of pairwise functions complemented with an additive +three-body potential term +0 +20 +40 +60 +Seed +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 + (kcal mol +1) +A +Exponential +Warm restart +0 +20 +40 +60 +Seed +1.4 +1.7 +2.0 +2.3 +2.6 +2.9 +3.2 +3.5 +3.8 +f (kcal mol +1 nm +1) +B +FIG. 3: Comparison of the root mean square error calculated +on the validation set for 60 replicas differing in the initial +seed of the training procedure using both an exponential de- +cay of the learning rate (points) and the warm restart method +(squares), for the energy (panel A) and for the forces (panel +B). For the forces, a significant improvement both in the av- +erage error and in its variance is found for the warm restart +schedule. +E = +� +i +� +j>i +U2(rij)+λ +� +i +� +j̸=i +� +j>k +U3 (rij, rik, θjik) (15) +where the two body contribution between two particles +i and j at relative distance rij is a generalized Lennard- +Jones potential +U2 (rij) = Aϵ +� +B +� σ +rij +�p +− +� σ +rij +�q� +exp +� +σ +rij − aσ +� +(16) +where the p = 12 and q = 6 powers are substituted +by q = 0 and p = 4, multiplied by an exponential cut- +off that brings the potential to zero at aσ, with a = +1.8 and σ = 2.3925 ˚A. Aϵ (with A = 7.049556277 and +ϵ = 6.189 kcal mol−1) controls the strength of the two +body part. B controls the two-body repulsion (with B = +0.6022245584). +The three body contribution is computed from all pos- +sible ordered triplets formed by the central particle with +the interacting neighbors (with the same cut-off aσ as the +two-body term) and favours the tetrahedral coordination +of the atoms via the following functional form +U3 (rij, rik, θjik) = ϵ [cos (θjik) − cos (θ0)]2 × +exp +� +γσ +rij − aσ +� +exp +� +γσ +rik − aσ +� +(17) +where θjik is the angle formed in the triplet jik and +γ = 1.2 controls the smoothness of the cut-off function +on approaching the cut-off. Finally, θ0 = 109.47◦ and +λ = 23.15 controls the strength of the angular part of +the potential. + +7 +The mW model, with its three-body terms centered +around a specific angle and non-monotonic radial interac- +tions, is based on a functional form which is quite differ- +ent from the radial and angular descriptors selected in the +NN model. The NN is thus agnostic with respect to the +functional form that describes the physical system (the +mW in this case). But having a reference model with ex- +plicit three body contributions offers a more challenging +target for the NN potential compared to potential models +built entirely from pairwise interactions. The mW model +is thus an excellent candidate to test the performance of +the proposed NN potential. +III. +RESULTS +A. +Training +We study two different NN models, indicated with the +labels NN1 and NN3, differing in the number of state +points included in the training set. These two models are +built with a cut-off of Rc = 4.545 ˚A +for the two-body +atomic descriptors and a cut-off of R′ +c = 4.306 ˚A +for +the three-body atomic descriptors. R′ +c is the same as the +mW cutoff while Rc was made slightly larger to miti- +gate the suppression of information at the boundaries by +the cutoff functions. +The NN1 model uses only train- +ing information based on mW equilibrium configurations +from one state point at ρ1 = 1.07 g cm−3, T1 = 270.9 K +where the stable phase is the liquid. +The NN3 model +uses training information based on mW liquid configura- +tions in three different state points, two state points at +ρ1 = 0.92 g cm−3, T1 = 221.1 K and ρ2 = 0.92 g cm−3, +T2 = 270.9 K where the stable solid phase is the clathrate +Si34/Si136 [73] and one state point at ρ3 = 1.15 g cm−3, +T2 = 270.9 K. +This choice of points in the phase diagram is aimed to +improve agreement with the low temperature-low density +as well as high density regions of the phase diagram. Im- +portantly, all configurations come from either stable or +metastable liquid state configurations. Indeed, the point +at ρ2 = 0.92 g cm−3, T2 = 270.9 K is quite close to +the limit of stability (respect to cavitation) of the liquid +state. +To generate the training set, we simulate a system of +N = 1000 mW particles with a standard molecular dy- +namics code in the NVT ensemble, where we use a time +step of 4 fs and run 107 steps for each state point. From +these trajectory, we randomly select 200 configurations +(frames) to create a dataset of positions, total energies +and forces. We then split the dataset in the training and +in the test data sets, the first one containing 80% of the +data. We then run the training for 4000 epochs with a +minibatch of 4 frames. At the end of every epoch, we +check if the validation loss is improved and we save the +model parameters. In Fig. 2 we plot the loss function for +the training and test datasets (B), the root mean square +error of the total energy per particle (C), and of the force +(D) for the NN3 model. The results show that the learn- +ing rate schedule of Eq. 14 is very effective in reducing +both the loss and error functions. +Interestingly, the neural network seems to avoid over- +fitting (i.e. the validation loss is decreasing at the same +rate as the loss on the training data), and the best model +(deepest local minimum explored), in a given window of +training steps, is always found at the end of that win- +dow, which also indicates that the accuracy could be fur- +ther improved by running more training steps. Indeed we +found that by increasing the number of training steps by +one order of magnitude the error in the forces decreases +by a further 30%. Similar accuracy of the training stage +is obtained also for the NN1 model (not shown). +The training procedure always terminates with an +error +on +the +test +set +equal +or +less +than +∆ϵ +≃ +0.01 kcal mol−1 (0.43 meV) for the energy, and of ∆f ≃ +1.55 kcal mol−1 nm−1 (6.72 meV ˚A−1) for the forces. +These values are comparable to the state-of-the-art NN +potentials [23, 54, 55, 61], and within the typical accuracy +of DFT calculations [74]. +We can compare the precision of our model with that +of alternative NN potentials trained on a range of water +models. An alternative mW neural network potential has +been trained on a dataset made of 1991 configurations +of 128 particles system at different pressure and tem- +perature (including both liquid and ice structures) with +Behler-Parinello symmetry functions [24]. +The train- +ing of this model (which uses more atomic fingerprints +and a larger cutoff radius) converged to an error in en- +ergy of ∆ϵ ≃ 0.0062 kcal mol−1 (0.27 meV), and ∆f ≃ +3.46 kcal mol−1 nm−1 (15.70 meV ˚A−1) for the forces. In +a recent work searching for liquid-liquid transition signa- +tures in an ab-initio water NN model [55], a dataset of +configurations spanning a temperature range of 0−600 K +and a pressure range of 0 − 50 GPa was selected. For +a system of 192 particles, the training converged to an +error in energy of ∆ϵ ≃ 0.010 kcal mol−1 (0.46 meV), +and ∆f +≃ +9.96 kcal mol−1 nm−1 (43.2 meV ˚A−1) +for the forces. +In the NN model of MB-POL [61], +a dataset spanning a temperature range from 198 K +to 368 K at ambient pressure was selected. +In this +case, for a system of 256 water molecule, an accu- +racy of ∆ϵ ≃ 0.01 kcal mol−1 (0.43 meV) and ∆f ≃ +10 kcal mol−1 nm−1 (43.36 meV ˚A−1) was reached. Fi- +nally, the NN for water at T = 300 K used in Ref. [54], +reached precisions of ∆ϵ ≃ 0.046 kcal mol−1 (2 meV) and +∆f ≃ 25.36 kcal mol−1 nm−1 (110 meV ˚A−1). +While a direct comparison between NN potentials +trained on different reference potentials is not a valid test +to rank the respective accuracies, the comparisons above +show that our NN potential reaches a similar precision +in energies, and possibly an improved error in the force +estimation. +The accuracy of the NN potential could be further +improved by extending the size of the dataset and the +choice of the state points. In fact, while the datasets in +Ref. [54, 55, 61] have been built with optimized proce- + +8 +dures, the dataset used in this study was prepared by +sampling just one (NN1) or three (NN3) state-points. +Also the size of the datasets used in the present work is +smaller or comparable to the ones of Ref. [54, 55, 61]. +In Fig. 3 we compare the error in the energies (A) +and the forces (B) between sixty independent training +runs using the standard exponential decay of the learn- +ing rate (points) and the warm restart protocol (squares). +The figure shows that while the errors in the energy com- +putations are comparable between the two methods, the +warm restart protocol allows the forces to be computed +with higher accuracy. Moreover we found that the warm +restart procedure is less dependent on the initial seed and +that it reaches deeper basins than the standard exponen- +tial cooling rate. +B. +Comparing NN1 with NN3 +The NN potential model was implemented in a custom +MD code that makes use of the tensorflow C API [75]. +We adopted the same time step (4 fs), the same number +of particles (N = 1000) and the same number of steps +(107) as for the simulations in the mW model. +As described in the Training Section, we compare the +accuracy of two different training strategies: NN1 which +was trained on a single state point, and NN3 which is +instead trained on three different state point. In Fig. 4 +we plot the energy error (∆ϵ) between the NN potential +and the mW model with both NN1 (panel A) and NN3 +(panel B). Starting from NN1, we see that the model +already provides an excellent accuracy for a large range +of temperatures and for densities close to the training +density. The biggest shortcoming of the NN1 model is +at densities lower than the trained density, where the +NN potential model cavitates and does not retain the +long-lived metastable liquid state displayed by the mW +model. We speculate that this behaviour is due to the +absence of low density configurations in the training set, +which prevents the NN potential model from correctly +reproducing the attractive tails of the mW potential. +To overcome this limitation we have included two ad- +ditional state points at low density in the NN3 model. In +this case, Fig. 4B shows that NN3 provides a quite ac- +curate reproduction of the energy in the entire explored +density and temperature window (despite being trained +only with data at ρ = 0.92 g cm−3 and ρ = 1.15 g cm−3). +We can also compare the accuracy obtained during +production runs against the accuracy reached during +training, which was ∆ϵ ≃ 0.01 kcal mol−1. Fig. 4B shows +the error is of the order of 0.032 kcal mol−1 (1.3 meV), for +density above the training set density. But in the density +region between 0.92 and 1.15, the error is even smaller, +around 0.017 kcal mol−1 (0.7 meV) at the lowest density +boundary. +We can thus conclude that the NN3 model, which adds +to the NN1 model information at lower density and tem- +perature, in the region where tetrahedality in the wa- +0.9 +1.0 +1.1 +1.2 + (g cm +3) +385 +365 +345 +325 +305 +285 +265 +245 +225 +T (K) +A +0.9 +1.0 +1.1 +1.2 + (g cm +3) +B +0.001 +0.010 +0.020 +0.030 +0.040 + (kcal mol +1) +FIG. 4: Comparison between the mW total energy and the +NN1 model (A) and NN3 model (B) for different temperatures +and densities. While the NN3 model is able to reproduce the +mW total energy with a good agreement in a wide region of +densities and temperatures, the NN1 provide a good repre- +sentation only in a limited region of density and temperature +values. Blue squares represent the state points used for build- +ing the NN models. +ter structure is enhanced, is indeed capable to represent, +with only three state points, a quite large region of the +phase space, encompassing dense and stretched liquid +states. This suggests that a training based on few state +points at the boundary of the density/temperature re- +gion which needs to be studied is sufficient to produce a +high quality NN model. In the following we focus entirely +on the NN3 model. +C. +Comparison of thermodynamic, structural and +dynamical quantities +In Fig. 5 we present a comparison of thermodynamic +data between the mW model (squares) and its NN poten- +tial representation (points) across a wide range of state +points. Fig. 5A plots the energy as function of density +for temperatures ranging from melting to deeply super- +cooled conditions. Perhaps the most interesting result is +that the NN potential is able to capture the energy min- +imum, also called the optimal network forming density, +which is a distinctive anomalous property of water and +other empty liquids [76]. +Fig. 5(B) shows the pressure as a function of the tem- +perature for different densities, comparing the mW with +the NN3 model. Also the pressure shows a good agree- +ment between the two models in the region of densities +between ρ = 0.92 g cm−3 and ρ = 1.15 g cm−3, which, as +for the energy, tends to deteriorate at ρ = 1.22 g cm−3. +In the large density region explored, the structure of +the liquid changes considerably. On increasing density, a +transition from tetrahedral coordinated local structure, +prevalent at low T and low ρ, towards denser local envi- + +9 +0.85 +0.90 +0.95 +1.00 +1.05 +1.10 +1.15 +1.20 +1.25 + (g cm +3) +10.0 +9.5 +9.0 +8.5 + (kcal mol +1) +221.1K +233.6K +246.0K +258.5K +271.0K +299.0K +311.4K +373.7K +A +mW +NN3 +200 +225 +250 +275 +300 +325 +350 +375 +T (K) +0 +1 +2 +3 +4 +P (GPa) +0.92 g cm +3 +0.99 g cm +3 +1.07 g cm +3 +1.15 g cm +3 +1.19 g cm +3 +1.22 g cm +3 +B +FIG. 5: Comparison between the mW total energy and the +NN3 total energy as a function of density along different +isotherm (A) and comparison between the mW pressure and +the NN3 pressure as a function of temperature along differ- +ent isochores (B). The relative error of the NN vs the mW +potential grows with density, but remains within 3% even for +densities larger than the densities used in the training set. +ronments with interstitial molecules included in the first +coordination shell takes place. This structural change is +well displayed in the radial distribution function, shown +for different densities at fixed temperature in Fig. 6. +Fig. 6 also shows the progressive onset of a peak around +3.5 ˚A +developing on increasing pressure, which signals +the growth of interstitial molecules, coexisting with open +tetrahedral local structures [77, 78]. At the highest den- +sity, the tetrahedral peak completely merges with the +interstitial peak. The NN3 model reproduces quite ac- +curately all features of the radial distribution functions, +maxima and minima positions and their relative ampli- +tudes, at all densities, from the tetrahedral-dominated to +the interstitial-dominated limits. In general, NN3 model +reproduces quite well the mW potential in energies, pres- +sure and structures and it appreciably deviates from mW +pressures and energies quantities only at densities (above +1.15 g/cm3) which are outside of the training region. +To assess the ability of NN potential to correctly de- +scribe also the crystal phases of the mW potential, we +compare in Fig. 7 the g(r) of mW with the g(r) of the +NN3 model for four different stable solid phases [73]: +hexagonal and cubic ice (ρ = 1.00 g cm−3 and T = +246 K), the dense crystal SC16 (ρ = 1.20 g cm−3 and T = +234 K) and the clathrate phase Si136 (ρ = 0.80 g cm−3 +and T = 221 K). The results, shown in Fig. 7, show that, +despite no crystal configurations have been included in +the training set, a quite accurate representation of the +crystal structure at finite temperature is provided by the +NN3 model for all distinct sampled lattices. +0 +2 +4 +6 +8 +10 +12 +14 +16 +R (Å) +0 +1 +2 +3 +4 +5 +6 +7 +8 +RDF +0.92 g cm +3 +0.99 g cm +3 +1.15 g cm +3 +1.22 g cm +3 +mW +NN3 +FIG. 6: +Comparison between the mW radial distribution +functions g(r) and the NN3 g(r) at T = 270.9 K for four +different densities. +The tetrahedral structure (signalled by +the peak at 4.54 ˚A ) progressively weakens in favour of an in- +terstitial peak progressively growing at 3.5 − 3.8 ˚A . Different +g(r) have been progressively shifted by two to improve clarity. +0 +5 +10 +15 +R (Å) +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +RDF +Hexagonal diamond +Cubic diamond +Si136 Clathrate +SC16 +mW +NN3 +FIG. 7: +Comparison between the mW radial distribution +functions g(r) and the NN3 g(r) for four different lattices: +(A) hexagonal diamond (the oxygen positions of the ice Ih); +(B) cubic diamond (the oxygen positions of the ice Ic; (C) +the SC16 crystal (the dense crystal form stable at large pres- +sures in the mW model) and (D) the Si136 clathrate structure, +which is stable at negative pressures in the mW model. Dif- +ferent g(r) have been progressively shifted by four to improve +clarity. +Finally, we compare in Fig. 8 the diffusion coefficient +(evaluated from the long time limit of the mean square +displacement) for the mW and the NN3 model, in a wide +range of temperatures and densities, where water displays +a diffusion anomaly. +Fig. 8 shows again that, also for + +10 +0.85 +0.90 +0.95 +1.00 +1.05 +1.10 +1.15 +1.20 + (g cm +3) +0 +100 +200 +300 +400 +500 +600 +700 +800 +D (Å2 ns +1) +221.1K +233.6K +246.0K +258.5K +271.0K +299.0K +311.4K +373.7K +mW +NN3 +FIG. 8: Comparison between the mW diffusion coefficient D +and the NN3 corresponding quantity for different tempera- +tures and densities, in the interval 221 − 271 K. In this dy- +namic quantity, the relative error is, for all temperatures, +around 8%. Note also that in this T window the diffusion +coefficient shows a clear maximum, reproducing one of the +well-know diffusion anomaly of water. Diffusion coefficients +have been calculated in the NVT ensemble using the same +Andersen thermostat algorithm [79] for mW and NN3 poten- +tial. +dynamical quantities, the NN potential offers an excellent +representation of the mW potential, despite the fact that +no dynamical quantity was included in the training set. A +comparison between fluctuations of energy and pressure +of mW and NN3 potential is reported in Appendix B. +IV. +CONCLUSIONS +In this work we have presented a novel neural net- +work (NN) potential based on a new set of atomic fin- +gerprints (AFs) built from two- and three-body local de- +scriptors that are combined in a permutation-invariant +way through an exponential filter (see Eq. 3-4). One of +the distinctive advantages of our scheme is that the AF’s +parameters are optimized during the training procedure, +making the present algorithm a self-training network that +automatically selects the best AFs for the potential of in- +terest. +We have shown that the added complexity in the con- +current training of the AFs and of the NN weights can +be overcome with an annealing procedure based on the +warm restart method [71], where the learning rate goes +through damped oscillatory ramps. +This strategy not +only gives better accuracy compared to the commonly +implemented exponential learning rate decay, but also +allows the training procedure to converge rapidly inde- +pendently from the initialisation strategies of the model’s +parameters. +Moreover we show in Appendix C that the potential +hyper-surface of the NN model has the same smoothness +as the target model, as confirmed by (i) the possibility to +use the same timestep in the NN and in the target model +when integrating the equation of motion and (ii) by the +possibility of simulate the NN model even in the NVE +ensemble with proper energy conservation. +We test the novel NN on the mW model [40], a +one-component model system commonly used to de- +scribe water in classical simulations. This model, a re- +parametrization of the Stillinger-Weber model for sili- +con [72], while treating the water molecule as a simple +point, is able to reproduce the characteristic tetrahedral +local structure of water (and its distortion on increasing +density) via the use of three-body interactions. Indeed +water changes from a liquid of tetrahedrally coordinated +molecules to a denser liquid, in which a relevant fraction +of interstitial molecules are present in the first nearest- +neighbour shell. The complexity of the mW model, both +due to its functional form as well as to the variety of dif- +ferent local structures which characterise water, makes it +an ideal benchmark system to test our NN potential. +We find that a training based on configurations ex- +tracted by three different state points is able to pro- +vide a quite accurate representation of the mW poten- +tial hyper-surface, when the densities and temperatures +of the training state points delimit the region of in which +the NN potential is expected to work. We also find that +the error in the NN estimate of the total energy is low, +always smaller than 0.03 kcal mol−1, with a mean error +of 0.013 kcal mol−1. The NN model reproduces very well +not only the thermodynamic properties but also struc- +tural properties, as quantified by the radial distribution +function, and the dynamic properties, as expressed by +the diffusion coefficient, in the extended density interval +from ρ = 0.92 g cm−3 to ρ = 1.22 g cm−3. +Interestingly, we find that the NN model, trained only +on disordered configurations, is also able to properly +describe the radial distribution of the ordered lattices +which characterise the mW phase diagram, encompass- +ing the cubic and hexagonal ices, the SC16 and the Si136 +clathrate structure [73]. In this respect, the ability of the +NN model to properly represent crystal states suggests +that, in the case of the mW, and as such probably in the +case of water, the geometrical information relevant to the +ordered structures is contained in the sampling of phase +space typical of the disordered liquid phase. 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Engel, and +B. Cheng, Nature communications 11, 1 (2020). +V. +APPENDIX A +In this appendix we discuss the effective spacial range +covered by a NN potential whose fingerprints are defined +based on pair information confined within a sphere of +cutoff radius Rc. +As noted in reference [31], multi-body potentials and +especially non-additive multibody potentials induce lo- +cal interactions beyond the cut-off radius, enlarging the +sphere of interaction. +Indeed, the force on particle i +comes from the derivative of the local field of i and of +all its neighbours with respect to the coordinates of par- +ticle i. +2 +1 +3 +2 +1 +3 +4 +5 +A +B +FIG. 9: (A) Two-body interactions and (B) three-body inter- +actions in a non linear local field model Ei. The non linearity +of the local field enlarges the interaction cut-off where a neigh- +bour particle (blue) makes a bridge between non-neighboring +particle (red and blue). +Fig. 9 graphically explains the effective role of Rc in +the NN potential. +In panel A, we describe particle 1 +with only one neighbour (particle 2) within Rc. We also +represent the sphere centered on particle 3, which also +includes particle 2 as one of its neighbour. In this case, +the energy of the system will be represented as a sum +over the local fields E1, E2 and E3. Due to the intrinsic +non-linearity of the NN, the field Ei mixes together the +AFs, and consequently the distances and angles entering +in the AFs are non-linearly mixed in Ei. The force on +atom 1 is then written as +f1ν = −∂E1(r12) +∂x1ν +− ∂E2(r21, r23) +∂x1ν += −∂E1(r12) +∂x1ν +−∂E2(r21, r23) +∂r21 +∂r21 +∂x1ν +− ∂E2(r21, r23) +∂r23 +∂r23 +∂x1ν +(A1) +While the last term vanishes, the next to the last retains + +13 +A +B +FIG. 10: (A) Standard deviation of total energy (normal- +ized with the number of particles) and (B) standard deviation +of virial pressure for both NN3 model (red) and mW model +(black). +an intrinsic dependence on the coordinates both of par- +ticle 2 as well as of particle 3, if the local field E2 is non +linear. Thus, even if particle 3 is further than Rc, it en- +ters in the determination of the force acting on particle +1. A similar effect is also present in the angular part of +the AFs, as shown graphically in panel B. Indeed, for the +angular component of the AF the force on particle 1 is +f1ν = −∂E1(θ512) +∂x1ν +− ∂E2(θ123, θ124, θ324) +∂x1ν +. +(A2) +Also in this case two contributions can be separated: (i) +the interaction of particle 1 with triplets 123 and 124 is +an effect of the three-body AF and it is present also in +additive-models such as the mW model, (ii) the inter- +action of particle 1 with triplet 324 is an effect of the +non-additive nature of the NN local field Ei. +VI. +APPENDIX B +In this Appendix we provide further thermodynam- +ics comparisons between mW and NN3 potential focus- +ing on the pressure and energy fluctuations. We depict +in Fig. 10 the standard deviations of the total energy +(normalized by N) in panel (A) and the standard devia- +tion of virial pressure in panel (B). Energy fluctuations +of NN3 follow qualitatively and quantitatively the trend +of mW potential. +Pressure fluctuations of NN3 are in +good agreement with the mW model but, as for the pres- +sure (Fig. 5.B), the accuracy decreases approaching state +points outside the density range used for the training. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Time (ns) +10.2 +10.0 +9.8 +9.6 +9.4 +(kcal mol +1) +NN3 Total Energy +NN3 Potential Energy +mW Total Energy +mW Potential Energy +FIG. 11: NVE molecular dynamics at T = 299 K and ρ = +1.07 g cm−3 for both NN3 and mW model. The time step is +dt = 4 fs for both models. +VII. +APPENDIX C +In this Appendix we show a comparison between the +mW and NN3 potentials in terms of the energy conser- +vation in the NVE ensemble. In Fig. 11 we depict both +total energy and potential energy for mW and NN3 po- +tential. The potential energy and total energy of the two +models are in good agreement. +VIII. +APPENDIX D +In this Appendix we investigate the efficiency of the +training over different choices for the number and types +of atomic fingerprints introduced in the Neural Network +Model section. We start by using only one three-body +(n3b = 1) and one two-body (n2b = 1) AF and subse- +quently increasing the number of the AF. For every com- +bination of n2b and n3b, we run a 4000 epochs training +and at the end of each training we extract the best model. +We summarized these results in table I where we com- +pare the error on forces over the all investigated model. +From table I it emerges that the choice of n3b = 5 and +n2b = 5 is the more convenient both for accuracy and +computational efficiency. +Doubling the number of the +three-body AF marginally improves the error on forces +while increases the computational cost due to the increase +in the size of the input layer of the first hidden layer and +due to the additional time to compute the three-body +AF. Moreover in the RESULTS section we show that the +choice n3b = 5 and n2b = 5 is sufficient to represent the +target potential. Finally the accuracy of the training af- +ter doubling the configurations in the dataset reaches an +error on forces of ∆f = 5.85 meV ˚A−1 that is 0.87 times +the error value found with a half of the dataset. + +14 +TABLE I: +Table of errors on forces at the end of the 4000 +epoch-long training procedure for different combination of the +number and type of the AF. +n3b +n2b +∆f (meV ˚A−1) +n3b +n2b +∆f (meV ˚A−1) +1 +1 +72.79 +5 +1 +16.53 +1 +2 +67.92 +5 +2 +7.53 +1 +5 +56.25 +5 +5 +6.72 +1 +10 +56.00 +5 +10 +6.87 +1 +15 +56.02 +5 +15 +6.95 +2 +1 +53.76 +10 +1 +7.98 +2 +2 +43.95 +10 +2 +7.17 +2 +5 +32.43 +10 +5 +5.79 +2 +10 +32.39 +10 +10 +6.55 +2 +15 +24.70 +10 +15 +6.19 + diff --git a/BdFJT4oBgHgl3EQfsS2B/content/tmp_files/load_file.txt b/BdFJT4oBgHgl3EQfsS2B/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..028f172c2ff15f4287e588f518dd0a3dcdd93a47 --- /dev/null +++ b/BdFJT4oBgHgl3EQfsS2B/content/tmp_files/load_file.txt @@ -0,0 +1,1083 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf,len=1082 +page_content='A neural network potential with self-trained atomic fingerprints: a test with the mW water potential Francesco Guidarelli Mattioli, Francesco Sciortino, and John Russo∗ Sapienza University of Rome, Piazzale Aldo Moro 2, 00185 Rome, Italy (Dated: January 30, 2023) We present a neural network (NN) potential based on a new set of atomic fingerprints built upon two- and three-body contributions that probe distances and local orientational order respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Compared to existing NN potentials, the atomic fingerprints depend on a small set of tuneable parameters which are trained together with the neural network weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' To tackle the simultaneous training of the atomic fingerprint parameters and neural network weights we adopt an annealing protocol that progressively cycles the learning rate, significantly improving the accuracy of the NN potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We test the performance of the network potential against the mW model of water, which is a classical three-body potential that well captures the anomalies of the liquid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Trained on just three state points, the NN potential is able to reproduce the mW model in a very wide range of densities and temperatures, from negative pressures to several GPa, capturing the transition from an open random tetrahedral network to a dense interpenetrated network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The NN potential also reproduces very well properties for which it was not explicitly trained, such as dynamical properties and the structure of the stable crystalline phases of mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' INTRODUCTION Machine learning (ML) potentials represent one of the emerging trends in condensed matter physics and are revolutionising the landscape of computational research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Nowadays, different methods to derive ML potentials have been proposed, providing a powerful methodology to model liquids and solid phases in a large variety of molecular systems [1–16, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Among these methods, probably the most successful representation of a ML po- tential so far is given by Neural Network (NN) potentials, where the potential energy surface is the output of a feed- forward neural network [18–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In short, the idea underlying NN potentials construc- tion is to train a neural network to represent the po- tential energy surface of a target system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The model is initially trained on a set of configurations generated ad-hoc, for which total energies and forces are known, by minimizing a suitable defined loss-function based on the error in the energy and force predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' If the training set is sufficiently broad and representative, the model can then be used to evaluate the total energy and forces of any related atomic configuration with an accu- racy comparable to the original potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Typically the original potential will include additional degrees of free- dom, such as the electron density for DFT calculations, or solvent atoms in protein simulations, which make the full computation very expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' By training the net- work only on a subset of the original degrees of freedom one obtaines a coarse-grained representation that can be simulated at a much reduced computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' NN potentials thus combine the best of two worlds, retain- ing the accuracy of the underlying potential model, at the much lower cost of coarse-grained classical molecu- ∗Corresponding author: john.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='russo@uniroma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='it lar dynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The accuracy of the NN po- tential depends crucially on how local atomic positions are encoded in the input of the neural network, which needs to retain the symmetries of the underlying Hamil- tonian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' rotational, translational, and index permu- tation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Several methods have been proposed in the literature [12, 36], such as the approaches based on the Behler-Parrinello (BP) symmetry functions [18], the Smooth Overlap of Atomic Positions (SOAP) [37], N-body iterative contraction of equivariants (NICE) [38] and polynomial symmetry functions [39], or frameworks like the DeepMD [23], SchNet [22] and RuNNer [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In all cases, atomic positions are transformed into atomic fingerprints (AFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The choice of the AFs is particularly relevant, as it greatly affects the accuracy and generality of the resulting NN potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We develop here a fully learnable NN potential in which the AFs, while retaining the simplicity of typi- cal local fingerprints, do not need to be fixed beforehand but instead are learned during the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The coupled training of the atomic fingerprint param- eters and of the network weights makes the NN train- ing process more efficient since the NN representation is spontaneously built on a variable atomic fingerprint rep- resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' To tackle the combined minimization of the AF parameters and of the network weights we adopt an efficient annealing procedure, that periodically cycles the learning rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' the step size of the minimization algo- rithm, resulting in a fast and accurate training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We validate the NN potential on the mW model of water [40], which is a one-site classical potential that has found widespread adoption to study water’s anoma- lies [41, 42] and crystallization phenomena [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Since the first pioneering MD simulations [45, 46], water is of- ten chosen as a prototypical case study, as the large num- ber of distinct local structures that are compatible with its tetrahedral coordination make it the molecule with the most complex thermodynamic behavior [47], for ex- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='11612v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='soft] 27 Jan 2023 2 ample displaying a liquid-liquid critical point at super- cooled conditions [48–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' NN potentials for water have been developed starting from density functional calcu- lations, with different levels of accuracy [53–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' NN potentials have also been proposed to parametrise accu- rate classical models for water with the aim of speeding up the calculations when multi-body interactions are in- cluded [61], as in the MBpol model [62–64] or for testing the relevance of the long range interactions, as for the SPC/E model [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We choose the mW potential as our benchmark system because its explicit three-body poten- tial term offers a challenge to the NN representation that is not found in molecular models built from pair-wise interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We stress that we train the NN-potential against data which can be generated easily and for which structural and dynamic properties are well known (or can be evaluated with small numerical errors) in a wide range of temperatures and densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In this way, we can perform a quantitative accurate comparison between the original mW model and the hereby proposed NN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Our results show that training the NN potential at even just one density-temperature state point provides an accurate description of the mW model in a surround- ing phase space region that is approximately a hundred kelvins wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' A training based on three different state points extends the convergence window extensively, ac- curately reproducing state points at extreme conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' large negative and (crushingly) positive pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We will show that the NN reproduces thermodynamic, structural and dynamical properties of the mW liquid state, as well as structural properties of all the stable crystalline phases of mW water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In Section II we de- scribe the new atomic fingerprints and the details about the Neural Network potential implementation, including the warm restart procedure used to train the weights and the fingerprints at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In Section III we present the results, which include the accuracy of the models built from training sets that include one or three state points, and a comparison of the thermodynamic, structural and dynamic properties with those of the orig- inal mW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We conclude in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' THE NEURAL NETWORK MODEL The most important step in the design of a feed- forward neural network potential is the choice on how to define the first and the last layers of the network, respec- tively named the input and output layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We start with the output layer, as it determines the NN potential ar- chitecture to be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Here we follow the Behler Parrinello NN potential architecture [18], in which the total energy of the system is decomposed as the sum of local fields (Ei), each one representing the contribution of a local environment centered around atom i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Being this a many-body contribution, it is important to note that Ei is not the energy of the single atom i, but of all its environment (see also the Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' With this choice, the total energy of the system is simply the sum over all atoms, E = � Ei, and the force ⃗fi acting on atom i is the negative gradient of the total energy with respect to the coordinates ν of atom i, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' fiν = ∂E/∂xiν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We have to point out that a NN potential is differentiable and hence it is possible to evaluate the gradient of the energy analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' This allows to compute forces of the NN potential in the same way of other force fields, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' by the negative gradient of the total potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The input layer is built from two-body (distances) and three-body (angles) descriptors of the local environment, ⃗D(i) and T (i) respectively, ensuring translational and ro- tational invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The first layer of the neural network is the Atomic Fingerprint Constructor (AFC), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 1, which applies an exponential weighting on the atomic descriptors, restoring the invariance under per- mutations of atomic indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The outputs of this first layer are the atomic fingerprints (AFs) and in turn these are given to the first hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We will show how this organization of the AFC layer allows for the inter- nal parameters of the exponential weighting to be trained together with the weights in the hidden layers of the net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In the following we describe in detail the construc- tion of the inputs and the calculation flow in the first layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The atomic fingerprints The choice of input layer presents considerably more freedom, and it is here that we deviate from previous NN potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The data in this layer should retain all the in- formation needed to properly evaluate forces and energies of the particles in the system, possibly exploiting the in- ternal symmetries of the Hamiltonian (which in isotropic fluids are the rotational, translational and permutational invariance) to reduce the number of degenerate inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Given that the output was chosen as Ei, the energy of the atomic environment surrounding atom i, the input uses an atom-centered representation of the local environment of atom i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In the input layer, we define an atom-centered repre- sentation of the local environment of atom i, consider- ing both the distances rij with the nearest neighbours j within a spatial cut-off Rc, and the angles θjik between atom i and the pair of neighbours jk that are within a cut-off Rc′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' More precisely, for each atom j within Rc from i we calculate the following descriptors D(i) j (rij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Rc) = � 1 2 � 1 + cos � π rij Rc �� rij ≤ Rc 0 rij > Rc (1) and, for each triplet j − i − k within Rc′ from i, T (i) jk (rij, rik, θjik) = (2) 1 2 [1 + cos (θjik)] D(i) j (rij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Rc ′) D(i) k (rik;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Rc ′) 3 C Ei α γ δ β Compression ⃗ D(i) ⃗ D(i) T(i) θjik rij ≡ ( ⃗ D(i), T(i)) A B Atomic Fingerprint Constructor Hidden layers FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 1: Schematic representation of the Neural Network Potential flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' (A) Starting from the relative distances and the triplets angles between neighbouring atoms, the input layer evaluates the atomic descriptors ⃗D(i) = {D(i) j } (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 1) and T (i) = {T (i) jk } (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' (B) The first layer is the Atomic Fingerprint Constructor (AFC) and it combines the atomic descriptors into atomic fingerprints, weighting them with an exponential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The red nodes perform the calculation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 5, where from the two- body descriptors a weighting vector ⃗D(i) w (α) = {eαD(i) j } is calculated (square with α) and then the scalar product ⃗D(i) · ⃗D(i) w (α) is computed (square with point) and finally a logarithm is applied (circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The blue nodes perform the calculation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 7, where two weighting vectors are calculated from the two-body descriptors namely ⃗D(i) w (γ) and ⃗D(i) w (δ) and one weighting matrix from the three-body descriptors T (i) w (β) = {eβT (i) jk /2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Finally in the compression unit (Eq 6) values are combined as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5[ ⃗D(i) ◦ ⃗D(i) w (γ)]T [T (i) ◦T (i) w (β)][ ⃗D(i) ◦ ⃗D(i) w (δ)] where we use the circle symbol for the element-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The output value of the compression unit is given to the logarithm function (circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The complete network (D) is made of ten AFC units and two hidden layers with 25 nodes per layer and here is depicted 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 times smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Here i indicates the label of i-th particle while in- dex j and k run over all other particles in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 1, D(i) j (rij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Rc) is a function that goes continu- ously to zero at the cut-off (including its derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The choice of this functional form guarantees that D(i) j is able to express contributions even from neighbours close to the cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Other choices, based on polyno- mials or other non-linear functions, have been tested in the past [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' For example, we tested a parabolic cutoff function which produced considerably worse re- sults than the cutoff function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The function T (i) jk (rij, rik, θjik) is also continuous at the triplet cutoff R′ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The angular function 1 2 [1 + cos (θjik)] guarantees that 0 ≤ T (i) jk (rij, rik, θjik) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We note that the use of relative distances and angles in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 1-2 guarantees trans- lational and rotational invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The pairs and triplets descriptors are then fed to the AFC layer to compute the atomic fingerprints, AFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' These are computed by projecting the D(i) j and T (i) jk de- scriptors on a exponential set of functions defined by 4 D (i)(α) = ln � �� j̸=i D(i) j eαD(i) j + ϵ � � − Zα (3) T (i)(β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' δ) = ln � � � j̸=k̸=i T (i) jk eβT (i) jk eγD(i) j eδD(i) k 2 + ϵ � �(4) −Zβγδ These AFs are built summing over all pairs and all triplets involving particle i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' making them invariant un- der permutations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' and multiplying each descriptor by an exponential filter whose parameters are called α for dis- tance AFs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' and β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' δ for the triplet AFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' These param- eters play the role of feature selectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' by choosing an appropriate list of α, β, γ, δ the AFs can extract the necessary information from the atomic descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The best choice of α, β, γ, δ will emerge automatically dur- ing the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 3-4, the number ϵ is set to 10−3 and fixes the value of energy in the rare event that no neighbors are found inside the cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Parameters Zα and Zβγδ are optimized during the training process, shift- ing the AFs towards positive or negative values, and act as normalization factors that improve the representation of the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The definitions in equations 3-4 can be reformulated in terms of product between vectors and matrices in the fol- lowing way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The descriptors in equations 1-2 for particle i can be represented as a vector ⃗D(i) = {D(i) j } and a matrix T (i) = {T (i) jk } respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Given a choice of α, β, γ and δ, three weighting vector ⃗D(i) w (α) = {eαD(i) j }, ⃗D(i) w (γ) = {eγD(i) j } and ⃗D(i) w (δ) = {eδD(i) j } and one weighting ma- trix T (i) w (β) = {eβT (i) jk /2} are calculated from ⃗D(i) and T (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The 2-body atomic fingerprint (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 3) is finally computed as D (i)(α) = ln � ⃗D(i) · ⃗D(i) w (α) + ϵ � − Zα (5) The 3-body atomic fingerprint (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 4) is computed first by what we call compression step in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 1 as T c (i) = [ ⃗D(i) ◦ ⃗D(i) w (γ)]T [T (i) ◦ T (i) w (β)][ ⃗D(i) ◦ ⃗D(i) w (δ)] 2 (6) and finally by T (i)(β, γ, δ) = ln � T c (i)(β, γ, δ) + ϵ � − Zβγδ (7) where we use the circle symbol for the element-wise mul- tiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The NN potential flow is depicted in Figure 1 following the vectorial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In summary, our AFs select the local descriptors use- ful for the reconstruction of the potential by weight- ing them with an exponential factor tuned with expo- nents α, β, γ, δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' A similar weighting procedure has been showed to be extremely powerful in the selection of com- plex patterns and is widely applied in the so-called atten- tion layer first introduced by Google Brain [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' However the AFC layer imposes additionally physically motivated constraints on the neural network representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We note that the expression for the system energy is a sum over the fields Ei, but the local fields Ei are not addi- tive energies, involving all the pair distances and triplets angles within the cut-off sphere centered on particle i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' This non-additive feature favours the NN ability to cap- ture higher order correlations (multi-body contribution to the energy), and has been shown to outperform ad- ditive models in complex datasets [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The NN non- additivity requires the derivative of the whole energy E (as opposed to Ei) to estimate the force on a particle i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In this way, contributions to the force on particle i come not only from the descriptors of i but also from the descrip- tors of all particles who have i as a neighbour, de facto enlarging the effective region in space where interaction between particles are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' This allows the network to include contributions from length-scales larger than the cutoffs that define the atomic descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The Ap- pendix A provides further information on this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Hidden layers We employ a standard feed-forward fully-connected neural network composed of two hidden layers with 25 nodes per layer and using the hyperbolic tangent (tanh) as the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The nodes of the first hidden layer are fully connected to the ones in the second layer, and these connections have associated weights W which are optimized during the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The input of the first hidden layer is given by the AFC layer where we used five nodes for the two-body AFs (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 3) and five nodes for the three-body AFs (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 4) for a total of 10 AFs for each atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We explore the performance of some combinations for the number of two- body and three-body AF in Appendix D and we find that the choice of five and five is the more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The output is the local field Ei, for each atomic envi- ronment i, whose sum E = �N i=1 Ei represents the NN estimate of the potential energy E of the whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Loss function and training strategy To train the NN-potential we minimize a loss function computed over nf frames, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' the number of independent configurations extracted from an equilibrium simulation of the liquid phase of the target potential (in our case the mW potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The loss function is the sum of two contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The first contribution, H[{∆ϵk, ∆f k iν}], expresses the difference in each frame k between the NN estimates and the target values for both the total potential energy (nor- malized by total number of atoms) ϵk and the atomic 5 forces f k iν acting in direction ν on atom i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The nf energy ϵk values and 3Nnf force f k iν values are combined in the following expression H[{∆ϵk, ∆f k iν}] = pe nf nf � k=1 hHuber(∆ϵk) + pf 3Nnf nf � k=1 N � i=1 3 � ν=1 hHuber(∆f k iν) (8) where pe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='1 and pf = 1 control the relative contri- bution of the energy and the forces to the loss function, and hHuber(x) is the so-called Huber function hHuber(x) = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5x2 if |x| ≤ 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 + (|x| − 1) if |x| > 1 (9) pe and pf are hyper-parameters of the model, and we se- lected them with some preliminary tests that found those values to be near the optimal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The Huber func- tion [68] is an optimal choice whenever the exploration of the loss function goes through large errors caused by outliers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' data points that differ significantly from pre- vious inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Indeed when a large deviation between the model and data occur, a mean square error minimization may gives rise to an anomalous trajectory in parameters space, largely affecting the stability of the training pro- cedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' This may happen especially in the first part of the training procedure when the parameter optimization, relaxing both on the energy and forces error surfaces may experience some instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The second contribution to the loss function is a reg- ularization function, R[{αl, βm, γm, δm}], that serves to limit the range of positive values of αl and of the triplets βm, γm, δm (where the indexes l and m run over the five different values of α and five different triplets of values for β, γ and δ) in the window −∞ to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' To this aim we select the commonly used relu function rrelu(x) = � x − 5 if x > 5 0 if x ≤ 5 (10) (11) and write R[{αl, βm, γm, δm}] = 5 � l=1 rrelu(αl) + 5 � m=1 [rrelu(βm) + rrelu(γm) + rrelu(δm)] (12) Thus, the R function is activated whenever one param- eters of the AFC layer becomes, during the minimization, larger than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' To summarize, the global loss function L used in the training of the NN is L[ϵ, f] = H[{∆ϵk, ∆f k iν}] + pbR[{αl, βm, γm, δm}] (13) where pb = 1 weights the relative contribution of R com- pared to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Compared to a standard NN-potential, we train not only the network weights W but also the AFs param- eters Σ ≡ {αl, βm, γm, δm} at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The si- multaneous optimization of the weights W and AFs Σ prevents possible bottleneck in the optimisation of W at fixed representation of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Other NN potential approaches implement a separate initial procedure to optimise the Σ parameters followed by the optimisation of W at fixed Σ [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The two-step procedure not only requires a spe- cific methodological choice for optimising Σ, but also may not result in the optimal values, compared to a search in the full parameter space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' both Σ and W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Since the complexity of the loss function has increased, we have investigated in some detail some efficient strategies that lead to a fast and accurate training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Firstly, we initial- ize the parameters W via the Xavier algorithm, in which the weights are extracted from a random uniform distri- bution [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' To initialize the Σ parameters we used a uniform distribution in interval [−5, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We then mini- mize the loss function using the warm restart procedure proposed in reference [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In this procedure, the learn- ing rate η is reinitialized at every cycle l and inside each cycle it decays as a function of the number of training steps t following η(l)(t) = Al �(1 − ξf) 2 � 1 + cos �πt Tl �� + ξf � (14) 0 ≤ t ≤ Tl where ξf = 10−7, Al = η0ξl 0 is the initial learning rate of the l-th cycle with η0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='01 and ξ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='9, Tl = bτ l is the period of the l-th cycle with τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='4 and b = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The absolute number of training steps n during cycle l can be calculated summing over the length of all previous cycles as n = τ + �l−1 m=0 Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We also select to evaluate the loss function for groups of four frames (mini-batch) and we randomly select 200 frames nf = 200 for a system of 1000 atoms and hence we split this dataset in 160 frames (%80) for the training set and the 40 frames (%20) for the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 2(A) we represent the typical decay of the learn- ing rate of the warm restart procedure, which will be compared to the standard exponential decay protocol in the Results section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The Target Model To test the quality of the proposed novel NN we train the NN with data produced with the mW [40] model 6 100 101 102 103 104 105 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='010 A 0 1000 2000 3000 ne 10 1 100 B Validation Loss Training Loss 0 1000 2000 3000 ne 10 2 10 1 100 101 (kcal mol 1) C 0 1000 2000 3000 ne 101 f (kcal mol 1 nm 1) D FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 2: Model convergence properties: (A) Learning rate schedule (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 14) as a function of the absolute training step n (one step is defined as an update of the network parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' (B) The training and validation loss (see L[ϵ, f] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 13) evo- lution during the training procedure, reported as a function of the number of epoch ne (an epoch is defined as a complete evaluation of the training dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Root mean square (RMS) error of the total potential energy per particle (C) and of the force cartesian components (D) during the training evaluated in the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Data in panels B-C-D refers to the NN3 model and the green point shows the best model location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' This potential, a re-parametrization of the Stillinger-Weber model for silicon [72], uses a combina- tion of pairwise functions complemented with an additive three-body potential term 0 20 40 60 Seed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='07 (kcal mol 1) A Exponential Warm restart 0 20 40 60 Seed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='8 f (kcal mol 1 nm 1) B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 3: Comparison of the root mean square error calculated on the validation set for 60 replicas differing in the initial seed of the training procedure using both an exponential de- cay of the learning rate (points) and the warm restart method (squares), for the energy (panel A) and for the forces (panel B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' For the forces, a significant improvement both in the av- erage error and in its variance is found for the warm restart schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' E = � i � j>i U2(rij)+λ � i � j̸=i � j>k U3 (rij, rik, θjik) (15) where the two body contribution between two particles i and j at relative distance rij is a generalized Lennard- Jones potential U2 (rij) = Aϵ � B � σ rij �p − � σ rij �q� exp � σ rij − aσ � (16) where the p = 12 and q = 6 powers are substituted by q = 0 and p = 4, multiplied by an exponential cut- off that brings the potential to zero at aσ, with a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='8 and σ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='3925 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Aϵ (with A = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='049556277 and ϵ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='189 kcal mol−1) controls the strength of the two body part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' B controls the two-body repulsion (with B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='6022245584).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The three body contribution is computed from all pos- sible ordered triplets formed by the central particle with the interacting neighbors (with the same cut-off aσ as the two-body term) and favours the tetrahedral coordination of the atoms via the following functional form U3 (rij, rik, θjik) = ϵ [cos (θjik) − cos (θ0)]2 × exp � γσ rij − aσ � exp � γσ rik − aσ � (17) where θjik is the angle formed in the triplet jik and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='2 controls the smoothness of the cut-off function on approaching the cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Finally, θ0 = 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='47◦ and λ = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='15 controls the strength of the angular part of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 7 The mW model, with its three-body terms centered around a specific angle and non-monotonic radial interac- tions, is based on a functional form which is quite differ- ent from the radial and angular descriptors selected in the NN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The NN is thus agnostic with respect to the functional form that describes the physical system (the mW in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' But having a reference model with ex- plicit three body contributions offers a more challenging target for the NN potential compared to potential models built entirely from pairwise interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The mW model is thus an excellent candidate to test the performance of the proposed NN potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Training We study two different NN models, indicated with the labels NN1 and NN3, differing in the number of state points included in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' These two models are built with a cut-off of Rc = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='545 ˚A for the two-body atomic descriptors and a cut-off of R′ c = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='306 ˚A for the three-body atomic descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' R′ c is the same as the mW cutoff while Rc was made slightly larger to miti- gate the suppression of information at the boundaries by the cutoff functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The NN1 model uses only train- ing information based on mW equilibrium configurations from one state point at ρ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='07 g cm−3, T1 = 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='9 K where the stable phase is the liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The NN3 model uses training information based on mW liquid configura- tions in three different state points, two state points at ρ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='92 g cm−3, T1 = 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='1 K and ρ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='92 g cm−3, T2 = 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='9 K where the stable solid phase is the clathrate Si34/Si136 [73] and one state point at ρ3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='15 g cm−3, T2 = 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='9 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' This choice of points in the phase diagram is aimed to improve agreement with the low temperature-low density as well as high density regions of the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Im- portantly, all configurations come from either stable or metastable liquid state configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Indeed, the point at ρ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='92 g cm−3, T2 = 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='9 K is quite close to the limit of stability (respect to cavitation) of the liquid state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' To generate the training set, we simulate a system of N = 1000 mW particles with a standard molecular dy- namics code in the NVT ensemble, where we use a time step of 4 fs and run 107 steps for each state point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' From these trajectory, we randomly select 200 configurations (frames) to create a dataset of positions, total energies and forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We then split the dataset in the training and in the test data sets, the first one containing 80% of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We then run the training for 4000 epochs with a minibatch of 4 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' At the end of every epoch, we check if the validation loss is improved and we save the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 2 we plot the loss function for the training and test datasets (B), the root mean square error of the total energy per particle (C), and of the force (D) for the NN3 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The results show that the learn- ing rate schedule of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 14 is very effective in reducing both the loss and error functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Interestingly, the neural network seems to avoid over- fitting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' the validation loss is decreasing at the same rate as the loss on the training data), and the best model (deepest local minimum explored), in a given window of training steps, is always found at the end of that win- dow, which also indicates that the accuracy could be fur- ther improved by running more training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Indeed we found that by increasing the number of training steps by one order of magnitude the error in the forces decreases by a further 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Similar accuracy of the training stage is obtained also for the NN1 model (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The training procedure always terminates with an error on the test set equal or less than ∆ϵ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='01 kcal mol−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='43 meV) for the energy, and of ∆f ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='55 kcal mol−1 nm−1 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='72 meV ˚A−1) for the forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' These values are comparable to the state-of-the-art NN potentials [23, 54, 55, 61], and within the typical accuracy of DFT calculations [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We can compare the precision of our model with that of alternative NN potentials trained on a range of water models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' An alternative mW neural network potential has been trained on a dataset made of 1991 configurations of 128 particles system at different pressure and tem- perature (including both liquid and ice structures) with Behler-Parinello symmetry functions [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The train- ing of this model (which uses more atomic fingerprints and a larger cutoff radius) converged to an error in en- ergy of ∆ϵ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0062 kcal mol−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='27 meV), and ∆f ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='46 kcal mol−1 nm−1 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='70 meV ˚A−1) for the forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In a recent work searching for liquid-liquid transition signa- tures in an ab-initio water NN model [55], a dataset of configurations spanning a temperature range of 0−600 K and a pressure range of 0 − 50 GPa was selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' For a system of 192 particles, the training converged to an error in energy of ∆ϵ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='010 kcal mol−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='46 meV), and ∆f ≃ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='96 kcal mol−1 nm−1 (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='2 meV ˚A−1) for the forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In the NN model of MB-POL [61], a dataset spanning a temperature range from 198 K to 368 K at ambient pressure was selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In this case, for a system of 256 water molecule, an accu- racy of ∆ϵ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='01 kcal mol−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='43 meV) and ∆f ≃ 10 kcal mol−1 nm−1 (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='36 meV ˚A−1) was reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Fi- nally, the NN for water at T = 300 K used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' [54], reached precisions of ∆ϵ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='046 kcal mol−1 (2 meV) and ∆f ≃ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='36 kcal mol−1 nm−1 (110 meV ˚A−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' While a direct comparison between NN potentials trained on different reference potentials is not a valid test to rank the respective accuracies, the comparisons above show that our NN potential reaches a similar precision in energies, and possibly an improved error in the force estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The accuracy of the NN potential could be further improved by extending the size of the dataset and the choice of the state points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In fact, while the datasets in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' [54, 55, 61] have been built with optimized proce- 8 dures, the dataset used in this study was prepared by sampling just one (NN1) or three (NN3) state-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Also the size of the datasets used in the present work is smaller or comparable to the ones of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' [54, 55, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 3 we compare the error in the energies (A) and the forces (B) between sixty independent training runs using the standard exponential decay of the learn- ing rate (points) and the warm restart protocol (squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The figure shows that while the errors in the energy com- putations are comparable between the two methods, the warm restart protocol allows the forces to be computed with higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Moreover we found that the warm restart procedure is less dependent on the initial seed and that it reaches deeper basins than the standard exponen- tial cooling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Comparing NN1 with NN3 The NN potential model was implemented in a custom MD code that makes use of the tensorflow C API [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We adopted the same time step (4 fs), the same number of particles (N = 1000) and the same number of steps (107) as for the simulations in the mW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' As described in the Training Section, we compare the accuracy of two different training strategies: NN1 which was trained on a single state point, and NN3 which is instead trained on three different state point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 4 we plot the energy error (∆ϵ) between the NN potential and the mW model with both NN1 (panel A) and NN3 (panel B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Starting from NN1, we see that the model already provides an excellent accuracy for a large range of temperatures and for densities close to the training density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The biggest shortcoming of the NN1 model is at densities lower than the trained density, where the NN potential model cavitates and does not retain the long-lived metastable liquid state displayed by the mW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We speculate that this behaviour is due to the absence of low density configurations in the training set, which prevents the NN potential model from correctly reproducing the attractive tails of the mW potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' To overcome this limitation we have included two ad- ditional state points at low density in the NN3 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In this case, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 4B shows that NN3 provides a quite ac- curate reproduction of the energy in the entire explored density and temperature window (despite being trained only with data at ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='92 g cm−3 and ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='15 g cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We can also compare the accuracy obtained during production runs against the accuracy reached during training, which was ∆ϵ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='01 kcal mol−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 4B shows the error is of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='032 kcal mol−1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='3 meV), for density above the training set density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' But in the density region between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='92 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='15, the error is even smaller, around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='017 kcal mol−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='7 meV) at the lowest density boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We can thus conclude that the NN3 model, which adds to the NN1 model information at lower density and tem- perature, in the region where tetrahedality in the wa- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='2 (g cm 3) 385 365 345 325 305 285 265 245 225 T (K) A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='2 (g cm 3) B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='040 (kcal mol 1) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 4: Comparison between the mW total energy and the NN1 model (A) and NN3 model (B) for different temperatures and densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' While the NN3 model is able to reproduce the mW total energy with a good agreement in a wide region of densities and temperatures, the NN1 provide a good repre- sentation only in a limited region of density and temperature values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Blue squares represent the state points used for build- ing the NN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' ter structure is enhanced, is indeed capable to represent, with only three state points, a quite large region of the phase space, encompassing dense and stretched liquid states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' This suggests that a training based on few state points at the boundary of the density/temperature re- gion which needs to be studied is sufficient to produce a high quality NN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In the following we focus entirely on the NN3 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Comparison of thermodynamic, structural and dynamical quantities In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 5 we present a comparison of thermodynamic data between the mW model (squares) and its NN poten- tial representation (points) across a wide range of state points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 5A plots the energy as function of density for temperatures ranging from melting to deeply super- cooled conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Perhaps the most interesting result is that the NN potential is able to capture the energy min- imum, also called the optimal network forming density, which is a distinctive anomalous property of water and other empty liquids [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 5(B) shows the pressure as a function of the tem- perature for different densities, comparing the mW with the NN3 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Also the pressure shows a good agree- ment between the two models in the region of densities between ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='92 g cm−3 and ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='15 g cm−3, which, as for the energy, tends to deteriorate at ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='22 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In the large density region explored, the structure of the liquid changes considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' On increasing density, a transition from tetrahedral coordinated local structure, prevalent at low T and low ρ, towards denser local envi- 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='25 (g cm 3) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 (kcal mol 1) 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='1K 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='6K 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0K 258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5K 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0K 299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0K 311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='4K 373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='7K A mW NN3 200 225 250 275 300 325 350 375 T (K) 0 1 2 3 4 P (GPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='92 g cm 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='99 g cm 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='07 g cm 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='15 g cm 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='19 g cm 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='22 g cm 3 B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 5: Comparison between the mW total energy and the NN3 total energy as a function of density along different isotherm (A) and comparison between the mW pressure and the NN3 pressure as a function of temperature along differ- ent isochores (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The relative error of the NN vs the mW potential grows with density, but remains within 3% even for densities larger than the densities used in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' ronments with interstitial molecules included in the first coordination shell takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' This structural change is well displayed in the radial distribution function, shown for different densities at fixed temperature in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 6 also shows the progressive onset of a peak around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 ˚A developing on increasing pressure, which signals the growth of interstitial molecules, coexisting with open tetrahedral local structures [77, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' At the highest den- sity, the tetrahedral peak completely merges with the interstitial peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The NN3 model reproduces quite ac- curately all features of the radial distribution functions, maxima and minima positions and their relative ampli- tudes, at all densities, from the tetrahedral-dominated to the interstitial-dominated limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In general, NN3 model reproduces quite well the mW potential in energies, pres- sure and structures and it appreciably deviates from mW pressures and energies quantities only at densities (above 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='15 g/cm3) which are outside of the training region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' To assess the ability of NN potential to correctly de- scribe also the crystal phases of the mW potential, we compare in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 7 the g(r) of mW with the g(r) of the NN3 model for four different stable solid phases [73]: hexagonal and cubic ice (ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='00 g cm−3 and T = 246 K), the dense crystal SC16 (ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='20 g cm−3 and T = 234 K) and the clathrate phase Si136 (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='80 g cm−3 and T = 221 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The results, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 7, show that, despite no crystal configurations have been included in the training set, a quite accurate representation of the crystal structure at finite temperature is provided by the NN3 model for all distinct sampled lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 16 R (Å) 0 1 2 3 4 5 6 7 8 RDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='92 g cm 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='99 g cm 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='15 g cm 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='22 g cm 3 mW NN3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 6: Comparison between the mW radial distribution functions g(r) and the NN3 g(r) at T = 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='9 K for four different densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The tetrahedral structure (signalled by the peak at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='54 ˚A ) progressively weakens in favour of an in- terstitial peak progressively growing at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='8 ˚A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Different g(r) have been progressively shifted by two to improve clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 0 5 10 15 R (Å) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 RDF Hexagonal diamond Cubic diamond Si136 Clathrate SC16 mW NN3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 7: Comparison between the mW radial distribution functions g(r) and the NN3 g(r) for four different lattices: (A) hexagonal diamond (the oxygen positions of the ice Ih);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' (B) cubic diamond (the oxygen positions of the ice Ic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' (C) the SC16 crystal (the dense crystal form stable at large pres- sures in the mW model) and (D) the Si136 clathrate structure, which is stable at negative pressures in the mW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Dif- ferent g(r) have been progressively shifted by four to improve clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Finally, we compare in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 8 the diffusion coefficient (evaluated from the long time limit of the mean square displacement) for the mW and the NN3 model, in a wide range of temperatures and densities, where water displays a diffusion anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 8 shows again that, also for 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='20 (g cm 3) 0 100 200 300 400 500 600 700 800 D (Å2 ns 1) 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='1K 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='6K 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0K 258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5K 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0K 299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0K 311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='4K 373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='7K mW NN3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 8: Comparison between the mW diffusion coefficient D and the NN3 corresponding quantity for different tempera- tures and densities, in the interval 221 − 271 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In this dy- namic quantity, the relative error is, for all temperatures, around 8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Note also that in this T window the diffusion coefficient shows a clear maximum, reproducing one of the well-know diffusion anomaly of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Diffusion coefficients have been calculated in the NVT ensemble using the same Andersen thermostat algorithm [79] for mW and NN3 poten- tial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' dynamical quantities, the NN potential offers an excellent representation of the mW potential, despite the fact that no dynamical quantity was included in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' A comparison between fluctuations of energy and pressure of mW and NN3 potential is reported in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' CONCLUSIONS In this work we have presented a novel neural net- work (NN) potential based on a new set of atomic fin- gerprints (AFs) built from two- and three-body local de- scriptors that are combined in a permutation-invariant way through an exponential filter (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 3-4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' One of the distinctive advantages of our scheme is that the AF’s parameters are optimized during the training procedure, making the present algorithm a self-training network that automatically selects the best AFs for the potential of in- terest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We have shown that the added complexity in the con- current training of the AFs and of the NN weights can be overcome with an annealing procedure based on the warm restart method [71], where the learning rate goes through damped oscillatory ramps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' This strategy not only gives better accuracy compared to the commonly implemented exponential learning rate decay, but also allows the training procedure to converge rapidly inde- pendently from the initialisation strategies of the model’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Moreover we show in Appendix C that the potential hyper-surface of the NN model has the same smoothness as the target model, as confirmed by (i) the possibility to use the same timestep in the NN and in the target model when integrating the equation of motion and (ii) by the possibility of simulate the NN model even in the NVE ensemble with proper energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We test the novel NN on the mW model [40], a one-component model system commonly used to de- scribe water in classical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' This model, a re- parametrization of the Stillinger-Weber model for sili- con [72], while treating the water molecule as a simple point, is able to reproduce the characteristic tetrahedral local structure of water (and its distortion on increasing density) via the use of three-body interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Indeed water changes from a liquid of tetrahedrally coordinated molecules to a denser liquid, in which a relevant fraction of interstitial molecules are present in the first nearest- neighbour shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The complexity of the mW model, both due to its functional form as well as to the variety of dif- ferent local structures which characterise water, makes it an ideal benchmark system to test our NN potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We find that a training based on configurations ex- tracted by three different state points is able to pro- vide a quite accurate representation of the mW poten- tial hyper-surface, when the densities and temperatures of the training state points delimit the region of in which the NN potential is expected to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We also find that the error in the NN estimate of the total energy is low, always smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='03 kcal mol−1, with a mean error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='013 kcal mol−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The NN model reproduces very well not only the thermodynamic properties but also struc- tural properties, as quantified by the radial distribution function, and the dynamic properties, as expressed by the diffusion coefficient, in the extended density interval from ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='92 g cm−3 to ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='22 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Interestingly, we find that the NN model, trained only on disordered configurations, is also able to properly describe the radial distribution of the ordered lattices which characterise the mW phase diagram, encompass- ing the cubic and hexagonal ices, the SC16 and the Si136 clathrate structure [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In this respect, the ability of the NN model to properly represent crystal states suggests that, in the case of the mW, and as such probably in the case of water, the geometrical information relevant to the ordered structures is contained in the sampling of phase space typical of the disordered liquid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' These find- ings have been recently discussed in reference [80] where it has been demonstrated that liquid water contains all the building blocks of diverse ice phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We conclude by noticing that the present approach can be generalized to multicomponent systems, following the same strategy implemented by previous approaches [18, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Work in this direction is underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Acknowledgments FGM and JR acknowledge support from the European Research Council Grant DLV-759187 and CINECA grant ISCRAB NNPROT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 11 [1] K.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Engel, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Cheng, Nature communications 11, 1 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' APPENDIX A In this appendix we discuss the effective spacial range covered by a NN potential whose fingerprints are defined based on pair information confined within a sphere of cutoff radius Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' As noted in reference [31], multi-body potentials and especially non-additive multibody potentials induce lo- cal interactions beyond the cut-off radius, enlarging the sphere of interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Indeed, the force on particle i comes from the derivative of the local field of i and of all its neighbours with respect to the coordinates of par- ticle i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 2 1 3 2 1 3 4 5 A B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 9: (A) Two-body interactions and (B) three-body inter- actions in a non linear local field model Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The non linearity of the local field enlarges the interaction cut-off where a neigh- bour particle (blue) makes a bridge between non-neighboring particle (red and blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 9 graphically explains the effective role of Rc in the NN potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In panel A, we describe particle 1 with only one neighbour (particle 2) within Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We also represent the sphere centered on particle 3, which also includes particle 2 as one of its neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In this case, the energy of the system will be represented as a sum over the local fields E1, E2 and E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Due to the intrinsic non-linearity of the NN, the field Ei mixes together the AFs, and consequently the distances and angles entering in the AFs are non-linearly mixed in Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The force on atom 1 is then written as f1ν = −∂E1(r12) ∂x1ν − ∂E2(r21, r23) ∂x1ν = −∂E1(r12) ∂x1ν −∂E2(r21, r23) ∂r21 ∂r21 ∂x1ν − ∂E2(r21, r23) ∂r23 ∂r23 ∂x1ν (A1) While the last term vanishes, the next to the last retains 13 A B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 10: (A) Standard deviation of total energy (normal- ized with the number of particles) and (B) standard deviation of virial pressure for both NN3 model (red) and mW model (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' an intrinsic dependence on the coordinates both of par- ticle 2 as well as of particle 3, if the local field E2 is non linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Thus, even if particle 3 is further than Rc, it en- ters in the determination of the force acting on particle 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' A similar effect is also present in the angular part of the AFs, as shown graphically in panel B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Indeed, for the angular component of the AF the force on particle 1 is f1ν = −∂E1(θ512) ∂x1ν − ∂E2(θ123, θ124, θ324) ∂x1ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' (A2) Also in this case two contributions can be separated: (i) the interaction of particle 1 with triplets 123 and 124 is an effect of the three-body AF and it is present also in additive-models such as the mW model, (ii) the inter- action of particle 1 with triplet 324 is an effect of the non-additive nature of the NN local field Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' APPENDIX B In this Appendix we provide further thermodynam- ics comparisons between mW and NN3 potential focus- ing on the pressure and energy fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We depict in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 10 the standard deviations of the total energy (normalized by N) in panel (A) and the standard devia- tion of virial pressure in panel (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Energy fluctuations of NN3 follow qualitatively and quantitatively the trend of mW potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Pressure fluctuations of NN3 are in good agreement with the mW model but, as for the pres- sure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='B), the accuracy decreases approaching state points outside the density range used for the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 Time (ns) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='4 (kcal mol 1) NN3 Total Energy NN3 Potential Energy mW Total Energy mW Potential Energy FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 11: NVE molecular dynamics at T = 299 K and ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='07 g cm−3 for both NN3 and mW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The time step is dt = 4 fs for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' APPENDIX C In this Appendix we show a comparison between the mW and NN3 potentials in terms of the energy conser- vation in the NVE ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 11 we depict both total energy and potential energy for mW and NN3 po- tential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' The potential energy and total energy of the two models are in good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' APPENDIX D In this Appendix we investigate the efficiency of the training over different choices for the number and types of atomic fingerprints introduced in the Neural Network Model section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We start by using only one three-body (n3b = 1) and one two-body (n2b = 1) AF and subse- quently increasing the number of the AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' For every com- bination of n2b and n3b, we run a 4000 epochs training and at the end of each training we extract the best model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' We summarized these results in table I where we com- pare the error on forces over the all investigated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' From table I it emerges that the choice of n3b = 5 and n2b = 5 is the more convenient both for accuracy and computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Doubling the number of the three-body AF marginally improves the error on forces while increases the computational cost due to the increase in the size of the input layer of the first hidden layer and due to the additional time to compute the three-body AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Moreover in the RESULTS section we show that the choice n3b = 5 and n2b = 5 is sufficient to represent the target potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' Finally the accuracy of the training af- ter doubling the configurations in the dataset reaches an error on forces of ∆f = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='85 meV ˚A−1 that is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='87 times the error value found with a half of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' 14 TABLE I: Table of errors on forces at the end of the 4000 epoch-long training procedure for different combination of the number and type of the AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content=' n3b n2b ∆f (meV ˚A−1) n3b n2b ∆f (meV ˚A−1) 1 1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='79 5 1 16.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='02 5 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='95 2 1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='76 10 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='98 2 2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='95 10 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='17 2 5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='43 10 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='79 2 10 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='39 10 10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='55 2 15 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='70 10 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} +page_content='19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFJT4oBgHgl3EQfsS2B/content/2301.11612v1.pdf'} diff --git a/DdAzT4oBgHgl3EQfGvsw/content/tmp_files/2301.01032v1.pdf.txt b/DdAzT4oBgHgl3EQfGvsw/content/tmp_files/2301.01032v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..882243a7ec94669eb5943a0d59057f30f4166a7d --- /dev/null +++ b/DdAzT4oBgHgl3EQfGvsw/content/tmp_files/2301.01032v1.pdf.txt @@ -0,0 +1,2566 @@ +ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A +METACYCLIC GROUP. +ARISTIDES KONTOGEORGIS AND ALEXIOS TEREZAKIS +Abstract. We give a necessary and sufficient condition for a modular rep- +resentation of a group G = Cph ⋊ Cm in a field of characteristic zero to be +lifted to a representation over local principal ideal domain of characteristic +zero containing the ph roots of unity. +1. Introduction +The lifting problem for a representation +ρ : G → GLn(k), +where k is a field of characteristic p > 0, is about finding a local ring R of char- +acteristic 0, with maximal ideal mR such that R/mR = k, so that the following +diagram is commutative: +GLn(R) +� +G +� +� +GLn(k) +Equivalently one asks if there is a free R-module V , which is also an R[G]-module +such that V ⊗RR/mR is the k[G]-module corresponding to our initial representation. +We know that projective k[G]-modules lift in characteristic zero, [16, chap. 15], but +for a general k[G]-module such a lifting is not always possible, for example, see [10, +prop. 15]. This article aims to study the lifting problem for the group G = Cq⋊Cm, +where Cq is a cyclic group of order ph and Cm is a cyclic group of order m, (p, m) = 1 +and give necessary and sufficient condition in order to lift. We assume that the local +ring R contains the q-roots of unity and k is algebraically closed, and we might need +to consider a ramified extension of R, in order to ensure that certain q-roots of unit +are distant in the mR-topology, see remark 35. An example of such a ring R is the +ring of Witt vectors W(k)[ζq] with the q-roots of unity adjoined to it. +We notice that a decomposable R[G]-module V gives rise to a decomposable +R-module modulo mR and also an indecomposable R[G]-module can break in the +reduction modulo mR into a direct sum of indecomposable k[G]-summands. We +also give a classification of k[Cq ⋊ Cm]-modules in terms of Jordan decomposition +and give the relation with the more usual uniserial description in terms of their +socle [1]. +Date: January 4, 2023. +Key words and phrases. Lifting of representations, modular representation theory, integral +representation theory, Generalized Oort conjecture, metacyclic groups. +1 +arXiv:2301.01032v1 [math.AG] 3 Jan 2023 + +2 +A. KONTOGEORGIS AND A. TEREZAKIS +Our interest to this problem comes from the problem of lifting local actions. The +local lifting problem considers the following question: Does there exist an extension +Λ/W(k), and a representation +˜ρ : G �→ Aut(Λ[[T]]), +such that if t is the reduction of T, then the action of G on Λ[[T]] reduces to the +action of G on k[[t]]? +If the answer to the above question is positive, then we say that the G-action +lifts to characteristic zero. A group G for which every local G-action on k[[t]] lifts to +characteristic zero is called a local Oort group for k. Notice that cyclic groups are +always local Oort groups. This result was known as the “Oort conjecture”, which +was recently proved by F. Pop [15] using the work of A. Obus and S. Wewers [14]. +There are a lot of obstructions that prevent a local action to lift in characteristic +zero. Probably the most important of these obstructions in the KGB-obstruction +[4]. It is believed that this is the only obstruction for the local lifting problem, see +[11], [12]. In [10, Thm. 3] the authors have given a criterion for the local lifting +which involves the lifting of a linear representation of the same group. +The case +G = Cq ⋊Cm and especially the case of dihedral groups Dq = Cq ⋊C2, is a problem +of current interest in the theory of local liftings, see [12], [6], [18]. For more details +on the local lifting problem we refer to [3], [4], [5], [11]. +Keep also in mind that the Cq ⋊ Cm groups were important to the study of +group actions in holomorphic differentials of curves defined over fields of positive +characteristic p, where the group involved has cyclic p-Sylow subgroup, see [2]. +Let us now describe the method of proof. For understanding the splitting of +indecomposable R[G]-modules modulo mR, we develop a version of Jordan normal +form in lemma 16 for endomorphisms T : V → V of order ph, where V is a free +module of rank d. We give a way to select this basis, by selecting an initial suitable +element E ∈ V , see lemma 15. +The normal form (as given in eq. +(9)) of the +element T of order q, determines the decomposition of the reduction. We show +that for every indecomposable summand Vi of V , we can select E as an eigenvalue +of the generator σ of Cm and then by forcing the relation ΓT = T αΓ to hold, we +see how the action of σ can be extended recursivelly to an action of σ on Vi, this is +done in lemma 24. Proving that this gives indeed a well defined action is a technical +computation and is done in lemmata 26, 27, 28, 32, 33. +The important thing here is that the definition of the action of σ on E is the +“initial condition” of a dynamical system that determines the action of Cm on the +indecomposable summand Vi. The R[Cq⋊Cm] indecomposable module Vi can break +into a direct sum Vα(ϵν, κν)-modules 1 ≤ ν ≤ s (for a precise definition of them +see definition 9, notice that κi denotes the dimension). The action of σ on each +Vα(ϵν, κν) can be uniquely determined by the action of σ on an initial basis element +as shown in section 3, again by a “dynamical system” approach, where we need s +initial conditions, one for each Vα(ϵν, κν). The lifting condition essentially means +that the indecomposable summands Vα(ϵ, κ) of the special fibre, should be able +to be rearranged in a suitable way, so that they can be obtained as reductions of +indecomposable R[Cq ⋊ Cq]-modules. The precise expression of our lifting criterion +is given in the following proposition: +Proposition 1. Consider a k[G]-module M which is decomposed as a direct sum +M = Vα(ϵ1, κ1) ⊕ · · · ⊕ Vα(ϵs, κs). + +ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP. +3 +The module lifts to an R[G]-module if and only if the set {1, . . . , s} can be written +as a disjoint union of sets Iν, 1 ≤ ν ≤ t so that +a � +µ∈Iν κµ ≤ q, for all 1 ≤ ν ≤ t. +b � +µ∈Iν κµ ≡ a modm for all 1 ≤ ν ≤ t, where a ∈ {0, 1}. +c For each ν, 1 ≤ ν ≤ t there is an enumeration σ : {1, . . . , #Iν} → Iν ⊂ +{1, .., s}, such that +ϵσ(2) = ϵσ(1)ακσ(1), ϵσ(3) = ϵσ(3)ακσ(3), . . . ϵσ(s) = ϵσ(s−1)ακσ(s−1) +In the above proposition, each set Iν corresponds to a collection of modules +Vα(ϵµ, κµ), µ ∈ Iν which come as the reduction of an indecomposable R[Cq ⋊ Cm]- +module Vν of V . +Aknowledgements A. Terezakis is a recipient of financial support in the context +of a doctoral thesis (grant number MIS-5113934). The implementation of the doc- +toral thesis was co-financed by Greece and the European Union (European Social +Fund-ESF) through the Operational Programme—Human Resources Development, +Education and Lifelong Learning—in the context of the Act—Enhancing Human +Resources Research Potential by undertaking a Doctoral Research—Sub-action 2: +IKY Scholarship Programme for Ph.D. candidates in the Greek Universities. +2. Notation +Let τ be a generator of the cyclic group Cq and σ be a generator of the cyclic +group Cm. The group G is given in terms of generators and relations as follows: +G = ⟨σ, τ|τ q = 1, σm = 1, στσ−1 = τ α for some α ∈ N, 1 ≤ α ≤ ph − 1, (α, p) = 1⟩. +The integer α satisfies the following congruence: +(1) +αm ≡ 1 modq +as one sees by computing τ = σmτσ−m = τ αm. Also the integer α can be seen as +an element in the finite field Fp, and it is a (p − 1)-th root of unity, not necessarily +primitive. In particular the following holds: +Lemma 2. Let ζm ∈ k be a fixed primitive m-th root of unity. There is a natural +number a0, 0 ≤ a0 < m − 1 such that α = ζa0 +m . +Proof. The integer α if we see it as an element in k is an element in the finite field +Fp ⊂ k, therefore αp−1 = 1 as an element in Fp. Let ordp(α) be the order of α in F∗ +p. +By eq. (1) we have that ordp(α) | p−1 and ordp(α) | m, that is ordp(α) | (p−1, m). +The primitive m-th root of unity ζm generates a finite field Fp(ζm) = Fpν for +some integer ν, which has cyclic multiplicative group Fpν\{0} containing both the +cyclic groups ⟨ζm⟩ and ⟨α⟩. Since for every divisor δ of the order of a cyclic group +C there is a unique subgroup C′ < C of order δ we have that α ∈ ⟨ζm⟩, and the +result follows. +□ +Definition 3. For each pi | q we define ordpiα to be the smallest natural number +o such that αo ≡ 1 modpi. + +EZNA +OperationalProgramme +HumanResourcesDevelopment +2014-2020 +士 +EducationandLifelongLearning +avantuen-epyaoia-aaanaeun +Eupwaikn'Evwon +Co-financed byGreece and the European Union +European Social Fund4 +A. KONTOGEORGIS AND A. TEREZAKIS +It is clear that for ν ∈ N +αν ≡ 1 modpi ⇒ αν ≡ 1 modpj for all j ≤ i. +Therefore +ordpjα | ordpiα for j ≤ i. +On the other hand α ∈ N and αp−1 ≡ 1 modp so ordpα | p − 1. +Also since +σtτσ−t = τ αt we have that αm ≡ 1 modph, therefore ordpα | ordpiα | ordphα | m, +for 1 ≤ i ≤ h. +Lemma 4. The center CentG(τ) = ⟨τ, σordphα⟩. Moreover +|CentG(τ)| +ph += +m +ordph(α) =: m′ +Proof. The result follows by observing (τ νσt)τ(τ νσt)−1 = τ αt, for all 1 ≤ ν ≤ q, +1 ≤ t ≤ m. +□ +Remark 5. If ordpα = m then ordpiα = m for all 1 ≤ i ≤ h. +Lemma 6. If the group G = Cq ⋊ Cm is a subgroup of Aut(k[[t]]), then all orders +ordpiα = m/m′, for all 1 ≤ i ≤ h. +Proof. We will use the notation of the book of J.P.Serre on local fields [17]. By +[13, Th.1.1b] we have that the first gap i0 in the lower ramification filtration of the +cyclic group Cq satisfies (m, i0) = m′. +The ramification relation [17, prop. 9 p. 69] +αθi0(τ) = θi0(τ α) = θi0(στσ−1) = θ0(σ)i0θi0(τ), +implies that θ0(σ)i0 = α ∈ N. From (m, i0) = m′ and the fact that ordθ0(σ) = m +we obtain +m +m′ = ordθ0(σ)i0 = ordp(α). +Thus +m +m′ = ordpα|ordpiα|ordphα = m +m′ . +Hence all orders ordpiα = m/m′. +□ +Remark 7. If the KGB-obstruction vanishes and α ̸= 1, then by [11][prop. 5.9] +i0 ≡ −1 modm and ordpiα = m for all 1 ≤ i ≤ h. +3. Indecomposable Cq ⋊ Cm modules, modular representation theory +In this section we will describe the indecomposable Cq ⋊ Cm-modules. We will +give two methods in studying them. The first one is needed since it is in accordance +to the method we will give in order to describe indecomposable R[Cq⋊Cm]-modules. +The second one, using the structure of the socle, is the standard method of describ- +ing k[Cq ⋊ Cm]-modules in modular representation theory. + +ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP. +5 +3.1. Linear algebra method. The indecomposable modules of the Cq are deter- +mined by the Jordan normal forms of the generator τ of the cyclic group Cq. So +for each 1 ≤ κ ≤ ph there is exactly one Cq indecomposable module denoted by +Jκ. Therefore we have the following decomposition of an indecomposable Cq ⋊Cm- +module M considered as a Cq-module. +(2) +M = Jκ1 ⊕ · · · ⊕ Jκr. +Lemma 8. In the indecomposable module Jκ for every element E such that +(τ − Idκi)κi−1E ̸= 0 +the elements B = {E, (τ − Idκ)E, . . . , (τ − Idκ)κ−1E} form a basis of Jκ such that +the matrix of τ with respect to this basis is given by +(3) +τ = Idκ + +� +� +� +� +� +� +� +� +� +0 +· · · +· · · +· · · +0 +1 +... +... +0 +... +... +... +... +... +1 +0 +... +0 +· · · +0 +1 +0 +� +� +� +� +� +� +� +� +� +. +Proof. Since the set B has k-elements it is enough to prove that it consists of linear +independent elements. Indeed, consider a linear relation +λ0E + λ1(τ − Idκ)E + · · · + λκ−1(τ − Idκ)κ−1E = 0. +By applying (τ −Idκ)κ−1 we obtain λ0(τ −Idκ)κ−1 = 0, which gives us λ0 = 0. We +then apply (τ − Idκ)κ−2 to the linear relation and by the same argument we obtain +λ1 = 0 and we continue this way proving that λ0 = · · · = λκ−1 = 0. The matrix +form of τ in this basis is immediate. +□ +We will now prove that σ acts on each Jκ of eq. +(2) proving that r = 1. +Since the field k is algebraically closed and (m, p) = 1 we know that there is +a basis of M consisting of eigenvectors of σ. +There is an eigenvector E of σ, +which is not in the kernel of (τ − Idκ)κ1−1. Then the elements of the set B = +{E, (τ − Idκ)E, . . . , (τ − Idκ)κ1−1E} are linearly independent and form a direct Cq +summand of M isomorphic to Jκ1. +We will now show that this module is an k[Cq ⋊ Cm]-module. For this, we have +to show that the generator σ of Cm acts on the basis B. Observe that for every +0 ≤ i ≤ κ1 − 1 < ph +σ(τ − 1)i−1 = (τ α − 1)i−1σ. +Set e = E1 and κ = κ1. This means that the action of σ on e determines the action +of σ on all other basis elements eν := (τ − 1)ν−1e, 1 ≤ ν ≤ κ1. +Let us compute: +σei+1 = σ(τ − 1)ie = (τ α − 1)iζλ +me + +6 +A. KONTOGEORGIS AND A. TEREZAKIS +On the basis {e1, . . . , eκ1} the matrix τ is given by eq. (3) hence using the binomial +formula we compute +(4) +τ α = +� +� +� +� +� +� +� +� +� +� +� +� +1 +0 +· · · +· · · +· · · +0 +�α +1 +� +1 +... +... +�α +2 +� +�α +1 +� +... +... +... +�α +3 +� +�α +2 +� +... +1 +... +... +... +... +... +�α +1 +� +1 +0 +�α +k +� +� α +k−1 +� +· · · +�α +2 +� +�α +1 +� +1 +� +� +� +� +� +� +� +� +� +� +� +� +. +Thus τ α − 1 is a nilpotent matrix A = (aij) of the form: +aij = +��α +µ +� +if j = i − µ for some µ, 1 ≤ µ ≤ κ +0 +if j ≥ i +The ℓ-th power Aℓ = (a(ℓ) +ij ) of A is then computed by (keep in mind that aij = 0 +for i ≤ j) +a(ℓ) +ij = +� +i<ν1<···<νℓ−1 ℓ in order to have aij ̸= 0. Moreover for i = j + ℓ (which +is the the first non zero diagonal below the main diagonal) we have +ai,i+ℓ = ai,i+1ai+1,i+2 · · · ai+ℓ−1,i+ℓ = +�α +1 +�ℓ += αℓ. +Therefore, the matrix of Aℓ is of the following form: +(5) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +k − ℓ +� +�� +� +0 +· · · +· · · +0 +ℓ +� +�� +� +0 +· · · +0 +... +... +... +... +0 +· · · +· · · +0 +0 +· · · +0 +αℓ +... +0 +... +... +∗ +αℓ +... +... +... +... +... +... +... +0 +... +... +∗ +· · · +∗ +αℓ +0 +· · · +0 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Definition 9. We will denote by Vα(λ, κ) the indecomposable κ-dimensional G- +module given by the basis elements {(τ − 1)νe, ν = 0, . . . , κ − 1}, where σe = ζλ +me. +This definition is close to the notation used in [9]. +Lemma 10. The action of σ on the basis element ei of Vα(λ, κ) is given by: +(6) +σei = αi−1ζλ +mei + +κ +� +ν=i+1 +aνeν, +for some coefficients ai ∈ k. In particular the matrix of σ with respect to the basis +e1, . . . , eκ is lower triangular. + +ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP. +7 +Proof. Recall that ei = (τ − 1)i−1e1. Therefore +σei = σ(τ − 1)i−1e1 = (τ α − 1)i−1σe1 = ζλ +m(τ α − 1)i−1e1. +The result follows by eq. (5) +□ +We have constructed a set of indecomposable modules Vα(λ, κ). +Apparently +Vα(λ, κ) can not be isomorphic to Vα(λ′, κ′) if κ ̸= κ′, since they have different +dimensions. +Assume now that κ = κ′. Can the modules Vα(λ, κ) and Vα(λ′, κ) be isomorphic +for λ ̸= λ′? +The eigenvalues of the prime to p generator σ on Vα(λ, κ)are +ζλ +m, αζλ +m, . . . , ακ−1ζλ +m. +Similarly the eigenvalues for σ when acting on Vα(λ′, κ) are +ζλ′ +m , αζλ′ +m , . . . , ακ−1ζλ′ +m . +If the two sets of eigenvalues are different then the modules can not be isomorphic. +But even if λ ̸= λ′ modn the two sets of eigenvalues can still be equal. Even in this +case the modules can not be isomorphic. +Lemma 11. The modules Vα(λ1, κ) and Vα(λ2, κ) are isomorphic if and only if +λ1 ≡ λ2 modm. +Proof. Indeed, the module Vα(λ1, κ) has an eigenvector for the action of σ which +generates the Vα(λ1, κ) by powers of (τ − 1), i.e. the vectors +(7) +e, (τ − 1)e, (τ − 1)2e, . . . , (τ − 1)κ−1e +form a basis of Vα(λ1, κ). +The elements E which can generate Vα(λ1, κ) by powers of (τ − 1) are linear +combinations +E = +κ−1 +� +ν=0 +λi(τ − 1)νe, +for λi ∈ k and λ0 ̸= 0. +On the other hand using eq. (6) we see that σ with respect to the basis given in +eq. (7) admits the matrix form: +� +� +� +� +� +� +� +� +ζλ +m +0 +· · · +· · · +0 +0 +αζλ +m +0 +· · · +0 +... +... +... +... +... +... +... +... +... +0 +· · · +· · · +0 +ακ−1ζλ +m +� +� +� +� +� +� +� +� +. +It is now easy to see from the above matrix that every eigenvector of the eigenvalue +ανλ1, ν > 1 is expressed as a linear combination of the basis given in eq. (7), where +the coefficient of e is zero. +Therefore, the eigenvector of the eigenvalue ανζm can not generate the module +Vα(λ, κ) by powers of (σ − 1)ν. +□ + +8 +A. KONTOGEORGIS AND A. TEREZAKIS +3.2. The uniserial description. We will now give an alternative description of +the indecomposable Cq ⋊ Cm-modules, which is used in [2]. +It is known that Aut(Cq) ∼= F∗ +p × Q, for some abelian p-group Q. The repre- +sentation ψ : Cm → Aut(Cq) given by the action of Cm on Cq is known to factor +through a character χ : Cm → F∗ +p. The order of χ divides p−1 and χp−1 = χ−(p−1) +is the trivial one dimensional character. +For all i ∈ Z, χi defines a simple k[Cm]-module of k dimension one, which we +will denote by Sχi. For 0 ≤ ℓ ≤ m − 1 denote by Sℓ the simple module where +on which σ acts as ζℓ +m. Both Sχi, Sℓ can be seen as k[Cq ⋊ Cm]-modules using +inflation. Finally for 0 ≤ ℓ ≤ m − 1 we define χi(ℓ) ∈ {0, 1, . . . , m − 1} such that +Sχi(ℓ) ∼= Sℓ ⊗k Sχi. +There are q · m isomorphism classes of indecomposable k[Cq ⋊ Cm]-modules and +are all uniserial. An indecomposable k[Cq ⋊ Cm]-module U is unique determined +by its socle, which is the kernel of the action of τ − 1 on U, and its k-dimension. +For 0 ≤ ℓ ≤ m − 1 and 1 ≤ µ ≤ q, let Uℓ,µ be the indecomposable k[Cq ⋊ Cm] +module with socle Sa and k-dimension µ. Then Uℓ,µ is uniserial and its µ ascending +composition factors are the first µ composition factors of the sequence +Sℓ, Sχ−1(ℓ), Sχ−2(ℓ), . . . , Sχ−(p−2)(ℓ), Sℓ, Sχ−1(ℓ), Sχ−2(ℓ), . . . , Sχ−(p−2)(ℓ). +Notice that in our notation Vα(λ, κ) = Uλ+κ,κ. +Remark 12. The condition ordpi = m for all 1 ≤ i ≤ h, is equivalent to requiring +that ψi : Cm → Aut(Cpi) is faithful for all i. +4. Lifting of representations +Proposition 13. Let G = Cq ⋊ Cm. Assume that for all 1 ≤ i ≤ h, ordpia = m. +If the G-module V lifts to an R[G]-module ˜V , where K = Quot(R) is a field of +characterstic zero, then +m | +� +dim( ˜V ⊗R K) − dim( ˜V ⊗R K)Cq� +. +Moreover, if ˜V (ζαiκ +q +) is the eigenspace of the eigenvalue ζαiκ +q +of T acting on ˜V , +then +dim ˜V (ζκ +q ) = dim ˜V (ζακ +q ) = dim ˜V (ζα2κ +q +) = · · · = dim ˜V (ζαm−1κ +q +). +Proof. Consider a lifting ˜V of V . The generator τ of the cyclic part Cq has eigen- +values λ1, . . . , λs which are pn-roots of unity. Let ζq be a primitive q-root of unity. +Consider any eigenvalue λ ̸= 1. It is of the form λ = ζκ +q for some κ ∈ N, q ∤ κ. If E +is an eigenvector of T corresponding to λ, that is τE = ζκ +q E then +τσ−1E = σ−1τ αE = ζκαm−1 +q +σ−1E +and we have a series of eigenvectors E, σ−1E, σ−2E, · · · with corresponding eigen- +values ζκ +q , ζκα +q , ζκa2 +q +· · · , ζκαo +q +, where o = ordq/(q,k). Indeed, the integer o satisfies +the +καo ≡ κ modq ⇒ αm ≡ 1 mod +q +(q, k). +Therefore the eigenvalues λ ̸= 1 form orbits of size m, while the eigenspace of the +eigenvalue 1 is just the invariant space V G and the result follows. +□ + +ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP. +9 +5. Indecomposable Cq ⋊ Cm modules, integral representation theory +From now on V be a free R-module, where R is an integral local principal ideal +domain with maximal ideal mR, R has characteristic zero and that R contains all +q-th roots of unity and has characteristic zero. Let K = Quot(R). +The indecomposable modules for a cyclic group both in the ordinary and in the +modular case are described by writing down the Jordan normal form of a generator +of the cyclic group. Since in integral representation theory there are infinitely many +non-isomorphic indecomposable Cq-modules for q = ph, h ≥ 3, one is not expecting +to have a theory of Jordan normal forms even if one works over complete local +principal ideal domains [7], [8]. +Lemma 14. Let T be an element of order q = ph in End(V ), then the minimal +polynomial of T has simple eigenvalues and T is diagonalizable when seen as an +element in End(V ⊗ K). +Proof. Since T q = IdV , the minimal polynomial of T divides xq − 1, which has +simple roots over a field of characteristic zero. This ensures that T ∈ End(V ⊗ K) +is diagonalizable. +□ +Lemma 15. Let f(x) = (x − λ1)(x − λ2) · · · (x − λd) be the minimal polynomial of +T on V . There is an element E ∈ V , such that +E, (T − λ1IdV )E, (T − λ2IdV )(T − λ1IdV )E, . . . , (T − λd−1IdV ) · · · (T − λ1IdV )E +are linear independent elements in V ⊗ K. +Proof. Consider the endomorphisms for i = 1, . . . , d +Πi = +d +� +ν=1 +ν̸=i +(T − λνIdV ). +In the above product notice that T − λiIdV , T − λjIdV are commuting endomor- +phisms. Since the minimal polynomial of T has degree d all R-modules KerΠi are +strictly less than V . Moreover there is an element E such that E ̸∈ Ker(Πi) for all +1 ≤ i ≤ d. Consider a relation +(8) +d +� +µ=0 +γµ +µ +� +ν=0 +(T − λµIdV )E, +where �0 +ν=0(T − λνIdV )E = E. We fist apply the operator �d +ν=2(T − λνIdV ) to +eq. (8) and we obtain +0 = γ0Π1E, +and by the selection of E we have that a0 = 0. We now apply �d +ν=3(T − λνIdV ) +to eq. (8). We obtain that +0 = γ1 +d +� +ν=3 +(T − λνIdV )(T − λ1IdV ) = γ1Π2E, +and by the selection of E we have that γ1 = 0. We now apply �d +ν=4(T − λνIdV ) +to eq. (8) and we obtain +0 = γ2 +d +� +ν=4 +(T − λνIdV )(T − λ2IdV )(T − λ1IdV )E = γ2Π3E + +10 +A. KONTOGEORGIS AND A. TEREZAKIS +and by the selection of E we obtain γ3 = 0. Continuing this way we finally arrive +at γ0 = γ1 = · · · = γd−1 = 0. +□ +Lemma 16. Let V be a free R-module of rank R acted on by an automorphism +T : V → V of order ph. Assume that the minimal polynomial of T is of degree d +and has roots λ1, . . . , λd. Then T can be written as a matrix with respect to the +basis as follows: +(9) +� +� +� +� +� +� +� +� +� +λ1 +0 +· · · +· · · +0 +a1 +λ2 +... +... +0 +a2 +λ3 +... +... +... +... +... +... +0 +0 +· · · +0 +ad−1 +λd +� +� +� +� +� +� +� +� +� +Proof. By lemma 15 the elements +E, (T − λ1IdV )E, (T − λ2IdV )(T − λ1IdV )E, . . . , (T − λd−1IdV ) · · · (T − λ1IdV )E +form a free submodule of V of rank d. The theory of submodules of principal ideal +domains, there is a basis E1, E2, . . . , Ed of the free module V such that +E1 = E, +(10) +a1E2 = (T − λ1IdV )E1, +a2E3 = (T − λ2IdV )E2, +. . . +as−1Ed = (T − λd−1IdV )Ed−1. +Let us consider the module V1 = ⟨E1, . . . , Ed⟩ ⊂ V . By construction, the map T +restricts to an automorphism V1 → V1 with respect to the basis E1, . . . , Ed has the +desired form. We then consider the free module V/V1 and we repeat the procedure +for the minimal polynomial of T, which again acts on V/V1. The desired result +follows. +□ +Remark 17. The element T as defined in eq. (9) has order equal to the higher +order of the eigenvalues λ1, . . . , λd involved. Indeed, since we have assumed that +the eigenvalues are different the matrix is diagonalizable in Quot(R) and has order +equal to the maximal order of the eigenvalues involved. In particular it has order q +if there is at least one λi that is a primitive q-root of unity. The statement about +the order of T is not necessarily true if some of the eigenvalues are the same. For +instance the matrix +� +1 +0 +1 +1 +� +has infinite order over a field of characteristic zero. +Remark 18. The number of indecomposable R[T]-summands of V is given by +#{i : ai = 0} + 1. +A lift of a sum of indecomposable kCq-modules Jκ1 ⊕ · · · ⊕ Jκn can form an +indecomposable RCq-module. For example the indecomposable module where the + +ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP. +11 +generator T of Cq has the form +T = +� +� +� +� +� +� +� +� +� +λ1 +0 +· · · +· · · +0 +a1 +λ2 +... +... +0 +a2 +λ3 +... +... +... +... +... +... +0 +0 +· · · +0 +as−1 +λd +� +� +� +� +� +� +� +� +� +where a1 = · · · = aκ1−1 = 1, aκ1 ∈ mR, aκ1+1, . . . , aκ2+κ1−1 = 1, aκ2+κ1 ∈ +mR , etc reduces to a decomposable direct sum of Jordan normal forms of sizes +Jκ1, Jκ2−κ1, · · · . +Remark 19. It is an interesting question to classify these matrices up to conju- +gation with a matrix in GLd(R). It seems that the valuation of elements ai should +also play a role. +Definition 20. Let hi(x1, . . . , xj) be the complete symmetric polynomial of degree +i in the variables x1, . . . , xj. For instance +h3(x1, x2, x3) = x3 +1 + x2 +1x2 + x2 +1x3 + x1x2 +2 + x1x2x3 + x1x2 +3 + x3 +2 + x2 +2x3 + x2x2 +3 + x3 +3. +Set +L(κ, j, ν) = hκ(λj, λj+1, . . . , λj+ν) +A(i, j) = +� +aiai+1 · · · ai+j +if j ≥ 0 +0 +if j < 0 +Lemma 21. The matrix T α = (t(α) +ij ) is given by the following formula: +t(α) +ij += +� +� +� +� +� +λα +i +if i = j +A(j, i − j − 1) · L(α − (i − j), j, i − j) +if j < i +0 +if j > i +Proof. For j ≥ i the proof is trivial. When j < i and α = 1 it is immediate, since +L(x, ·, ·) ≡ 0, for every x ≤ 0. Assume this holds for α = n. If α = n + 1, +t(n+1) +ij += t(n) +ij tij = +r +� +k=1 +t(α) +ik tkj = λjt(α) +ij ++ ajt(α) +ij+1 = λjA(j, i − j − 1)L(α − (i − j), j, i − j)+ ++ ajA(j + 1, i − j − 2)L(α − (i − j − 1), j + 1, i − j − 1) = += A(j, i − j − 1) +� +λjhα−(i−j)(λj, . . . , λj) + hα−(i−j)+1(λj+1, . . . , λi) +� += += A(j, i − j − 1)hα−(i−j)+1(λj, . . . , λi) = += A(j, i − j − 1)L(α − (i − j) + 1, i, i − j). +□ +Remark 22. The space of homogeneous polynomials of degree k in n-variables +has dimension +�n−1+c +n−1 +� +. Since all q-roots of unity are reduced to 1 modulo mR the +quantity L(α − (i − j), j, i − j) is reduced to n = (i − j) + 1, c = α − (i − j) +�n − 1 + c +n − 1 +� += +� α +i − j +� +. +This equation is compatible with the computation of τ α given in eq. (4). + +12 +A. KONTOGEORGIS AND A. TEREZAKIS +Lemma 23. There is an eigenvector E of the generator σ of the cyclic group Cm +which is not an element in +s� +i=1 +Ker(Πi ⊗ K). +Proof. The eigenvectors E1, . . . , Ed of σ form a basis of the space V ⊗ K. +By +multiplying by certain elements in R, if necessary, we can assume that all Ei are in +V and their reductions Ei ⊗ R/mR, 1 ≤ i ≤ d give rise to a basis of eigenvectors of +a generator of the cyclic group Cm acting on V ⊗ R/mR. If every eigenvector Ei is +an element of some Ker(Πν) for 1 ≤ i ≤ d, then their reductions will be elements +in Ker(T − 1)d−1, a contradiction since the later kernel has dimension < d. +□ +Lemma 24. Let V be a free Cq ⋊ Cm-module, which is indecomposable as a Cq- +module. Consider the basis given in lemma 16. Then the value of σ(E1) determines +σ(Ei) for 2 ≤ i ≤ d. +Proof. Let σ be a generator of the cyclic group Cm. We will use the notation of +lemma 15. We use lemma 23 in order to select a suitable eigenvector of E1 of σ +and then form the basis E1, E2, . . . , Ed as given in eq. (10). We can compute the +action of σ on all basis elements Ei by +(11) +σ(ai−1Ei) = σ(T − λi−1IdV )Ei−1 = (T a − λi−1IdV )σ(Ei−1). +This means that one can define recursively the action of σ on all elements Ei. +Indeed, assume that +σ(Ei−1) = +d +� +ν=1 +γν,i−1Eν. +We now have +(T a − λi−1IdV )Eν = +d +� +µ=1 +t(α) +µ,νEµ − λi−1Eν += (λα +ν − λi−1)Eν + +d +� +µ=ν+1 +t(α) +µ,νEµ +We combine all the above to +ai−1σ(Ei) = +d +� +ν=1 +γν,i−1(λα +ν − λi−1)Eν + +d +� +ν=1 +γν,i−1 +d +� +µ=ν+1 +t(α) +µ,νEµ += +d +� +ν=1 +˜γν,iEν, +(12) +for a selection of elements γν,i ∈ R, which can be explicitly computed by collecting +the coefficients of the basis elements E1, . . . , Ed. +Observe that the quantity on the right hand side of eq. (12) must be divisible +by ai−1. Indeed, let v be the valuation of the local principal ideal domain R. Set +e0 = min +1≤ν≤d{v(˜γν,i)}. +If e0 < v(ai−1) then we divide eq. (12) by πe0 where π is the local uniformizer of +R, that is mR = πR. We then consider the divided equation modulo mR to obtain + +ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP. +13 +a linear dependence relation among the elements Ei ⊗ k, which is a contradiction. +Therefore e0 ≥ v(ai−1) and we obtain an equation +σ(Ei) = +d +� +ν=1 +˜γν,i +ai−1 +Eν = +d +� +ν=1 +γν,iEν. +□ +For example σ(E1) = ζϵ +mE1. We compute that +a1σ(E2) = (T α − λ1Id)σ(E1) +and +σ(E2) = (λα +1 − λ1) +a1 +ζϵ +µE1 + ζϵ +m +d +� +µ=2 +t(α) +µ,1 +a1 +Eµ += (λα +1 − λ1) +a1 +ζϵ +µE1 + ζϵ +m +d +� +µ=2 +A(1, µ − 2)L(α − (µ − 1), 1, µ − 1) +a1 +Eµ += (λα +1 − λ1) +a1 +ζϵ +µE1 + ζϵ +m +d +� +µ=2 +a1a2 · · · aµ−1hα−(µ−1)(λ1, λ2, . . . , λµ) +a1 +Eµ. +Proposition 25. Assume that no element a1, . . . , ad−1 given in eq. (9) is zero. +Given α ∈ N, α ≥ 1 and an element E1, which is not an element in �d +i=1 Ker(Πi ⊗ +K), if there is a matrix Γ = (γij), such that ΓTΓ−1 = T α and ΓE1 = ζϵ +mE1, then +this matrix Γ is unique. +Proof. We will use the idea leading to equation (11) replacing σ with Γ. We will +compute recursively and uniquely the entries γµ,i, arriving at the explicit formula +of eq. (18). +Observe that trivially γν,1 = 0 for all ν < 1 since we only allow 1 ≤ ν ≤ d. We +compute +˜γµ,i = γµ,i−1(λα +µ − λi−1) + +µ−1 +� +ν=1 +γν,i−1t(α) +µ,ν +(13) += γµ,i−1(λα +µ − λi−1) + +µ−1 +� +ν=1 +γν,i−1A(ν, µ − ν − 1)L +� +α − (µ − ν), ν, µ − ν) += γµ,i−1(λα +µ − λi−1) + +µ−1 +� +ν=1 +γν,i−1aνaν+1 · · · aµ−1hα−µ+ν(λν, λν+1, . . . , λµ) +Define +[λα +m − λx]j +i = +j� +x=i +(λα +µ − λx) +[a]j +i = +j� +x=i +ax +for i ≤ j. If i > j then both of the above quantities are defined to be equal to 1. + +14 +A. KONTOGEORGIS AND A. TEREZAKIS +Observe that for µ = 1 eq. (13) becomes +(14) +γ1,i = +1 +ai−1 +γ1,i−1(λα +1 − λi−1) +and we arrive at (assuming that Γ(E1) = ζϵ +mE1) +(15) +γ1,i = +ζϵ +m +a1a2 · · · ai−1 +i−1 +� +x=1 +(λα +1 − λx) = +ζϵ +m +a1a2 · · · ai−1 +[λα +1 − λx]i−1 +1 +. +For µ ≥ 2 we have γµ,1 = 0, since by assumption TE1 = ζϵ +mE1. Therefore eq. (13) +gives us +γµ,i = +i−2 +� +κ1=0 +[λα +µ − λx]i−1 +i−κ1 +[a]i−1 +i−1−κ1 +µ−1 +� +µ2=1 +γµ2,i−1−κ1[a]µ−1 +µ2 hα−µ+µ2(λµ2, . . . , λµ) += +µ−1 +� +µ2=1 +[a]µ−1 +µ2 hα−µ+µ2(λµ2, . . . , λµ) +i−2 +� +κ1=0 +[λα +µ − λx]i−1 +i−κ1 +[a]i−1 +i−1−κ1 +γµ2,i−1−κ1. +(16) +We will now prove eq. (16) by induction on i. For i = 2, µ ≥ 2 we have +γµ,2 = 1 +a1 +γµ,1(λα +µ − λ1) + 1 +a1 +µ−1 +� +µ2=1 +γµ2,1[a]µ−1 +µ2 hα−µ+µ2(λµ2, . . . , λµ) += 1 +a1 +[a]µ−1 +1 +hα−µ+1(λ1, . . . , λµ)γ1,1. + +ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP. +15 +Assume now that eq. (16) holds for computing γµ,i−1. We will treat the γµ,i case. +We have +γµ,i = (λα +µ − λi−1) +ai−1 +γµ,i−1 + +1 +ai−1 +µ−1 +� +µ2=1 +γµ2,i−1[a]µ−1 +µ2 hα−µ+µ2(λµ2, . . . , λµ) += (λα +µ − λi−1) +ai−1 +µ−1 +� +µ2=1 +[a]µ−1 +µ2 hα−µ+µ2(λµ2, . . . , λµ) +i−3 +� +κ1=0 +[λα +µ − λx]i−2 +i−1−κ1 +[a]i−2 +i−2−κ1 +γµ2,i−2−κ1 ++ +1 +ai−1 +µ−1 +� +µ2=1 +γµ2,i−1[a]µ−1 +µ2 hα−µ+µ2(λµ2, . . . , λµ) += +µ−1 +� +µ2=1 +[a]µ−1 +µ2 hα−µ+µ2(λµ2, . . . , λµ) +i−3 +� +κ1=0 +[λα +µ − λx]i−1 +i−1−κ1 +[a]i−1 +i−2−κ1 +γµ2,i−2−κ1 ++ +1 +ai−1 +µ−1 +� +µ2=1 +γµ2,i−1[a]µ−1 +µ2 hα−µ+µ2(λµ2, . . . , λµ) += +µ−1 +� +µ2=1 +[a]µ−1 +µ2 hα−µ+µ2(λµ2, . . . , λµ) +i−2 +� +κ1=1 +[λα +µ − λx]i−1 +i−κ1 +[a]i−1 +i−1−κ1 +γµ2,i−1−κ1 ++ +µ−1 +� +µ2=1 +[a]µ−1 +µ2 hα−µ+µ2(λµ2, . . . , λµ) +1 +ai−1 +γµ2,i−1 += +µ−1 +� +µ2=1 +[a]µ−1 +µ2 hα−µ+µ2(λµ2, . . . , λµ) +i−2 +� +κ1=0 +[λα +µ − λx]i−1 +i−κ1 +[a]i−1 +i−1−κ1 +γµ2,i−1−κ1 +and equation (16) is now proved. +We proceed recursively applying eq. (16) to each of the summands γµ2,i−1−κ1 +if µ2 > 1 and i − 1 − κ1 > 1. If µ2 = 1, then γµ2,i−1−κ1 is computed by eq. (14) +and if µ2 > 1 and i − 1 − κ1 ≤ 1 then γµ2,i−1−κ1 = 0. We can classify all iterations +needed by the set Σµ of sequences (µs, µs−1, . . . , µ3, µ2) such that +(17) +1 = µs < µs−1 < · · · < µ3 < µ2 < µ = µ1. +For example for µ = 5 the set of such sequences is given by +Σµ = {(1), (1, 2), (1, 3), (1, 2, 3), (1, 4), (1, 2, 4), (1, 3, 4), (1, 2, 3, 4)} +corresponding to the tree of iterations given in figure 1. The length of the sequence +(µs, µs−1, . . . , µ2) is given in eq. (17) is s − 1. In each iteration the i changes to +i − 1 − k thus we have the following sequence of indices +i1 = i → i2 = i−1−κ1 → i3 = i−2−(κ1+κ2) → · · · → is = i−(s−1)−(κ1+· · ·+κs−1) +For the sequence i1, i2, . . . , we might have it = 1 for t < s − 1. But in this case, +we will arrive at the element γµt+1,it = γµt,1 = 0 since µt > 1. This means that we +will have to consider only selections κ1, . . . , κs−1 such that is−1 ≥ 1. Therefore we + +16 +A. KONTOGEORGIS AND A. TEREZAKIS +µ = 5 +µ2 = 1 +µ2 = 2 +µ3 = 1 +µ2 = 3 +µ3 = 1 +µ3 = 2 +µ4 = 1 +µ2 = 4 +µ3 = 1 +µ3 = 2 +µ4 = 1 +µ3 = 3 +µ4 = 1 +µ4 = 2 +µ5 = 1 +Figure 1. +Iteration tree for µ = 5 +arrive at the following expression for µ ≥ 2 +γµ,i = +� +(µs,...,µ2)∈Σµ +[a]µ−1 +µ2 [a]µ2−1 +µ3 +· · · [a]µs−1−1 +µs +s +� +ν=2 +hα−µν−1+µν(λµν, . . . , λµν−1) +· +� +i=i1>i2>···>is≥1 +s−1 +� +ν=1 +[λα +µν − λx]iν−1 +iν+1+1 +[a]iν−1 +iν+1 +· γ1,is. += +� +(µs,...,µ2)∈Σµ +s +� +ν=2 +hα−µν−1+µν(λµν, . . . , λµν−1) +· +� +i=i1>i2>···>is≥1 +[a]µ−1 +1 +[a]i−1 +is +s−1 +� +ν=1 +[λα +µν − λx]iν−1 +iν+1+1 +ζϵ +m[λα +1 − λx]is−1 +1 +[a]is−1 +1 += +� +(µs,...,µ2)∈Σµ +s +� +ν=2 +hα−µν−1+µν(λµν, . . . , λµν−1)[a]µ−1 +1 +[a]i−1 +1 +ζϵ +m +� +i=i1>i2>···>is≥1 +s +� +ν=1 +[λα +µν − λx]iν−1 +iν+1+1 +(18) +where is+1 + 1 = 1 that is is+1 = 0. +□ +We will now prove that the matrix Γ of lemma 25 exists by cheking that ΓT = +T αΓ. Set (aµ,i) = ΓT, (bµ,i) = T αΓ. For i < d we have +aµ,i = +d +� +ν=1 +γµ,νtν,i = γµ,itii + γµ,i+1ti+1,i += γµ,iλi + γµ,i(λα +µ − λi) + +µ−1 +� +ν=1 +γν,it(α) +µ,ν += γµ,iλα +µ + +µ−1 +� +ν=1 +γν,it(α) +µ,ν = +µ +� +ν=1 +t(α) +µ,νγν,i = bµ,i. +For i = d we have: +aµ,d = +d +� +ν=1 +γµ,νtν,d = γµ,dtd,d = γµ,dλd + +ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP. +17 +while +bµ,d = +d +� +ν=1 +t(α) +µ,νγν,d = +µ−1 +� +ν=1 +t(α) +µ,νγν,d + λα +µγµ,d +This gives us the relation +(19) +(λd − λa +µ)γµ,d = +µ−1 +� +ν=1 +t(α) +µ,νγν,d +For µ = 1 using eq. (15) we have +γ1,dλd = γ1,dλα +1 ⇒ [λα +1 − λx]d +1 = 0. +This relation is satisfied if λα +1 is one of {λ1, . . . , λd}. Without loss of generality we +assume that +(20) +λ(a) +i += +� +λi+1 +if m ∤ i +λi−m+1 +if m | i +We have the following conditions: +µ = 2 +(λd − λα +2 )γ2,d = t(α) +2,1 γ1,d +µ = 3 +(λd − λα +3 )γ3,d = t(α) +3,1 γ1,d + t(α) +3,2 γ2,d +µ = 4 +(λd − λα +4 )γ4,d = t(α) +4,1 γ1,d + t(α) +4,2 γ2,d + t(α) +4,3 γ3,d +... +... +µ = d − 1 +(λd − λα +d−1)γd−1,d = t(α) +d−1,1γ1,d + t(α) +d−1,2γ2,d + · · · + t(α) +d−1,d−2γd−1,d +All these equations are true provided that γ1,d, . . . , γd−2,d = 0. Finally, for µ = d, +we have +(21) +(λd − λα +d )γd,d = +d−1 +� +ν=1 +t(α) +d,νγν,d +which is true provided that (λd − λα +d )γd,d = t(a) +d,d−1γd−1,d. +Lemma 26. For n ≥ 2 the vertical sum Sn of the products of every line of the +following array +y +1 +1 +(x1 − x2) +(x1 − x3) +· · · +· · · +(x1 − xn) +2 +(z − x1) +1 +(x1 − x3) +· · · +· · · +(x1 − xn) +3 +(z − x1) +(z − x2) +1 +... +... +... +... +... +... +... +... +... +... +... +... +... +n − 1 +(z − x1) +(z − x2) +· · · +(z − xn−2) +1 +(x1 − xn) +n +(z − x1) +(z − x2) +· · · +(z − xn−2) +(z − xn−1) +1 +is given by +Sn = +n +� +y=1 +n +� +ν=y+1 +(x1 − xν) +y−1 +� +µ=1 +(z − xµ) = (z − x2) · · · (z − xn). + +18 +A. KONTOGEORGIS AND A. TEREZAKIS +In particular when z = xn the sum is zero. +Proof. We will prove the lemma by induction. For n = 2 we have S2 = (x1 − x2) + +(z−x1) = z−x2. Assume that the equality holds for n. The sum Sn+1 corresponds +to the array: +y +1 +1 +(x1 − x2) +(x1 − x3) +· · · +(x1 − xn) +(x1 − xn+1) +2 +(z − x1) +1 +(x1 − x3) +· · · +(x1 − xn) +(x1 − xn+1) +3 +(z − x1) +(z − x2) +1 +... +... +... +... +... +... +... +... +... +n − 1 +(z − x1) +· · · +(z − xn−2) +1 +(x1 − xn) +(x1 − xn+1) +n +(z − x1) +(z − x2) +· · · +(z − xn−1) +1 +(x1 − xn+1) +n + 1 +(z − x1) +(z − x2) +· · · +(z − xn−1) +(z − xn) +1 +We have by definition Sn+1 = Sn(x1 − xn+1) + (z − x1)(z − x2) · · · (z − xn), which +by induction gives +Sn+1 = (z − x2) · · · (z − xn)(x1 − xn+1) + (z − x1)(z − x2) · · · (z − xn) += (z − x2) · · · (z − xn)(x1 − xn+1 + z − x1) +and gives the desired result. +□ +Lemma 27. Consider A < l < L < B. The quantity +� +l≤y≤L +[λa − λx]y−1 +A +· [λb − λx]B +y+1 +equals to +[λa − λx]l−1 +A +· [λb − λx]B +L+1 · [λa − λx]L +l − [λb − λx]L +l +(λa − λb) +Proof. We write +� +l≤y≤L +[λa − λx]y−1 +A +· [λb − λx]B +y+1 += [λa − λx]l−1 +A +· [λb − λx]B +L+1 · +� +l≤y≤L +[λa − λx]y−1 +l +· [λb − λx]L +y+1 +The last sum can be read as the vertical sum S of the products of every line in the +following array: +y +l +1 +(λb − λl+1)(λb − λl+2) +· · · +(λb − λL−1)(λb − λL) +l + 1 (λa − λl) +1 +(λb − λl+2) +· · · +(λb − λL−1)(λb − λL) +l + 2 (λa − λl)(λa − λl+1) +1 +... +... +... +... +... +... +... +... +... +L − 2(λa − λl)(λa − λl+1) +· · · +1 +(λb − λL−1)(λb − λL) +L − 1(λa − λl)(λa − λl+1) +· · · +(λa − λL−2) +1 +(λb − λL) +L +(λa − λl)(λa − λl+1) +· · · +(λa − λL−2)(λa − λL−1) +1 +If l = b, then lemma 26 implies that S = [λa − λx]L +b+1. Furthermore, if L = a then +S = 0. + +ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP. +19 +The quantity S cannot be directly computed using lemma 26, if l ̸= b. +We +proceed by forming the array: +y +b +1 +(λb − λb+1) +· · · +(λb − λl) +· · · +· · · +· · · +· · · +(λb − λL) +... +... +... +l − 1 (λa − λb) +· · · +1 +(λb − λl) +· · · +· · · +· · · +· · · +(λb − λL) +l +(λa − λb) +· · · +(λa − λl−1) +1 +(λb − λl+1)(λb − λl+2) +· · · +(λb − λL−1)(λb − λL) +l + 1 (λa − λb) +· · · +(λa − λl−1)(λa − λl) +1 +(λb − λl+2) +· · · +(λb − λL−1)(λb − λL) +l + 2 (λa − λb) +· · · +(λa − λl−1)(λa − λl)(λa − λl+1) +1 +... +... +... +... +... +... +... +... +... +L − 2(λa − λb) +· · · +(λa − λl−1)(λa − λl)(λa − λl+1) +· · · +1 +(λb − λL−1)(λb − λL) +L − 1(λa − λb) +· · · +(λa − λl−1)(λa − λl)(λa − λl+1) +· · · +(λa − λL−2) +1 +(λb − λL) +L +(λa − λb) +· · · +(λa − λl−1)(λa − λl)(λa − λl+1) +· · · +(λa − λL−2)(λa − λL−1) +1 +The value of this array is computed using lemma 26 to be equal to [λa −λx]L +b+1. We +observe that the sum of the products of the top left array can be computed using +lemma 26, while the sum of the products of the lower right array is S. +[λa − λx]l−1 +b +· S + [λa − λx]l−1 +b+1 · [λb − λx]L +l = [λa − λx]L +b+1 +we arrive at +[λa − λx]l−1 +b +S = [λa − λx]l−1 +b+1 +� +[λa − λx]L +l − [λb − λx]L +l +� +or equivalently +(λa − λb) · S = [λa − λx]L +l − [λb − λx]L +l +□ +Lemma 28. For all 1 ≤ µ ≤ d − 2 we have γµ,d = 0. +Proof. Let µ1 = µ > µ2 > · · · > µs = 1 ∈ Σµ be a selection of iterations and +d = i1 > i2 > · · · · · · is ≥ 1 > is+1 = 0 be the sequence of i’s. Using eq. (20) we +see that the quantity [λα +µν − λx]iν−1 +iν+1+1 ̸= 0 if and only if one of the following two +inequalities hold: +either +iν+1 >µν − mf(µν) +(22) +or +iν <µν + 2 − mf(µν), +(23) +where +f(x) = +� +1 +if m | x +0 +if m ∤ x +We will denote the above two inequalities by (22)ν,(23)ν when applied for the +integer ν. Assume, that for all 1 ≤ ν ≤ s one of the two inequalities (22)ν,(23)ν +hold, that is [λα +µν − λx]iν−1 +iν+1+1 ̸= 0. Inequality (22)s can not hold for ν = s since it +gives us 0 = is+1 > 1 = µs, we have m ∤ 1 = µs. +We will keep the sequence ¯µ : µ1 > µ2 > · · · > µs fixed and we will sum over all +possible selections of sequences of i1 > · · · is > is+1 = 0, that is we will show that +the sum +(24) +Γ¯µ,i := +� +i=i1>i2>···>is≥1 +s +� +ν=1 +[λα +µν − λx]iν−1 +iν+1+1 +is zero, which will show that γµ,d = 0 using eq. (18). + +20 +A. KONTOGEORGIS AND A. TEREZAKIS +Observe now that if (23)ν holds and m ∤ ν, ν −1, then (23)ν−1 also holds. Indeed +the combination of (23)ν and (22)ν−1 gives the impossible inequality +µν + 2 +(23)ν +> +iν +(22)ν−1 +> +µν−1. +Assume now that m | ν and (23)ν holds, then (23)ν−1 also holds. +Indeed the +combination of (23)ν and (22)ν−1 gives us +µν + 2 − m +(23)ν +> +iν +(22)ν−1 +> +µν−1 − mf(µν−1). +If m ∤ µν−1, then the above inequality is impossible since it implies that +µν + 2 − m > µν−1 > µν. +If m | µν−1, then the inequality is also impossible since it implies that µν + 2 > +µν−1 so if we write µν−1 = k′m and µν = km, k, k′ ∈ N, k′ > k, we arrive at +2 > (k′ − k)m ≥ m. This proves the following +Lemma 29. The inequality (22)ν−1 might be correct only in cases where m | µν−1, +m ∤ µν. +Assume that for all ν inequality (23) holds. Then for ν = 1 it gives us (recall +that µ ≤ d − 2) +(25) +µ + 2 ≤ d = i1 < µ1 + 2 − mf(µ1) = µ + 2 − mf(µ), +which is impossible. Therefore either there are ν such that none of the two inequal- +ities (22)ν, (23)ν hold (in this case the contribution to the sum is zero) or there are +cases where (22) holds. +The sumands appearing in eq. (24) can be zero, for example the sequence µ1 = +m > µ2 = 1 with i2 = 2 < i1 = d, s = 2 give the contribution +[λα +µ2 − λx]i2−1 +1 +[λα +µ1 − λx]d−1 +i2 += [λα +1 − λx]1 +1[λα +m − λx]d−1 +i2+1 = (λ2 − λ1)[λ1 − λx]d−1 +3 +while for i2 = 1 < i1 = d it gives the contribution +[λα +µ2 − λx]i2−1 +1 +[λα +µ1 − λx]d−1 +i2+1 = [λα +1 − λx]0 +1[λα +m − λx]d−1 +2 += [λ1 − λx]d−1 +2 +It is clear that these non-zero contributions cancel out when added. +Lemma 30. Assume that m | µν0−1 and m ∤ µν0, where (23)ν0 and (22)ν0−1 hold. +Then, we can eliminate µν0−1 and iν0 from both selections of the sequence of µ’s +and i’s, i.e. we can form the sequence of length s − 1 +¯µs−1 = µs < ¯µs−2 = µs−1 < · · · < ¯µν0−1 = µν0 < ¯µν0−2 = µν0−2 < · · · < ¯µ1 = µ1. +and the corresponding sequence of equal length +¯is−1 = is < ¯is−2 = is−1 < · · · < ¯iν0−1 = iν0−1 < ¯iν0 = iν0+1 < · · · < ¯i1 = i1 = d, +so that +Γ¯µ,i = +� +i1>···>is +s +� +ν=1 +[λα +µν − λx]iν−1 +iν+1+1 = (⋆) +� +¯i1>···>¯is−1 +s +� +ν=1 +ν̸=ν0−1 +[λα +µν − λx]iν−1 +iν+1+1, +where (⋆) is a non zero element. + +ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP. +21 +Proof. (of lemma 30) We are in the case m | µν0−1 and m ∤ µν0, where (23)ν0 and +(22)ν0−1 hold, +(26) +µν0−1 − m +(22)ν0−1 +< +iν0 +(23)ν0 +< +µν0 + 2, +or equivalently +µ0 := µν0−1 − m + 1 ≤ iν0 ≤ µν0 + 1 +For iν0+1 the inequality (22)ν0 iν0+1 > µν0 − mf(µν0) can not hold, since it implies +iν0+1 < iν0 +(23)ν0 +< +µν0 + 2 < iν0+1 + 2. +Observe that also +iν0+1 + 1 ≤ iν0 ≤ iν0−1 − 1. +Set l = max{µ0, iν0+1 + 1} and L = min{µν0 + 1, iν0−1 − 1}. Then y = iν0 satisfies +l ≤ y ≤ L. +By lemma 27 the quantity +� +l≤y≤L +[λµν0+1 − λx]y−1 +iν0+1+1 · [λµ0 − λx] +iν0−1−1 +y+1 +equals to +[λµν0+1 − λx]l−1 +iν0+1+1 · [λµ0 − λx] +iν0−1−1 +L+1 +· [λµν0+1 − λx]L +l − [λµ0 − λx]L +l +(λµν0+1 − λµ0) +(27) +[λµν0+1 − λx]L +iν0+1+1 · [λµ0 − λx] +iν0−1−1 +L+1 +− [λµν0+1 − λx]l−1 +iν0+1+1 · [λµ0 − λx] +iν0−1−1 +l +(λµν0+1 − λµ0) +Case A1 l = µ0 ≥ iν0+1 + 1. Then [λµ0 − λx]L +l = 0. +Case A2 l = iν0+1 + 1 > µ0. We set z := iν0+1, which is bounded by eq. (23)ν0+1 +that is +µ0 +Case A2 +≤ +z +(23)ν0+1 +≤ +µν0+1 + 1. +Notice that in this case m ∤ µν0+1. Indeed, we have assumed that inequality (23)ν0+1 +holds wich gives us +µν0−1 − m = µ0 − 1 +(Case A2) +< +iν0+1 +(23)ν0+1 +< +µν0+1 + 2 − m, +which implies that µν0−1 < µν0+1 + 2, a contradiction. Thus for l = z + 1 we +compute +� +µ0≤z≤µν0+1+1 +[λα +µν0+1 − λx] +iν0+1−1 +iν0+2+1 · [λµ0 − λx]L +l = += +� +µ0≤z≤µν0+1+1 +[λµν0+1+1 − λx]z−1 +iν0+2+1 · [λµ0 − λx]L +z+1 = += (⋆) · [λµν0+1+1 − λx] +µν0+1+1 +µ0 +− [λµ0 − λx] +µν0+1+1 +µ0 +λµν0+1+1 − λµ0+1 += 0. +Case B1 L = µν0 + 1 ≤ iν0−1 − 1. In this case [λµν0+1 − λx]L +l = 0. + +22 +A. KONTOGEORGIS AND A. TEREZAKIS +Case B2 L = iν0−1 − 1 < µν0 + 1. In this case eq. (27) is reduced to +[λµν0+1 − λx] +iν0−1−1 +iν0+1+1 +(λµν0+1 − λµ0) +This means that we have erased the µν0−1 from the product and we have +� +i1>···>is +s +� +ν=1 +[λα +µν − λx]iν−1 +iν+1+1 = (⋆) +� +i1>···>is +s +� +ν=1 +ν̸=ν0−1 +[λα +µν − λx]iν−1 +iν+1+1, +where (⋆) is a non zero element. This procedure gives us that the original quantity +[λα +µν0 − λx] +iν0−1 +iν0+1+1 · [λα +µν0−1 − λx] +iν0−1−1 +iν0+1 +after summing over iν0 becomes the quantity +[λα +µν0 − λx] +iν0−1−1 +iν0+1+1 = [λα +¯µν0−1 − λx] +¯iν0−1−1 +¯iν0+1 +, +that is we have eliminated the µν0−1 and iν0 from both selections of the sequence +of µ’s and i’s, i.e. we have the sequence of length s − 1 +¯µs−1 = µs < ¯µs−2 = µs−1 < · · · < ¯µν0−1 = µν0 < ¯µν0−2 = µν0−2 < · · · < ¯µ1 = µ1. +and the corresponding sequence of equal length +¯is−1 = is < ¯is−2 = is−1 < · · · < ¯iν0−1 = iν0−1 < ¯iν0 = iν0+1 < · · · < ¯i1 = i1 = d. +□ +Remark 31. One should be careful here since ¯iν0−1 = iν0−1 > iν0 > ¯iν0 = iν0+1, +so ¯iν0−1 > ¯iν0 + 1. This means that the new sequence of ¯is−1 > · · · > ¯i1 satisfies a +stronger inequality in the ν0 position, unless ν0 − 1 = d in the computation of γd,d. +Consider the set s, s − 1, . . . , ν0 such that m ∤ µν for s ≥ ν ≥ ν0 and assume +that m | µν0−1 and (23)ν0 and (22)ν0−1 hold. We apply lemma 30 and we obtain +a new sequence of µ’s with µν0−1 removed, provided that ν0 − 1 > 1. We continue +this way and in the sequence of µ’s we eliminate all possible inequalities like (26) +obtaining a series of µ which involves only inequalities of type (23). But this is not +possible if µ ≤ d − 2, according to equation (25). This proves that all γµ,d = 0 for +1 ≤ µ ≤ d − 2, this completes the proof of lemma 28. +□ +Lemma 32. If µ2 ̸= d − 1, then the contribution of the corresponding summand +Γ¯µ,i to γd,d is zero. +Proof. We are in the case µ = d = i. We begin the procedure of eliminating all +sequences of inequalities of the form (23)ν0, (22)ν0−1, where m | ν0−1, m ∤ ν0, using +lemma 30. For ν = 1 inequality (23)1 can not hold since it implies the impossible +inequality d = i1 < d + 2 − m. Therefore, (22)1 holds, that is i2 > d − m. On the +other hand we can assume that (23)2 holds by the elimination process, so we have +d − m +(22)1 +< i2 +(23)2 +< µ2 + 2. +Following the analysis of the proof of lemma 28 we see that the contribution to γd,d +is non zero if case B2 holds, that is (ν0 = 2 in this case) d − 1 = iν0−1 − 1 < µ2 + 1, +obtaining that µ2 = d − 1. +□ + +ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP. +23 +Lemma 33. Equation (21) holds, that is +(λd − λα +d )γd,d = +d−1 +� +ν=1 +t(α) +d,νγν,d = t(α) +d,d−1γd−1,d. +Proof. We will use the procedure of the proof of lemma 30. We recall that for +each fixed sequence of µs > · · · > µ1 we summed over all possible sequences i1 > +· · · > is+1 = 0. +In the final step the inequality (26) appears, for ν0 = 2, and +µν0 = µ2 = d − 1 and ν0 − 1 = 1 and µν0−1 = µ = d, that is: +0 = µν0−1 − m +(22)2 +< iν0 +(23)1 +< µν0 + 2 = d + 1. +As in the proof of lemma 30 we sum over y = iν and the result is either zero in case +B1 or in the B2 case, where µν0 = µ2 = d − 1 and µ0 = µν0−1 − m + 1 = d − m + 1, +the contribution is computed to be equal to +[λα +µν0+1 − λx] +iν0−1−1 +iν0+1+1 +(λµν0+1 − λm0) += [λα +d − λx]d−1 +i3+1 +λd − λα +d +. +The last µν0−1 = µ1 = d is eliminated in the above expression. This means that +for a fixed sequence µ1 > . . . > µs the contribution of the inner sum in eq. (24) is +given by +1 +λd − λα +d +· +� +d−1=i2>i3>···>is≥1 +s +� +ν=2 +[λα +µν − λx]iν−1 +iν+1+1. +Observe that µ1 = d does not appear in this expression and this expression corre- +sponds to the sequence ¯µ1 = µ2 = d − 1 > ¯µ2 = µ3 > · · · > ¯µs−1 = ¯µs = 1. Notice, +also that the problem described in remark 31 does not appear here, sence we erased +i1 which is not between some i’s but the first one. Therefore, we can relate it to +a similar expression that contributes to γd−1,d. Conversely every contribution of +γd−1,d gives rise to a contribution in γd,d, by multiplying by λd − λα +d . The desired +result follows by the expression of γµ,d given in eq. (18). +□ +We have shown so far how to construct matrices Γ, T so that +(28) +T q = 1, ΓTΓ−1 = T α. +We will now prove that Γ has order m. By equation (28) Γk should satisfy equation +ΓkTΓ−k = T αk. +Using proposition 25 asserting the uniqueness of such Γk with α replaced by αk we +have that the matrix multiplication of the entries of Γ giving rise to (γ(k) +µ,i ) = Γk +coincide to the values by the the recursive method of proposition (28) applied +for Γ′ = Γk, α′ = αk and Γ′E1 = ζϵk +m E1. +In particular for k = m, we have +αm ≡ 1 modpν for all 1 ≤ ν ≤ h, that is the matrix Γk should be recursively +constructed using proposition (28) for the relation ΓmTΓm = T, ΓmE1 = E1, +leading to the conclusion Γm = Id. +Notice that the first eigenvalue of Γ is a +primitive root of unity, therefore Γ has order exactly m. +By lemma 10 the action of σ in the special fibre is given by a lower triangular +matrix. Therefore, we must have +(29) +γν,i ∈ mr for ν < i. + +24 +A. KONTOGEORGIS AND A. TEREZAKIS +Proposition 34. If +(30) +v(λi − λj) > v(aν) for all 1 ≤ i, j ≤ d and 1 ≤ ν ≤ d − 1 +then the matrix (γµ,i) has entries in the ring R and is lower triangular modulo mR. +Proof. Assume that the condition of eq. (30) holds. In equation (18) we compute +the fraction +(31) +[a]µ−1 +1 +[a]i−1 +1 += +� +� +� +� +� +1 +[a]i−1 +µ +if i > µ +1 +if i = µ +[a]µ−1 +i +if i < µ +The number of (λα +µ − λx) factors in the numerator is equal to (recall that is+1 = 0) +s +� +ν=1 +(iν − 1 − iν+1 − 1 + 1) = i − s, +and i > µ ≥ s, so i − s > 0. Therefore, for the upper part of the matrix i > µ we +have i − s factors of the form (λα +i − λj) in the numerator and i − µ factors ax in +the denominator. Their difference is equal to (i − s) − (i − µ) = µ − s ≥ 0. By +assumption the matrix reduces to an upper triangular matrix modulo mR. +□ +Remark 35. The condition given in equation (30) can be satisfied in the following +way: It is clear that λi − λj ∈ mR. Even in the case vmR(λi − λj) = 1 we can +consider a ramified extension R′ of the ring R with ramification index e, in order to +make the valuation vmR′ (λi − λj) = e and then there is space to select vmR′ (ai) < +vmR′ (λi − λj). +Proposition 36. We have that +(32) +γi,i ≡ ζϵ +mαi−1 modmR +Let A = {a1, . . . , ad−1} ∈ R be the set of elements below the diagonal in eq. (9). If +ai ∈ mR, then +γµ,i ∈ mR for µ ̸= i, +that is Ei is an eigenvector for the reduced action of Γ modulo mR. If aκ1, . . . , aκr +the elements of the set A which are in mR, then the reduced matrix of Γ has the +form: +� +� +� +� +� +� +Γ1 +0 +· · · +0 +0 +Γ2 +... +... +... +... +... +0 +0 +· · · +0 +Γr +� +� +� +� +� +� +where Γ1, Γ2, . . . , Γr+1 for 1 ≤ ν ≤ r + 1 are (κν − κν−1) × (κν − κν−1) lower +triangular matrices (we set κ0 = 0, κr+1 = d). + +ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP. +25 +Proof. Consider the matrix Γ: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +γ11 +... +... +0 +γκ1,1 +· · · +γκ1,κ1 +γ11 +γκ1+1,κ1+1 +... +... +γµ,i +γκ2,κ1+1 +· · · +γκ2,κ2 +γκ1+1,κ1+1 +γµ,i +... +γκr+1,κr+1 +· · · +... +... +γκ1,κ1 +γκ2,κ2 +γd,κr+1 +· · · γd,d +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 ≤ i ≤ κ1 < m ≤ d +κ1 < i ≤ κ2 < µ ≤ d +We have that µ = i and the only element in Σµ which does not have any factor of +the form (λα +y − λx) is the sequence +1 = µs = µs−1 − 1 < µs−1 < · · · < µ2 = µ1 − 1 < µ1 = µ +For this sequence eq. (18) becomes +γi,i = +s +� +ν=2 +hα−1(λµν, λµν−1)ζϵ +m modmR, +which gives the desired result since hα−1(λµν, λµν−1) ≡ +�α +1 +� += α modmR. +For proving that all entries γµ,i ∈ mR for κν < i ≤ κν+1 < µ ≤ d, that is for +all entries bellow the central blocks, we observe that from equation (18) combined +with eq. (31) that γµ,i is divisible by [a]µ−1 +i += aiai+1 · · · aκν+1 · · · aµ−1 ∈ mR. +□ +Recall that by lemma 2 there is an 1 ≤ a0 ≤ m such that α = ζa0 +m . +Proposition 37. The indecomposable module V modulo mR breaks into a direct +sum of r + 1 indecomposable k[Cq ⋊ Cm] modules Vν, 1 ≤ ν ≤ r + 1. Each Vν is +isomorphic to Vα(ϵ + a0κν−1, κν − κν−1). +Proof. By eq. (32) the first eigenvalue of the reduced matrix block Γν is +ζϵ +mακν−1 = ζϵ+(κν−1)a0 +m +. +Since that first eigenvalue together with the size of the block determine the last +eigenvalue, that is the action of Cm on the socle the reduced block is uniquely +determined up to isomorphism. +□ +This way we arrive at a new obstruction. +Assume that the indecomposable +representation given by the matrix T as in lemma 16 reduces modulo mR to a sum +of Jordan blocks. Then the σ action on the leading elements of each Jordan block +in the special fibre should be described by the corresponding action of σ on the +leading eigenvector E of V . The corresponding actions on the special fibre should +be compatible. +This observation is formally given in proposition 1, which we now prove: Each set +Iν, 1 ≤ ν ≤ t corresponds to an indecomposable R[G]-module, which decomposes +to the indecomposables Vα(ϵµ, κµ), ν ∈ Iν of the special fiber. Indecomposable + +26 +A. KONTOGEORGIS AND A. TEREZAKIS +summands have different roots of unity in R, therefore � +µ∈Iν kν ≤ q, this is con- +dition (1.a). The second condition (1.b) comes from proposition 13. If 1 is one of +the possible eigenvalues of the lift T, then � +µ∈Iν κµ ≡ 1 modm. If all eigenvalues +of the lift T are different than one, then � +µ∈Iν κµ ≡ 0 modm. If #Iν = q, then +there is one zero eigenvalue and the sum equals 1 modm. +It is clear by eq. (32) that condition (1.c) is a necessary condition. On the other +hand if (1.c) is satisfied we can write (after a permutation if necessary) the set +{1, . . . , S}, S = �t +ν=1 #Iν as +J1 = {1, 2, . . . , κ(1) +1 , κ(1) +1 ++ 1, . . . , κ(1) +1 ++ κ(1) +2 , . . . , +r1 +� +j=1 +κ(1) +j += b1}, I1 = {κ(1) +1 , . . . , κ(1) +r1 } +J2 = {b1 + 1, b1 + 2, . . . , b2 = b1 + +r2 +� +j=1 +κ(2) +j }, I2 = {κ(2) +1 , . . . , κ(2) +r2 } +· · · · · · +Js = {bs−1 + 1, bt−1 + 2, . . . , bt = S}, Is = {κ(s) +1 , . . . , κ(s) +rs } +The matrix given in eq. (9), where +ai = +� +� +� +� +� +0 +if i ∈ {b1, . . . , bs−1} +π +if i ∈ {κ(ν) +1 , κ(ν) +1 ++ κ(ν) +2 , κ(ν) +1 ++ κ(ν) +2 ++ κ(ν) +3 , . . . , κ(ν) +1 ++ κ(ν) +2 ++ · · · + κ(ν) +rν−1} +1 +otherwise +lifts the τ generator, and by (12) there is a well defined extended action of the σ +as well. +Example: Consider the group q = 52, m = 4, α = 7, +G = C52 ⋊ C4 = ⟨σ, τ|σ4 = τ 25 = 1, στσ−1 = τ 7⟩. +Observe that ord57 = ord527 = 4. +• The module V (ϵ, 25) is projective and is known to lift in characteristic zero. +This fits well with proposition 1, since 4 | 25 − 1 = 4 · 6. +• The modules V (ϵ, κ) do not lift in characteristic zero if 4 ∤ κ or 4 ∤ κ − 1. +Therefore only V (ϵ, 1), V (ϵ, 4), V (ϵ, 5), V (ϵ, 8), V (ϵ, 9), V (ϵ, 12), V (ϵ, 13), +V (ϵ, 16), V (ϵ, 17), V (ϵ, 20), V (ϵ, 21), V (ϵ, 24), V (ϵ, 25) lift. +• The module V (1, 2) ⊕ V (3, 2) lift to characteristic zero, where the matrix +of T with respect to a basis E1, E2, E3, E4 is given by +T = +� +� +� +� +ζq +0 +0 +0 +1 +ζ2 +q +0 +0 +0 +π +ζ3 +q +0 +0 +0 +1 +ζ4 +q +� +� +� +� +and σ(E1) = ζqE1. +• The module V (1, 2) ⊕ V (1, 2) does not lift in characteristic zero. There is +no way to permute the direct summands so that the eigenvalues of σ are +given by ζϵ +m, αζϵ +m, α2ζϵ +m, α3ζϵ +m. Notice that α = 2 = ζm. +• The module V (ϵ1, 21)⊕V (221·ϵ1, 23) does not lift in characteristic zero. The +sum 21+24 is divisible by 4, ϵ2 = 221ϵ1 is compatible, but 21+23 = 44 > 25 +so the representation of T in the supposed indecomposable module formed + +ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP. +27 +by their sum can not have different eigenvalues which should be 25-th roots +of unity. +References +[1] J. L. Alperin. Local representation theory, volume 11 of Cambridge Studies in Advanced +Mathematics. Cambridge University Press, Cambridge, 1986. Modular representations as an +introduction to the local representation theory of finite groups. +[2] Frauke M. Bleher, Ted Chinburg, and Aristides Kontogeorgis. Galois structure of the holo- +morphic differentials of curves. J. Number Theory, 216:1–68, 2020. +[3] T. Chinburg, R. Guralnick, and D. Harbater. Oort groups and lifting problems. Compos. +Math., 144(4):849–866, 2008. +[4] Ted Chinburg, Robert Guralnick, and David Harbater. The local lifting problem for actions +of finite groups on curves. Ann. Sci. ´Ec. Norm. Sup´er. (4), 44(4):537–605, 2011. +[5] Ted Chinburg, Robert Guralnick, and David Harbater. Global Oort groups. J. Algebra, +473:374–396, 2017. +[6] Huy Dang, Soumyadip Das, Kostas Karagiannis, Andrew Obus, and Vaidehee Thatte. Local +oort groups and the isolated differential data criterion, 2019. +[7] A. Heller and I. Reiner. Representations of cyclic groups in rings of integers. I. Ann. of Math. +(2), 76:73–92, 1962. +[8] A. Heller and I. Reiner. Representations of cyclic groups in rings of integers. II. Ann. of Math. +(2), 77:318–328, 1963. +[9] Sotiris Karanikolopoulos and Aristides Kontogeorgis. Representation of cyclic groups in pos- +itive characteristic and Weierstrass semigroups. J. Number Theory, 133(1):158–175, 2013. +[10] Aristides Kontogeorgis and Alexios Terezakis. The canonical ideal and the deformation theory +of curves with automorphisms, 2021. +[11] Andrew Obus. The (local) lifting problem for curves. In Galois-Teichm¨uller theory and arith- +metic geometry, volume 63 of Adv. Stud. Pure Math., pages 359–412. Math. Soc. Japan, +Tokyo, 2012. +[12] Andrew Obus. A generalization of the Oort conjecture. Comment. Math. Helv., 92(3):551– +620, 2017. +[13] Andrew Obus and Rachel Pries. Wild tame-by-cyclic extensions. J. Pure Appl. Algebra, +214(5):565–573, 2010. +[14] Andrew Obus and Stefan Wewers. Cyclic extensions and the local lifting problem. Ann. of +Math. (2), 180(1):233–284, 2014. +[15] Florian Pop. The Oort conjecture on lifting covers of curves. Ann. of Math. (2), 180(1):285– +322, 2014. +[16] Jean-Pierre Serre. Linear representations of finite groups. Springer-Verlag, New York, 1977. +Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathe- +matics, Vol. 42. +[17] Jean-Pierre Serre. Local fields. Springer-Verlag, New York, 1979. Translated from the French +by Marvin Jay Greenberg. +[18] Bradley Weaver. The local lifting problem for D4. Israel J. Math., 228(2):587–626, 2018. +Department of Mathematics, National and Kapodistrian University of Athens Pane- +pistimioupolis, 15784 Athens, Greece +Email address: kontogar@math.uoa.gr +Department of Mathematics, National and Kapodistrian University of Athens, Panepis- +timioupolis, 15784 Athens, Greece +Email address: aleksistere@math.uoa.gr + diff --git a/DdAzT4oBgHgl3EQfGvsw/content/tmp_files/load_file.txt b/DdAzT4oBgHgl3EQfGvsw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f6f806a23cde0d14d4301b0be929482fb4c4297 --- /dev/null +++ b/DdAzT4oBgHgl3EQfGvsw/content/tmp_files/load_file.txt @@ -0,0 +1,1116 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf,len=1115 +page_content='ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' ARISTIDES KONTOGEORGIS AND ALEXIOS TEREZAKIS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We give a necessary and sufficient condition for a modular rep- resentation of a group G = Cph ⋊ Cm in a field of characteristic zero to be lifted to a representation over local principal ideal domain of characteristic zero containing the ph roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Introduction The lifting problem for a representation ρ : G → GLn(k), where k is a field of characteristic p > 0, is about finding a local ring R of char- acteristic 0, with maximal ideal mR such that R/mR = k, so that the following diagram is commutative: GLn(R) � G � � GLn(k) Equivalently one asks if there is a free R-module V , which is also an R[G]-module such that V ⊗RR/mR is the k[G]-module corresponding to our initial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We know that projective k[G]-modules lift in characteristic zero, [16, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 15], but for a general k[G]-module such a lifting is not always possible, for example, see [10, prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This article aims to study the lifting problem for the group G = Cq⋊Cm, where Cq is a cyclic group of order ph and Cm is a cyclic group of order m, (p, m) = 1 and give necessary and sufficient condition in order to lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We assume that the local ring R contains the q-roots of unity and k is algebraically closed, and we might need to consider a ramified extension of R, in order to ensure that certain q-roots of unit are distant in the mR-topology, see remark 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' An example of such a ring R is the ring of Witt vectors W(k)[ζq] with the q-roots of unity adjoined to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We notice that a decomposable R[G]-module V gives rise to a decomposable R-module modulo mR and also an indecomposable R[G]-module can break in the reduction modulo mR into a direct sum of indecomposable k[G]-summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We also give a classification of k[Cq ⋊ Cm]-modules in terms of Jordan decomposition and give the relation with the more usual uniserial description in terms of their socle [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Date: January 4, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Lifting of representations, modular representation theory, integral representation theory, Generalized Oort conjecture, metacyclic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='01032v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='AG] 3 Jan 2023 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' KONTOGEORGIS AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' TEREZAKIS Our interest to this problem comes from the problem of lifting local actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The local lifting problem considers the following question: Does there exist an extension Λ/W(k), and a representation ˜ρ : G �→ Aut(Λ[[T]]), such that if t is the reduction of T, then the action of G on Λ[[T]] reduces to the action of G on k[[t]]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If the answer to the above question is positive, then we say that the G-action lifts to characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' A group G for which every local G-action on k[[t]] lifts to characteristic zero is called a local Oort group for k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Notice that cyclic groups are always local Oort groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This result was known as the “Oort conjecture”, which was recently proved by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Pop [15] using the work of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Obus and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Wewers [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' There are a lot of obstructions that prevent a local action to lift in characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Probably the most important of these obstructions in the KGB-obstruction [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' It is believed that this is the only obstruction for the local lifting problem, see [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' In [10, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 3] the authors have given a criterion for the local lifting which involves the lifting of a linear representation of the same group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The case G = Cq ⋊Cm and especially the case of dihedral groups Dq = Cq ⋊C2, is a problem of current interest in the theory of local liftings, see [12], [6], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For more details on the local lifting problem we refer to [3], [4], [5], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Keep also in mind that the Cq ⋊ Cm groups were important to the study of group actions in holomorphic differentials of curves defined over fields of positive characteristic p, where the group involved has cyclic p-Sylow subgroup, see [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let us now describe the method of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For understanding the splitting of indecomposable R[G]-modules modulo mR, we develop a version of Jordan normal form in lemma 16 for endomorphisms T : V → V of order ph, where V is a free module of rank d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We give a way to select this basis, by selecting an initial suitable element E ∈ V , see lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The normal form (as given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (9)) of the element T of order q, determines the decomposition of the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We show that for every indecomposable summand Vi of V , we can select E as an eigenvalue of the generator σ of Cm and then by forcing the relation ΓT = T αΓ to hold, we see how the action of σ can be extended recursivelly to an action of σ on Vi, this is done in lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proving that this gives indeed a well defined action is a technical computation and is done in lemmata 26, 27, 28, 32, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The important thing here is that the definition of the action of σ on E is the “initial condition” of a dynamical system that determines the action of Cm on the indecomposable summand Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The R[Cq⋊Cm] indecomposable module Vi can break into a direct sum Vα(ϵν, κν)-modules 1 ≤ ν ≤ s (for a precise definition of them see definition 9, notice that κi denotes the dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The action of σ on each Vα(ϵν, κν) can be uniquely determined by the action of σ on an initial basis element as shown in section 3, again by a “dynamical system” approach, where we need s initial conditions, one for each Vα(ϵν, κν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The lifting condition essentially means that the indecomposable summands Vα(ϵ, κ) of the special fibre, should be able to be rearranged in a suitable way, so that they can be obtained as reductions of indecomposable R[Cq ⋊ Cq]-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The precise expression of our lifting criterion is given in the following proposition: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Consider a k[G]-module M which is decomposed as a direct sum M = Vα(ϵ1, κ1) ⊕ · · · ⊕ Vα(ϵs, κs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 3 The module lifts to an R[G]-module if and only if the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , s} can be written as a disjoint union of sets Iν, 1 ≤ ν ≤ t so that a � µ∈Iν κµ ≤ q, for all 1 ≤ ν ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' b � µ∈Iν κµ ≡ a modm for all 1 ≤ ν ≤ t, where a ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' c For each ν, 1 ≤ ν ≤ t there is an enumeration σ : {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , #Iν} → Iν ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='., s}, such that ϵσ(2) = ϵσ(1)ακσ(1), ϵσ(3) = ϵσ(3)ακσ(3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' ϵσ(s) = ϵσ(s−1)ακσ(s−1) In the above proposition, each set Iν corresponds to a collection of modules Vα(ϵµ, κµ), µ ∈ Iν which come as the reduction of an indecomposable R[Cq ⋊ Cm]- module Vν of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Aknowledgements A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Terezakis is a recipient of financial support in the context of a doctoral thesis (grant number MIS-5113934).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The implementation of the doc- toral thesis was co-financed by Greece and the European Union (European Social Fund-ESF) through the Operational Programme—Human Resources Development, Education and Lifelong Learning—in the context of the Act—Enhancing Human Resources Research Potential by undertaking a Doctoral Research—Sub-action 2: IKY Scholarship Programme for Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' candidates in the Greek Universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Notation Let τ be a generator of the cyclic group Cq and σ be a generator of the cyclic group Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The group G is given in terms of generators and relations as follows: G = ⟨σ, τ|τ q = 1, σm = 1, στσ−1 = τ α for some α ∈ N, 1 ≤ α ≤ ph − 1, (α, p) = 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The integer α satisfies the following congruence: (1) αm ≡ 1 modq as one sees by computing τ = σmτσ−m = τ αm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Also the integer α can be seen as an element in the finite field Fp, and it is a (p − 1)-th root of unity, not necessarily primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' In particular the following holds: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let ζm ∈ k be a fixed primitive m-th root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' There is a natural number a0, 0 ≤ a0 < m − 1 such that α = ζa0 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The integer α if we see it as an element in k is an element in the finite field Fp ⊂ k, therefore αp−1 = 1 as an element in Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let ordp(α) be the order of α in F∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' By eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (1) we have that ordp(α) | p−1 and ordp(α) | m, that is ordp(α) | (p−1, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The primitive m-th root of unity ζm generates a finite field Fp(ζm) = Fpν for some integer ν, which has cyclic multiplicative group Fpν\\{0} containing both the cyclic groups ⟨ζm⟩ and ⟨α⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Since for every divisor δ of the order of a cyclic group C there is a unique subgroup C′ < C of order δ we have that α ∈ ⟨ζm⟩, and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For each pi | q we define ordpiα to be the smallest natural number o such that αo ≡ 1 modpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=" EZNA OperationalProgramme HumanResourcesDevelopment 2014-2020 士 EducationandLifelongLearning avantuen-epyaoia-aaanaeun Eupwaikn'Evwon Co-financed byGreece and the European Union European Social Fund4 A." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' KONTOGEORGIS AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' TEREZAKIS It is clear that for ν ∈ N αν ≡ 1 modpi ⇒ αν ≡ 1 modpj for all j ≤ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore ordpjα | ordpiα for j ≤ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' On the other hand α ∈ N and αp−1 ≡ 1 modp so ordpα | p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Also since σtτσ−t = τ αt we have that αm ≡ 1 modph, therefore ordpα | ordpiα | ordphα | m, for 1 ≤ i ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The center CentG(τ) = ⟨τ, σordphα⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Moreover |CentG(τ)| ph = m ordph(α) =: m′ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The result follows by observing (τ νσt)τ(τ νσt)−1 = τ αt, for all 1 ≤ ν ≤ q, 1 ≤ t ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If ordpα = m then ordpiα = m for all 1 ≤ i ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If the group G = Cq ⋊ Cm is a subgroup of Aut(k[[t]]), then all orders ordpiα = m/m′, for all 1 ≤ i ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We will use the notation of the book of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='Serre on local fields [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' By [13, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='1b] we have that the first gap i0 in the lower ramification filtration of the cyclic group Cq satisfies (m, i0) = m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The ramification relation [17, prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 9 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 69] αθi0(τ) = θi0(τ α) = θi0(στσ−1) = θ0(σ)i0θi0(τ), implies that θ0(σ)i0 = α ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' From (m, i0) = m′ and the fact that ordθ0(σ) = m we obtain m m′ = ordθ0(σ)i0 = ordp(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Thus m m′ = ordpα|ordpiα|ordphα = m m′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Hence all orders ordpiα = m/m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If the KGB-obstruction vanishes and α ̸= 1, then by [11][prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='9] i0 ≡ −1 modm and ordpiα = m for all 1 ≤ i ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Indecomposable Cq ⋊ Cm modules, modular representation theory In this section we will describe the indecomposable Cq ⋊ Cm-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We will give two methods in studying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The first one is needed since it is in accordance to the method we will give in order to describe indecomposable R[Cq⋊Cm]-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The second one, using the structure of the socle, is the standard method of describ- ing k[Cq ⋊ Cm]-modules in modular representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Linear algebra method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The indecomposable modules of the Cq are deter- mined by the Jordan normal forms of the generator τ of the cyclic group Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' So for each 1 ≤ κ ≤ ph there is exactly one Cq indecomposable module denoted by Jκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore we have the following decomposition of an indecomposable Cq ⋊Cm- module M considered as a Cq-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (2) M = Jκ1 ⊕ · · · ⊕ Jκr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' In the indecomposable module Jκ for every element E such that (τ − Idκi)κi−1E ̸= 0 the elements B = {E, (τ − Idκ)E, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , (τ − Idκ)κ−1E} form a basis of Jκ such that the matrix of τ with respect to this basis is given by (3) τ = Idκ + � � � � � � � � � 0 · · · · · · 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 0 · · 0 1 0 � � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Since the set B has k-elements it is enough to prove that it consists of linear independent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Indeed, consider a linear relation λ0E + λ1(τ − Idκ)E + · · · + λκ−1(τ − Idκ)κ−1E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' By applying (τ −Idκ)κ−1 we obtain λ0(τ −Idκ)κ−1 = 0, which gives us λ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We then apply (τ − Idκ)κ−2 to the linear relation and by the same argument we obtain λ1 = 0 and we continue this way proving that λ0 = · · · = λκ−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The matrix form of τ in this basis is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ We will now prove that σ acts on each Jκ of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (2) proving that r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Since the field k is algebraically closed and (m, p) = 1 we know that there is a basis of M consisting of eigenvectors of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' There is an eigenvector E of σ, which is not in the kernel of (τ − Idκ)κ1−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Then the elements of the set B = {E, (τ − Idκ)E, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , (τ − Idκ)κ1−1E} are linearly independent and form a direct Cq summand of M isomorphic to Jκ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We will now show that this module is an k[Cq ⋊ Cm]-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For this, we have to show that the generator σ of Cm acts on the basis B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Observe that for every 0 ≤ i ≤ κ1 − 1 < ph σ(τ − 1)i−1 = (τ α − 1)i−1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Set e = E1 and κ = κ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This means that the action of σ on e determines the action of σ on all other basis elements eν := (τ − 1)ν−1e, 1 ≤ ν ≤ κ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let us compute: σei+1 = σ(τ − 1)ie = (τ α − 1)iζλ me 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' KONTOGEORGIS AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' TEREZAKIS On the basis {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , eκ1} the matrix τ is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (3) hence using the binomial formula we compute (4) τ α = � � � � � � � � � � � � 1 0 · · · · · · 0 �α 1 � 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' �α 2 � �α 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' �α 3 � �α 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' �α 1 � 1 0 �α k � � α k−1 � · · �α 2 � �α 1 � 1 � � � � � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Thus τ α − 1 is a nilpotent matrix A = (aij) of the form: aij = ��α µ � if j = i − µ for some µ, 1 ≤ µ ≤ κ 0 if j ≥ i The ℓ-th power Aℓ = (a(ℓ) ij ) of A is then computed by (keep in mind that aij = 0 for i ≤ j) a(ℓ) ij = � i<ν1<···<νℓ−1 ℓ in order to have aij ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Moreover for i = j + ℓ (which is the the first non zero diagonal below the main diagonal) we have ai,i+ℓ = ai,i+1ai+1,i+2 · · · ai+ℓ−1,i+ℓ = �α 1 �ℓ = αℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore, the matrix of Aℓ is of the following form: (5) � � � � � � � � � � � � � � k − ℓ � �� � 0 · · · · 0 ℓ � �� � 0 · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 0 · · · · 0 0 · · 0 αℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' ∗ αℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' ∗ · · ∗ αℓ 0 · · 0 � � � � � � � � � � � � � � Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We will denote by Vα(λ, κ) the indecomposable κ-dimensional G- module given by the basis elements {(τ − 1)νe, ν = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , κ − 1}, where σe = ζλ me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This definition is close to the notation used in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The action of σ on the basis element ei of Vα(λ, κ) is given by: (6) σei = αi−1ζλ mei + κ � ν=i+1 aνeν, for some coefficients ai ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' In particular the matrix of σ with respect to the basis e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , eκ is lower triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Recall that ei = (τ − 1)i−1e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore σei = σ(τ − 1)i−1e1 = (τ α − 1)i−1σe1 = ζλ m(τ α − 1)i−1e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The result follows by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (5) □ We have constructed a set of indecomposable modules Vα(λ, κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Apparently Vα(λ, κ) can not be isomorphic to Vα(λ′, κ′) if κ ̸= κ′, since they have different dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Assume now that κ = κ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Can the modules Vα(λ, κ) and Vα(λ′, κ) be isomorphic for λ ̸= λ′?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The eigenvalues of the prime to p generator σ on Vα(λ, κ)are ζλ m, αζλ m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , ακ−1ζλ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Similarly the eigenvalues for σ when acting on Vα(λ′, κ) are ζλ′ m , αζλ′ m , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , ακ−1ζλ′ m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If the two sets of eigenvalues are different then the modules can not be isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' But even if λ ̸= λ′ modn the two sets of eigenvalues can still be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Even in this case the modules can not be isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The modules Vα(λ1, κ) and Vα(λ2, κ) are isomorphic if and only if λ1 ≡ λ2 modm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Indeed, the module Vα(λ1, κ) has an eigenvector for the action of σ which generates the Vα(λ1, κ) by powers of (τ − 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' the vectors (7) e, (τ − 1)e, (τ − 1)2e, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , (τ − 1)κ−1e form a basis of Vα(λ1, κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The elements E which can generate Vα(λ1, κ) by powers of (τ − 1) are linear combinations E = κ−1 � ν=0 λi(τ − 1)νe, for λi ∈ k and λ0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' On the other hand using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (6) we see that σ with respect to the basis given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (7) admits the matrix form: � � � � � � � � ζλ m 0 · · · · 0 0 αζλ m 0 · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 0 · · · · 0 ακ−1ζλ m � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' It is now easy to see from the above matrix that every eigenvector of the eigenvalue ανλ1, ν > 1 is expressed as a linear combination of the basis given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (7), where the coefficient of e is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore, the eigenvector of the eigenvalue ανζm can not generate the module Vα(λ, κ) by powers of (σ − 1)ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' KONTOGEORGIS AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' TEREZAKIS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The uniserial description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We will now give an alternative description of the indecomposable Cq ⋊ Cm-modules, which is used in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' It is known that Aut(Cq) ∼= F∗ p × Q, for some abelian p-group Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The repre- sentation ψ : Cm → Aut(Cq) given by the action of Cm on Cq is known to factor through a character χ : Cm → F∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The order of χ divides p−1 and χp−1 = χ−(p−1) is the trivial one dimensional character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For all i ∈ Z, χi defines a simple k[Cm]-module of k dimension one, which we will denote by Sχi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For 0 ≤ ℓ ≤ m − 1 denote by Sℓ the simple module where on which σ acts as ζℓ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Both Sχi, Sℓ can be seen as k[Cq ⋊ Cm]-modules using inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Finally for 0 ≤ ℓ ≤ m − 1 we define χi(ℓ) ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , m − 1} such that Sχi(ℓ) ∼= Sℓ ⊗k Sχi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' There are q · m isomorphism classes of indecomposable k[Cq ⋊ Cm]-modules and are all uniserial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' An indecomposable k[Cq ⋊ Cm]-module U is unique determined by its socle, which is the kernel of the action of τ − 1 on U, and its k-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For 0 ≤ ℓ ≤ m − 1 and 1 ≤ µ ≤ q, let Uℓ,µ be the indecomposable k[Cq ⋊ Cm] module with socle Sa and k-dimension µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Then Uℓ,µ is uniserial and its µ ascending composition factors are the first µ composition factors of the sequence Sℓ, Sχ−1(ℓ), Sχ−2(ℓ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , Sχ−(p−2)(ℓ), Sℓ, Sχ−1(ℓ), Sχ−2(ℓ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , Sχ−(p−2)(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Notice that in our notation Vα(λ, κ) = Uλ+κ,κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The condition ordpi = m for all 1 ≤ i ≤ h, is equivalent to requiring that ψi : Cm → Aut(Cpi) is faithful for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Lifting of representations Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let G = Cq ⋊ Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Assume that for all 1 ≤ i ≤ h, ordpia = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If the G-module V lifts to an R[G]-module ˜V , where K = Quot(R) is a field of characterstic zero, then m | � dim( ˜V ⊗R K) − dim( ˜V ⊗R K)Cq� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Moreover, if ˜V (ζαiκ q ) is the eigenspace of the eigenvalue ζαiκ q of T acting on ˜V , then dim ˜V (ζκ q ) = dim ˜V (ζακ q ) = dim ˜V (ζα2κ q ) = · · · = dim ˜V (ζαm−1κ q ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Consider a lifting ˜V of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The generator τ of the cyclic part Cq has eigen- values λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λs which are pn-roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let ζq be a primitive q-root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Consider any eigenvalue λ ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' It is of the form λ = ζκ q for some κ ∈ N, q ∤ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If E is an eigenvector of T corresponding to λ, that is τE = ζκ q E then τσ−1E = σ−1τ αE = ζκαm−1 q σ−1E and we have a series of eigenvectors E, σ−1E, σ−2E, · · · with corresponding eigen- values ζκ q , ζκα q , ζκa2 q · · , ζκαo q , where o = ordq/(q,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Indeed, the integer o satisfies the καo ≡ κ modq ⇒ αm ≡ 1 mod q (q, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore the eigenvalues λ ̸= 1 form orbits of size m, while the eigenspace of the eigenvalue 1 is just the invariant space V G and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Indecomposable Cq ⋊ Cm modules, integral representation theory From now on V be a free R-module, where R is an integral local principal ideal domain with maximal ideal mR, R has characteristic zero and that R contains all q-th roots of unity and has characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let K = Quot(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The indecomposable modules for a cyclic group both in the ordinary and in the modular case are described by writing down the Jordan normal form of a generator of the cyclic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Since in integral representation theory there are infinitely many non-isomorphic indecomposable Cq-modules for q = ph, h ≥ 3, one is not expecting to have a theory of Jordan normal forms even if one works over complete local principal ideal domains [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let T be an element of order q = ph in End(V ), then the minimal polynomial of T has simple eigenvalues and T is diagonalizable when seen as an element in End(V ⊗ K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Since T q = IdV , the minimal polynomial of T divides xq − 1, which has simple roots over a field of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This ensures that T ∈ End(V ⊗ K) is diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let f(x) = (x − λ1)(x − λ2) · · · (x − λd) be the minimal polynomial of T on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' There is an element E ∈ V , such that E, (T − λ1IdV )E, (T − λ2IdV )(T − λ1IdV )E, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , (T − λd−1IdV ) · · · (T − λ1IdV )E are linear independent elements in V ⊗ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Consider the endomorphisms for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , d Πi = d � ν=1 ν̸=i (T − λνIdV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' In the above product notice that T − λiIdV , T − λjIdV are commuting endomor- phisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Since the minimal polynomial of T has degree d all R-modules KerΠi are strictly less than V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Moreover there is an element E such that E ̸∈ Ker(Πi) for all 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Consider a relation (8) d � µ=0 γµ µ � ν=0 (T − λµIdV )E, where �0 ν=0(T − λνIdV )E = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We fist apply the operator �d ν=2(T − λνIdV ) to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (8) and we obtain 0 = γ0Π1E, and by the selection of E we have that a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We now apply �d ν=3(T − λνIdV ) to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We obtain that 0 = γ1 d � ν=3 (T − λνIdV )(T − λ1IdV ) = γ1Π2E, and by the selection of E we have that γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We now apply �d ν=4(T − λνIdV ) to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (8) and we obtain 0 = γ2 d � ν=4 (T − λνIdV )(T − λ2IdV )(T − λ1IdV )E = γ2Π3E 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' KONTOGEORGIS AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' TEREZAKIS and by the selection of E we obtain γ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Continuing this way we finally arrive at γ0 = γ1 = · · · = γd−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let V be a free R-module of rank R acted on by an automorphism T : V → V of order ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Assume that the minimal polynomial of T is of degree d and has roots λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Then T can be written as a matrix with respect to the basis as follows: (9) � � � � � � � � � λ1 0 · · · · 0 a1 λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 0 a2 λ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 0 0 · · 0 ad−1 λd � � � � � � � � � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' By lemma 15 the elements E, (T − λ1IdV )E, (T − λ2IdV )(T − λ1IdV )E, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , (T − λd−1IdV ) · · · (T − λ1IdV )E form a free submodule of V of rank d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The theory of submodules of principal ideal domains, there is a basis E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , Ed of the free module V such that E1 = E, (10) a1E2 = (T − λ1IdV )E1, a2E3 = (T − λ2IdV )E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' as−1Ed = (T − λd−1IdV )Ed−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let us consider the module V1 = ⟨E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , Ed⟩ ⊂ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' By construction, the map T restricts to an automorphism V1 → V1 with respect to the basis E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , Ed has the desired form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We then consider the free module V/V1 and we repeat the procedure for the minimal polynomial of T, which again acts on V/V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The desired result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ Remark 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The element T as defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (9) has order equal to the higher order of the eigenvalues λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λd involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Indeed, since we have assumed that the eigenvalues are different the matrix is diagonalizable in Quot(R) and has order equal to the maximal order of the eigenvalues involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' In particular it has order q if there is at least one λi that is a primitive q-root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The statement about the order of T is not necessarily true if some of the eigenvalues are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For instance the matrix � 1 0 1 1 � has infinite order over a field of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Remark 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The number of indecomposable R[T]-summands of V is given by #{i : ai = 0} + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' A lift of a sum of indecomposable kCq-modules Jκ1 ⊕ · · · ⊕ Jκn can form an indecomposable RCq-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For example the indecomposable module where the ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 11 generator T of Cq has the form T = � � � � � � � � � λ1 0 · · · · 0 a1 λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 0 a2 λ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 0 0 · · 0 as−1 λd � � � � � � � � � where a1 = · · · = aκ1−1 = 1, aκ1 ∈ mR, aκ1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , aκ2+κ1−1 = 1, aκ2+κ1 ∈ mR , etc reduces to a decomposable direct sum of Jordan normal forms of sizes Jκ1, Jκ2−κ1, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Remark 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' It is an interesting question to classify these matrices up to conju- gation with a matrix in GLd(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' It seems that the valuation of elements ai should also play a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Definition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let hi(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , xj) be the complete symmetric polynomial of degree i in the variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For instance h3(x1, x2, x3) = x3 1 + x2 1x2 + x2 1x3 + x1x2 2 + x1x2x3 + x1x2 3 + x3 2 + x2 2x3 + x2x2 3 + x3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Set L(κ, j, ν) = hκ(λj, λj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λj+ν) A(i, j) = � aiai+1 · · · ai+j if j ≥ 0 0 if j < 0 Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The matrix T α = (t(α) ij ) is given by the following formula: t(α) ij = � � � � � λα i if i = j A(j, i − j − 1) · L(α − (i − j), j, i − j) if j < i 0 if j > i Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For j ≥ i the proof is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' When j < i and α = 1 it is immediate, since L(x, ·, ·) ≡ 0, for every x ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Assume this holds for α = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If α = n + 1, t(n+1) ij = t(n) ij tij = r � k=1 t(α) ik tkj = λjt(α) ij + ajt(α) ij+1 = λjA(j, i − j − 1)L(α − (i − j), j, i − j)+ + ajA(j + 1, i − j − 2)L(α − (i − j − 1), j + 1, i − j − 1) = = A(j, i − j − 1) � λjhα−(i−j)(λj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λj) + hα−(i−j)+1(λj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λi) � = = A(j, i − j − 1)hα−(i−j)+1(λj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λi) = = A(j, i − j − 1)L(α − (i − j) + 1, i, i − j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ Remark 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The space of homogeneous polynomials of degree k in n-variables has dimension �n−1+c n−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Since all q-roots of unity are reduced to 1 modulo mR the quantity L(α − (i − j), j, i − j) is reduced to n = (i − j) + 1, c = α − (i − j) �n − 1 + c n − 1 � = � α i − j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This equation is compatible with the computation of τ α given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' KONTOGEORGIS AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' TEREZAKIS Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' There is an eigenvector E of the generator σ of the cyclic group Cm which is not an element in s� i=1 Ker(Πi ⊗ K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The eigenvectors E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , Ed of σ form a basis of the space V ⊗ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' By multiplying by certain elements in R, if necessary, we can assume that all Ei are in V and their reductions Ei ⊗ R/mR, 1 ≤ i ≤ d give rise to a basis of eigenvectors of a generator of the cyclic group Cm acting on V ⊗ R/mR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If every eigenvector Ei is an element of some Ker(Πν) for 1 ≤ i ≤ d, then their reductions will be elements in Ker(T − 1)d−1, a contradiction since the later kernel has dimension < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let V be a free Cq ⋊ Cm-module, which is indecomposable as a Cq- module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Consider the basis given in lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Then the value of σ(E1) determines σ(Ei) for 2 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let σ be a generator of the cyclic group Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We will use the notation of lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We use lemma 23 in order to select a suitable eigenvector of E1 of σ and then form the basis E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , Ed as given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We can compute the action of σ on all basis elements Ei by (11) σ(ai−1Ei) = σ(T − λi−1IdV )Ei−1 = (T a − λi−1IdV )σ(Ei−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This means that one can define recursively the action of σ on all elements Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Indeed, assume that σ(Ei−1) = d � ν=1 γν,i−1Eν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We now have (T a − λi−1IdV )Eν = d � µ=1 t(α) µ,νEµ − λi−1Eν = (λα ν − λi−1)Eν + d � µ=ν+1 t(α) µ,νEµ We combine all the above to ai−1σ(Ei) = d � ν=1 γν,i−1(λα ν − λi−1)Eν + d � ν=1 γν,i−1 d � µ=ν+1 t(α) µ,νEµ = d � ν=1 ˜γν,iEν, (12) for a selection of elements γν,i ∈ R, which can be explicitly computed by collecting the coefficients of the basis elements E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Observe that the quantity on the right hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (12) must be divisible by ai−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Indeed, let v be the valuation of the local principal ideal domain R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Set e0 = min 1≤ν≤d{v(˜γν,i)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If e0 < v(ai−1) then we divide eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (12) by πe0 where π is the local uniformizer of R, that is mR = πR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We then consider the divided equation modulo mR to obtain ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 13 a linear dependence relation among the elements Ei ⊗ k, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore e0 ≥ v(ai−1) and we obtain an equation σ(Ei) = d � ν=1 ˜γν,i ai−1 Eν = d � ν=1 γν,iEν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ For example σ(E1) = ζϵ mE1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We compute that a1σ(E2) = (T α − λ1Id)σ(E1) and σ(E2) = (λα 1 − λ1) a1 ζϵ µE1 + ζϵ m d � µ=2 t(α) µ,1 a1 Eµ = (λα 1 − λ1) a1 ζϵ µE1 + ζϵ m d � µ=2 A(1, µ − 2)L(α − (µ − 1), 1, µ − 1) a1 Eµ = (λα 1 − λ1) a1 ζϵ µE1 + ζϵ m d � µ=2 a1a2 · · · aµ−1hα−(µ−1)(λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµ) a1 Eµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proposition 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Assume that no element a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , ad−1 given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (9) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Given α ∈ N, α ≥ 1 and an element E1, which is not an element in �d i=1 Ker(Πi ⊗ K), if there is a matrix Γ = (γij), such that ΓTΓ−1 = T α and ΓE1 = ζϵ mE1, then this matrix Γ is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We will use the idea leading to equation (11) replacing σ with Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We will compute recursively and uniquely the entries γµ,i, arriving at the explicit formula of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Observe that trivially γν,1 = 0 for all ν < 1 since we only allow 1 ≤ ν ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We compute ˜γµ,i = γµ,i−1(λα µ − λi−1) + µ−1 � ν=1 γν,i−1t(α) µ,ν (13) = γµ,i−1(λα µ − λi−1) + µ−1 � ν=1 γν,i−1A(ν, µ − ν − 1)L � α − (µ − ν), ν, µ − ν) = γµ,i−1(λα µ − λi−1) + µ−1 � ν=1 γν,i−1aνaν+1 · · · aµ−1hα−µ+ν(λν, λν+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµ) Define [λα m − λx]j i = j� x=i (λα µ − λx) [a]j i = j� x=i ax for i ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If i > j then both of the above quantities are defined to be equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' KONTOGEORGIS AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' TEREZAKIS Observe that for µ = 1 eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (13) becomes (14) γ1,i = 1 ai−1 γ1,i−1(λα 1 − λi−1) and we arrive at (assuming that Γ(E1) = ζϵ mE1) (15) γ1,i = ζϵ m a1a2 · · · ai−1 i−1 � x=1 (λα 1 − λx) = ζϵ m a1a2 · · · ai−1 [λα 1 − λx]i−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For µ ≥ 2 we have γµ,1 = 0, since by assumption TE1 = ζϵ mE1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (13) gives us γµ,i = i−2 � κ1=0 [λα µ − λx]i−1 i−κ1 [a]i−1 i−1−κ1 µ−1 � µ2=1 γµ2,i−1−κ1[a]µ−1 µ2 hα−µ+µ2(λµ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµ) = µ−1 � µ2=1 [a]µ−1 µ2 hα−µ+µ2(λµ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµ) i−2 � κ1=0 [λα µ − λx]i−1 i−κ1 [a]i−1 i−1−κ1 γµ2,i−1−κ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (16) We will now prove eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (16) by induction on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For i = 2, µ ≥ 2 we have γµ,2 = 1 a1 γµ,1(λα µ − λ1) + 1 a1 µ−1 � µ2=1 γµ2,1[a]µ−1 µ2 hα−µ+µ2(λµ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµ) = 1 a1 [a]µ−1 1 hα−µ+1(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµ)γ1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 15 Assume now that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (16) holds for computing γµ,i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We will treat the γµ,i case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We have γµ,i = (λα µ − λi−1) ai−1 γµ,i−1 + 1 ai−1 µ−1 � µ2=1 γµ2,i−1[a]µ−1 µ2 hα−µ+µ2(λµ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµ) = (λα µ − λi−1) ai−1 µ−1 � µ2=1 [a]µ−1 µ2 hα−µ+µ2(λµ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµ) i−3 � κ1=0 [λα µ − λx]i−2 i−1−κ1 [a]i−2 i−2−κ1 γµ2,i−2−κ1 + 1 ai−1 µ−1 � µ2=1 γµ2,i−1[a]µ−1 µ2 hα−µ+µ2(λµ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµ) = µ−1 � µ2=1 [a]µ−1 µ2 hα−µ+µ2(λµ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµ) i−3 � κ1=0 [λα µ − λx]i−1 i−1−κ1 [a]i−1 i−2−κ1 γµ2,i−2−κ1 + 1 ai−1 µ−1 � µ2=1 γµ2,i−1[a]µ−1 µ2 hα−µ+µ2(λµ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµ) = µ−1 � µ2=1 [a]µ−1 µ2 hα−µ+µ2(λµ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµ) i−2 � κ1=1 [λα µ − λx]i−1 i−κ1 [a]i−1 i−1−κ1 γµ2,i−1−κ1 + µ−1 � µ2=1 [a]µ−1 µ2 hα−µ+µ2(λµ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµ) 1 ai−1 γµ2,i−1 = µ−1 � µ2=1 [a]µ−1 µ2 hα−µ+µ2(λµ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµ) i−2 � κ1=0 [λα µ − λx]i−1 i−κ1 [a]i−1 i−1−κ1 γµ2,i−1−κ1 and equation (16) is now proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We proceed recursively applying eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (16) to each of the summands γµ2,i−1−κ1 if µ2 > 1 and i − 1 − κ1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If µ2 = 1, then γµ2,i−1−κ1 is computed by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (14) and if µ2 > 1 and i − 1 − κ1 ≤ 1 then γµ2,i−1−κ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We can classify all iterations needed by the set Σµ of sequences (µs, µs−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , µ3, µ2) such that (17) 1 = µs < µs−1 < · · · < µ3 < µ2 < µ = µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For example for µ = 5 the set of such sequences is given by Σµ = {(1), (1, 2), (1, 3), (1, 2, 3), (1, 4), (1, 2, 4), (1, 3, 4), (1, 2, 3, 4)} corresponding to the tree of iterations given in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The length of the sequence (µs, µs−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , µ2) is given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (17) is s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' In each iteration the i changes to i − 1 − k thus we have the following sequence of indices i1 = i → i2 = i−1−κ1 → i3 = i−2−(κ1+κ2) → · · · → is = i−(s−1)−(κ1+· · ·+κs−1) For the sequence i1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , we might have it = 1 for t < s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' But in this case, we will arrive at the element γµt+1,it = γµt,1 = 0 since µt > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This means that we will have to consider only selections κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , κs−1 such that is−1 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore we 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' KONTOGEORGIS AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' TEREZAKIS µ = 5 µ2 = 1 µ2 = 2 µ3 = 1 µ2 = 3 µ3 = 1 µ3 = 2 µ4 = 1 µ2 = 4 µ3 = 1 µ3 = 2 µ4 = 1 µ3 = 3 µ4 = 1 µ4 = 2 µ5 = 1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Iteration tree for µ = 5 arrive at the following expression for µ ≥ 2 γµ,i = � (µs,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=',µ2)∈Σµ [a]µ−1 µ2 [a]µ2−1 µ3 · · [a]µs−1−1 µs s � ν=2 hα−µν−1+µν(λµν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµν−1) � i=i1>i2>···>is≥1 s−1 � ν=1 [λα µν − λx]iν−1 iν+1+1 [a]iν−1 iν+1 γ1,is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' = � (µs,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=',µ2)∈Σµ s � ν=2 hα−µν−1+µν(λµν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµν−1) � i=i1>i2>···>is≥1 [a]µ−1 1 [a]i−1 is s−1 � ν=1 [λα µν − λx]iν−1 iν+1+1 ζϵ m[λα 1 − λx]is−1 1 [a]is−1 1 = � (µs,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=',µ2)∈Σµ s � ν=2 hα−µν−1+µν(λµν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λµν−1)[a]µ−1 1 [a]i−1 1 ζϵ m � i=i1>i2>···>is≥1 s � ν=1 [λα µν − λx]iν−1 iν+1+1 (18) where is+1 + 1 = 1 that is is+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ We will now prove that the matrix Γ of lemma 25 exists by cheking that ΓT = T αΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Set (aµ,i) = ΓT, (bµ,i) = T αΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For i < d we have aµ,i = d � ν=1 γµ,νtν,i = γµ,itii + γµ,i+1ti+1,i = γµ,iλi + γµ,i(λα µ − λi) + µ−1 � ν=1 γν,it(α) µ,ν = γµ,iλα µ + µ−1 � ν=1 γν,it(α) µ,ν = µ � ν=1 t(α) µ,νγν,i = bµ,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For i = d we have: aµ,d = d � ν=1 γµ,νtν,d = γµ,dtd,d = γµ,dλd ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 17 while bµ,d = d � ν=1 t(α) µ,νγν,d = µ−1 � ν=1 t(α) µ,νγν,d + λα µγµ,d This gives us the relation (19) (λd − λa µ)γµ,d = µ−1 � ν=1 t(α) µ,νγν,d For µ = 1 using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (15) we have γ1,dλd = γ1,dλα 1 ⇒ [λα 1 − λx]d 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This relation is satisfied if λα 1 is one of {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , λd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Without loss of generality we assume that (20) λ(a) i = � λi+1 if m ∤ i λi−m+1 if m | i We have the following conditions: µ = 2 (λd − λα 2 )γ2,d = t(α) 2,1 γ1,d µ = 3 (λd − λα 3 )γ3,d = t(α) 3,1 γ1,d + t(α) 3,2 γ2,d µ = 4 (λd − λα 4 )γ4,d = t(α) 4,1 γ1,d + t(α) 4,2 γ2,d + t(α) 4,3 γ3,d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' µ = d − 1 (λd − λα d−1)γd−1,d = t(α) d−1,1γ1,d + t(α) d−1,2γ2,d + · · · + t(α) d−1,d−2γd−1,d All these equations are true provided that γ1,d, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , γd−2,d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Finally, for µ = d, we have (21) (λd − λα d )γd,d = d−1 � ν=1 t(α) d,νγν,d which is true provided that (λd − λα d )γd,d = t(a) d,d−1γd−1,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For n ≥ 2 the vertical sum Sn of the products of every line of the following array y 1 1 (x1 − x2) (x1 − x3) · · · · (x1 − xn) 2 (z − x1) 1 (x1 − x3) · · · · (x1 − xn) 3 (z − x1) (z − x2) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' n − 1 (z − x1) (z − x2) · · (z − xn−2) 1 (x1 − xn) n (z − x1) (z − x2) · · (z − xn−2) (z − xn−1) 1 is given by Sn = n � y=1 n � ν=y+1 (x1 − xν) y−1 � µ=1 (z − xµ) = (z − x2) · · · (z − xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' KONTOGEORGIS AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' TEREZAKIS In particular when z = xn the sum is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We will prove the lemma by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For n = 2 we have S2 = (x1 − x2) + (z−x1) = z−x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Assume that the equality holds for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The sum Sn+1 corresponds to the array: y 1 1 (x1 − x2) (x1 − x3) · · (x1 − xn) (x1 − xn+1) 2 (z − x1) 1 (x1 − x3) · · (x1 − xn) (x1 − xn+1) 3 (z − x1) (z − x2) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' n − 1 (z − x1) · · (z − xn−2) 1 (x1 − xn) (x1 − xn+1) n (z − x1) (z − x2) · · (z − xn−1) 1 (x1 − xn+1) n + 1 (z − x1) (z − x2) · · (z − xn−1) (z − xn) 1 We have by definition Sn+1 = Sn(x1 − xn+1) + (z − x1)(z − x2) · · · (z − xn), which by induction gives Sn+1 = (z − x2) · · · (z − xn)(x1 − xn+1) + (z − x1)(z − x2) · · · (z − xn) = (z − x2) · · · (z − xn)(x1 − xn+1 + z − x1) and gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ Lemma 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Consider A < l < L < B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The quantity � l≤y≤L [λa − λx]y−1 A [λb − λx]B y+1 equals to [λa − λx]l−1 A [λb − λx]B L+1 · [λa − λx]L l − [λb − λx]L l (λa − λb) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We write � l≤y≤L [λa − λx]y−1 A [λb − λx]B y+1 = [λa − λx]l−1 A [λb − λx]B L+1 · � l≤y≤L [λa − λx]y−1 l [λb − λx]L y+1 The last sum can be read as the vertical sum S of the products of every line in the following array: y l 1 (λb − λl+1)(λb − λl+2) · · (λb − λL−1)(λb − λL) l + 1 (λa − λl) 1 (λb − λl+2) · · (λb − λL−1)(λb − λL) l + 2 (λa − λl)(λa − λl+1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' L − 2(λa − λl)(λa − λl+1) · · 1 (λb − λL−1)(λb − λL) L − 1(λa − λl)(λa − λl+1) · · (λa − λL−2) 1 (λb − λL) L (λa − λl)(λa − λl+1) · · (λa − λL−2)(λa − λL−1) 1 If l = b, then lemma 26 implies that S = [λa − λx]L b+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Furthermore, if L = a then S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 19 The quantity S cannot be directly computed using lemma 26, if l ̸= b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We proceed by forming the array: y b 1 (λb − λb+1) · · (λb − λl) · · · · · · · · (λb − λL) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' l − 1 (λa − λb) · · 1 (λb − λl) · · · · · · · · (λb − λL) l (λa − λb) · · (λa − λl−1) 1 (λb − λl+1)(λb − λl+2) · · (λb − λL−1)(λb − λL) l + 1 (λa − λb) · · (λa − λl−1)(λa − λl) 1 (λb − λl+2) · · (λb − λL−1)(λb − λL) l + 2 (λa − λb) · · (λa − λl−1)(λa − λl)(λa − λl+1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' L − 2(λa − λb) · · (λa − λl−1)(λa − λl)(λa − λl+1) · · 1 (λb − λL−1)(λb − λL) L − 1(λa − λb) · · (λa − λl−1)(λa − λl)(λa − λl+1) · · (λa − λL−2) 1 (λb − λL) L (λa − λb) · · (λa − λl−1)(λa − λl)(λa − λl+1) · · (λa − λL−2)(λa − λL−1) 1 The value of this array is computed using lemma 26 to be equal to [λa −λx]L b+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We observe that the sum of the products of the top left array can be computed using lemma 26, while the sum of the products of the lower right array is S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' [λa − λx]l−1 b S + [λa − λx]l−1 b+1 · [λb − λx]L l = [λa − λx]L b+1 we arrive at [λa − λx]l−1 b S = [λa − λx]l−1 b+1 � [λa − λx]L l − [λb − λx]L l � or equivalently (λa − λb) · S = [λa − λx]L l − [λb − λx]L l □ Lemma 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For all 1 ≤ µ ≤ d − 2 we have γµ,d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Let µ1 = µ > µ2 > · · · > µs = 1 ∈ Σµ be a selection of iterations and d = i1 > i2 > · · · · · · is ≥ 1 > is+1 = 0 be the sequence of i’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (20) we see that the quantity [λα µν − λx]iν−1 iν+1+1 ̸= 0 if and only if one of the following two inequalities hold: either iν+1 >µν − mf(µν) (22) or iν <µν + 2 − mf(µν), (23) where f(x) = � 1 if m | x 0 if m ∤ x We will denote the above two inequalities by (22)ν,(23)ν when applied for the integer ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Assume, that for all 1 ≤ ν ≤ s one of the two inequalities (22)ν,(23)ν hold, that is [λα µν − λx]iν−1 iν+1+1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Inequality (22)s can not hold for ν = s since it gives us 0 = is+1 > 1 = µs, we have m ∤ 1 = µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We will keep the sequence ¯µ : µ1 > µ2 > · · · > µs fixed and we will sum over all possible selections of sequences of i1 > · · · is > is+1 = 0, that is we will show that the sum (24) Γ¯µ,i := � i=i1>i2>···>is≥1 s � ν=1 [λα µν − λx]iν−1 iν+1+1 is zero, which will show that γµ,d = 0 using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' KONTOGEORGIS AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' TEREZAKIS Observe now that if (23)ν holds and m ∤ ν, ν −1, then (23)ν−1 also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Indeed the combination of (23)ν and (22)ν−1 gives the impossible inequality µν + 2 (23)ν > iν (22)ν−1 > µν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Assume now that m | ν and (23)ν holds, then (23)ν−1 also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Indeed the combination of (23)ν and (22)ν−1 gives us µν + 2 − m (23)ν > iν (22)ν−1 > µν−1 − mf(µν−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If m ∤ µν−1, then the above inequality is impossible since it implies that µν + 2 − m > µν−1 > µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If m | µν−1, then the inequality is also impossible since it implies that µν + 2 > µν−1 so if we write µν−1 = k′m and µν = km, k, k′ ∈ N, k′ > k, we arrive at 2 > (k′ − k)m ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This proves the following Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The inequality (22)ν−1 might be correct only in cases where m | µν−1, m ∤ µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Assume that for all ν inequality (23) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Then for ν = 1 it gives us (recall that µ ≤ d − 2) (25) µ + 2 ≤ d = i1 < µ1 + 2 − mf(µ1) = µ + 2 − mf(µ), which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore either there are ν such that none of the two inequal- ities (22)ν, (23)ν hold (in this case the contribution to the sum is zero) or there are cases where (22) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The sumands appearing in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (24) can be zero, for example the sequence µ1 = m > µ2 = 1 with i2 = 2 < i1 = d, s = 2 give the contribution [λα µ2 − λx]i2−1 1 [λα µ1 − λx]d−1 i2 = [λα 1 − λx]1 1[λα m − λx]d−1 i2+1 = (λ2 − λ1)[λ1 − λx]d−1 3 while for i2 = 1 < i1 = d it gives the contribution [λα µ2 − λx]i2−1 1 [λα µ1 − λx]d−1 i2+1 = [λα 1 − λx]0 1[λα m − λx]d−1 2 = [λ1 − λx]d−1 2 It is clear that these non-zero contributions cancel out when added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Lemma 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Assume that m | µν0−1 and m ∤ µν0, where (23)ν0 and (22)ν0−1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Then, we can eliminate µν0−1 and iν0 from both selections of the sequence of µ’s and i’s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' we can form the sequence of length s − 1 ¯µs−1 = µs < ¯µs−2 = µs−1 < · · · < ¯µν0−1 = µν0 < ¯µν0−2 = µν0−2 < · · · < ¯µ1 = µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' and the corresponding sequence of equal length ¯is−1 = is < ¯is−2 = is−1 < · · · < ¯iν0−1 = iν0−1 < ¯iν0 = iν0+1 < · · · < ¯i1 = i1 = d, so that Γ¯µ,i = � i1>···>is s � ν=1 [λα µν − λx]iν−1 iν+1+1 = (⋆) � ¯i1>···>¯is−1 s � ν=1 ν̸=ν0−1 [λα µν − λx]iν−1 iν+1+1, where (⋆) is a non zero element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (of lemma 30) We are in the case m | µν0−1 and m ∤ µν0, where (23)ν0 and (22)ν0−1 hold, (26) µν0−1 − m (22)ν0−1 < iν0 (23)ν0 < µν0 + 2, or equivalently µ0 := µν0−1 − m + 1 ≤ iν0 ≤ µν0 + 1 For iν0+1 the inequality (22)ν0 iν0+1 > µν0 − mf(µν0) can not hold, since it implies iν0+1 < iν0 (23)ν0 < µν0 + 2 < iν0+1 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Observe that also iν0+1 + 1 ≤ iν0 ≤ iν0−1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Set l = max{µ0, iν0+1 + 1} and L = min{µν0 + 1, iν0−1 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Then y = iν0 satisfies l ≤ y ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' By lemma 27 the quantity � l≤y≤L [λµν0+1 − λx]y−1 iν0+1+1 · [λµ0 − λx] iν0−1−1 y+1 equals to [λµν0+1 − λx]l−1 iν0+1+1 · [λµ0 − λx] iν0−1−1 L+1 [λµν0+1 − λx]L l − [λµ0 − λx]L l (λµν0+1 − λµ0) (27) [λµν0+1 − λx]L iν0+1+1 · [λµ0 − λx] iν0−1−1 L+1 − [λµν0+1 − λx]l−1 iν0+1+1 · [λµ0 − λx] iν0−1−1 l (λµν0+1 − λµ0) Case A1 l = µ0 ≥ iν0+1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Then [λµ0 − λx]L l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Case A2 l = iν0+1 + 1 > µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We set z := iν0+1, which is bounded by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (23)ν0+1 that is µ0 Case A2 ≤ z (23)ν0+1 ≤ µν0+1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Notice that in this case m ∤ µν0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Indeed, we have assumed that inequality (23)ν0+1 holds wich gives us µν0−1 − m = µ0 − 1 (Case A2) < iν0+1 (23)ν0+1 < µν0+1 + 2 − m, which implies that µν0−1 < µν0+1 + 2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Thus for l = z + 1 we compute � µ0≤z≤µν0+1+1 [λα µν0+1 − λx] iν0+1−1 iν0+2+1 · [λµ0 − λx]L l = = � µ0≤z≤µν0+1+1 [λµν0+1+1 − λx]z−1 iν0+2+1 · [λµ0 − λx]L z+1 = = (⋆) · [λµν0+1+1 − λx] µν0+1+1 µ0 − [λµ0 − λx] µν0+1+1 µ0 λµν0+1+1 − λµ0+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Case B1 L = µν0 + 1 ≤ iν0−1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' In this case [λµν0+1 − λx]L l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 22 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' KONTOGEORGIS AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' TEREZAKIS Case B2 L = iν0−1 − 1 < µν0 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' In this case eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (27) is reduced to [λµν0+1 − λx] iν0−1−1 iν0+1+1 (λµν0+1 − λµ0) This means that we have erased the µν0−1 from the product and we have � i1>···>is s � ν=1 [λα µν − λx]iν−1 iν+1+1 = (⋆) � i1>···>is s � ν=1 ν̸=ν0−1 [λα µν − λx]iν−1 iν+1+1, where (⋆) is a non zero element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This procedure gives us that the original quantity [λα µν0 − λx] iν0−1 iν0+1+1 · [λα µν0−1 − λx] iν0−1−1 iν0+1 after summing over iν0 becomes the quantity [λα µν0 − λx] iν0−1−1 iν0+1+1 = [λα ¯µν0−1 − λx] ¯iν0−1−1 ¯iν0+1 , that is we have eliminated the µν0−1 and iν0 from both selections of the sequence of µ’s and i’s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' we have the sequence of length s − 1 ¯µs−1 = µs < ¯µs−2 = µs−1 < · · · < ¯µν0−1 = µν0 < ¯µν0−2 = µν0−2 < · · · < ¯µ1 = µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' and the corresponding sequence of equal length ¯is−1 = is < ¯is−2 = is−1 < · · · < ¯iν0−1 = iν0−1 < ¯iν0 = iν0+1 < · · · < ¯i1 = i1 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ Remark 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' One should be careful here since ¯iν0−1 = iν0−1 > iν0 > ¯iν0 = iν0+1, so ¯iν0−1 > ¯iν0 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This means that the new sequence of ¯is−1 > · · · > ¯i1 satisfies a stronger inequality in the ν0 position, unless ν0 − 1 = d in the computation of γd,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Consider the set s, s − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , ν0 such that m ∤ µν for s ≥ ν ≥ ν0 and assume that m | µν0−1 and (23)ν0 and (22)ν0−1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We apply lemma 30 and we obtain a new sequence of µ’s with µν0−1 removed, provided that ν0 − 1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We continue this way and in the sequence of µ’s we eliminate all possible inequalities like (26) obtaining a series of µ which involves only inequalities of type (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' But this is not possible if µ ≤ d − 2, according to equation (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This proves that all γµ,d = 0 for 1 ≤ µ ≤ d − 2, this completes the proof of lemma 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If µ2 ̸= d − 1, then the contribution of the corresponding summand Γ¯µ,i to γd,d is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We are in the case µ = d = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We begin the procedure of eliminating all sequences of inequalities of the form (23)ν0, (22)ν0−1, where m | ν0−1, m ∤ ν0, using lemma 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For ν = 1 inequality (23)1 can not hold since it implies the impossible inequality d = i1 < d + 2 − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore, (22)1 holds, that is i2 > d − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' On the other hand we can assume that (23)2 holds by the elimination process, so we have d − m (22)1 < i2 (23)2 < µ2 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Following the analysis of the proof of lemma 28 we see that the contribution to γd,d is non zero if case B2 holds, that is (ν0 = 2 in this case) d − 1 = iν0−1 − 1 < µ2 + 1, obtaining that µ2 = d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 23 Lemma 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Equation (21) holds, that is (λd − λα d )γd,d = d−1 � ν=1 t(α) d,νγν,d = t(α) d,d−1γd−1,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We will use the procedure of the proof of lemma 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We recall that for each fixed sequence of µs > · · · > µ1 we summed over all possible sequences i1 > · · > is+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' In the final step the inequality (26) appears, for ν0 = 2, and µν0 = µ2 = d − 1 and ν0 − 1 = 1 and µν0−1 = µ = d, that is: 0 = µν0−1 − m (22)2 < iν0 (23)1 < µν0 + 2 = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' As in the proof of lemma 30 we sum over y = iν and the result is either zero in case B1 or in the B2 case, where µν0 = µ2 = d − 1 and µ0 = µν0−1 − m + 1 = d − m + 1, the contribution is computed to be equal to [λα µν0+1 − λx] iν0−1−1 iν0+1+1 (λµν0+1 − λm0) = [λα d − λx]d−1 i3+1 λd − λα d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The last µν0−1 = µ1 = d is eliminated in the above expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This means that for a fixed sequence µ1 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' > µs the contribution of the inner sum in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (24) is given by 1 λd − λα d � d−1=i2>i3>···>is≥1 s � ν=2 [λα µν − λx]iν−1 iν+1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Observe that µ1 = d does not appear in this expression and this expression corre- sponds to the sequence ¯µ1 = µ2 = d − 1 > ¯µ2 = µ3 > · · · > ¯µs−1 = ¯µs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Notice, also that the problem described in remark 31 does not appear here, sence we erased i1 which is not between some i’s but the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore, we can relate it to a similar expression that contributes to γd−1,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Conversely every contribution of γd−1,d gives rise to a contribution in γd,d, by multiplying by λd − λα d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The desired result follows by the expression of γµ,d given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ We have shown so far how to construct matrices Γ, T so that (28) T q = 1, ΓTΓ−1 = T α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We will now prove that Γ has order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' By equation (28) Γk should satisfy equation ΓkTΓ−k = T αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Using proposition 25 asserting the uniqueness of such Γk with α replaced by αk we have that the matrix multiplication of the entries of Γ giving rise to (γ(k) µ,i ) = Γk coincide to the values by the the recursive method of proposition (28) applied for Γ′ = Γk, α′ = αk and Γ′E1 = ζϵk m E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' In particular for k = m, we have αm ≡ 1 modpν for all 1 ≤ ν ≤ h, that is the matrix Γk should be recursively constructed using proposition (28) for the relation ΓmTΓm = T, ΓmE1 = E1, leading to the conclusion Γm = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Notice that the first eigenvalue of Γ is a primitive root of unity, therefore Γ has order exactly m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' By lemma 10 the action of σ in the special fibre is given by a lower triangular matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore, we must have (29) γν,i ∈ mr for ν < i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' KONTOGEORGIS AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' TEREZAKIS Proposition 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If (30) v(λi − λj) > v(aν) for all 1 ≤ i, j ≤ d and 1 ≤ ν ≤ d − 1 then the matrix (γµ,i) has entries in the ring R and is lower triangular modulo mR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Assume that the condition of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (30) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' In equation (18) we compute the fraction (31) [a]µ−1 1 [a]i−1 1 = � � � � � 1 [a]i−1 µ if i > µ 1 if i = µ [a]µ−1 i if i < µ The number of (λα µ − λx) factors in the numerator is equal to (recall that is+1 = 0) s � ν=1 (iν − 1 − iν+1 − 1 + 1) = i − s, and i > µ ≥ s, so i − s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore, for the upper part of the matrix i > µ we have i − s factors of the form (λα i − λj) in the numerator and i − µ factors ax in the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Their difference is equal to (i − s) − (i − µ) = µ − s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' By assumption the matrix reduces to an upper triangular matrix modulo mR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ Remark 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The condition given in equation (30) can be satisfied in the following way: It is clear that λi − λj ∈ mR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Even in the case vmR(λi − λj) = 1 we can consider a ramified extension R′ of the ring R with ramification index e, in order to make the valuation vmR′ (λi − λj) = e and then there is space to select vmR′ (ai) < vmR′ (λi − λj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proposition 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' We have that (32) γi,i ≡ ζϵ mαi−1 modmR Let A = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , ad−1} ∈ R be the set of elements below the diagonal in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If ai ∈ mR, then γµ,i ∈ mR for µ ̸= i, that is Ei is an eigenvector for the reduced action of Γ modulo mR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If aκ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , aκr the elements of the set A which are in mR, then the reduced matrix of Γ has the form: � � � � � � Γ1 0 · · 0 0 Γ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 0 0 · · 0 Γr � � � � � � where Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , Γr+1 for 1 ≤ ν ≤ r + 1 are (κν − κν−1) × (κν − κν−1) lower triangular matrices (we set κ0 = 0, κr+1 = d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Consider the matrix Γ: � � � � � � � � � � � � � � � � � � � � γ11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 0 γκ1,1 · · γκ1,κ1 γ11 γκ1+1,κ1+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' γµ,i γκ2,κ1+1 · · γκ2,κ2 γκ1+1,κ1+1 γµ,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' γκr+1,κr+1 · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' γκ1,κ1 γκ2,κ2 γd,κr+1 · · γd,d � � � � � � � � � � � � � � � � � � � � 1 ≤ i ≤ κ1 < m ≤ d κ1 < i ≤ κ2 < µ ≤ d We have that µ = i and the only element in Σµ which does not have any factor of the form (λα y − λx) is the sequence 1 = µs = µs−1 − 1 < µs−1 < · · · < µ2 = µ1 − 1 < µ1 = µ For this sequence eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (18) becomes γi,i = s � ν=2 hα−1(λµν, λµν−1)ζϵ m modmR, which gives the desired result since hα−1(λµν, λµν−1) ≡ �α 1 � = α modmR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' For proving that all entries γµ,i ∈ mR for κν < i ≤ κν+1 < µ ≤ d, that is for all entries bellow the central blocks, we observe that from equation (18) combined with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (31) that γµ,i is divisible by [a]µ−1 i = aiai+1 · · · aκν+1 · · · aµ−1 ∈ mR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ Recall that by lemma 2 there is an 1 ≤ a0 ≤ m such that α = ζa0 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proposition 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The indecomposable module V modulo mR breaks into a direct sum of r + 1 indecomposable k[Cq ⋊ Cm] modules Vν, 1 ≤ ν ≤ r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Each Vν is isomorphic to Vα(ϵ + a0κν−1, κν − κν−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' By eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (32) the first eigenvalue of the reduced matrix block Γν is ζϵ mακν−1 = ζϵ+(κν−1)a0 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Since that first eigenvalue together with the size of the block determine the last eigenvalue, that is the action of Cm on the socle the reduced block is uniquely determined up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' □ This way we arrive at a new obstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Assume that the indecomposable representation given by the matrix T as in lemma 16 reduces modulo mR to a sum of Jordan blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Then the σ action on the leading elements of each Jordan block in the special fibre should be described by the corresponding action of σ on the leading eigenvector E of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The corresponding actions on the special fibre should be compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This observation is formally given in proposition 1, which we now prove: Each set Iν, 1 ≤ ν ≤ t corresponds to an indecomposable R[G]-module, which decomposes to the indecomposables Vα(ϵµ, κµ), ν ∈ Iν of the special fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Indecomposable 26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' KONTOGEORGIS AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' TEREZAKIS summands have different roots of unity in R, therefore � µ∈Iν kν ≤ q, this is con- dition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The second condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='b) comes from proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If 1 is one of the possible eigenvalues of the lift T, then � µ∈Iν κµ ≡ 1 modm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If all eigenvalues of the lift T are different than one, then � µ∈Iν κµ ≡ 0 modm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' If #Iν = q, then there is one zero eigenvalue and the sum equals 1 modm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' It is clear by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (32) that condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='c) is a necessary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' On the other hand if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='c) is satisfied we can write (after a permutation if necessary) the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , S}, S = �t ν=1 #Iν as J1 = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , κ(1) 1 , κ(1) 1 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , κ(1) 1 + κ(1) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , r1 � j=1 κ(1) j = b1}, I1 = {κ(1) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , κ(1) r1 } J2 = {b1 + 1, b1 + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , b2 = b1 + r2 � j=1 κ(2) j }, I2 = {κ(2) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , κ(2) r2 } · · · · · Js = {bs−1 + 1, bt−1 + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , bt = S}, Is = {κ(s) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , κ(s) rs } The matrix given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' (9), where ai = � � � � � 0 if i ∈ {b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , bs−1} π if i ∈ {κ(ν) 1 , κ(ν) 1 + κ(ν) 2 , κ(ν) 1 + κ(ν) 2 + κ(ν) 3 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' , κ(ν) 1 + κ(ν) 2 + · · · + κ(ν) rν−1} 1 otherwise lifts the τ generator, and by (12) there is a well defined extended action of the σ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Example: Consider the group q = 52, m = 4, α = 7, G = C52 ⋊ C4 = ⟨σ, τ|σ4 = τ 25 = 1, στσ−1 = τ 7⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Observe that ord57 = ord527 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The module V (ϵ, 25) is projective and is known to lift in characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' This fits well with proposition 1, since 4 | 25 − 1 = 4 · 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The modules V (ϵ, κ) do not lift in characteristic zero if 4 ∤ κ or 4 ∤ κ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Therefore only V (ϵ, 1), V (ϵ, 4), V (ϵ, 5), V (ϵ, 8), V (ϵ, 9), V (ϵ, 12), V (ϵ, 13), V (ϵ, 16), V (ϵ, 17), V (ϵ, 20), V (ϵ, 21), V (ϵ, 24), V (ϵ, 25) lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The module V (1, 2) ⊕ V (3, 2) lift to characteristic zero, where the matrix of T with respect to a basis E1, E2, E3, E4 is given by T = � � � � ζq 0 0 0 1 ζ2 q 0 0 0 π ζ3 q 0 0 0 1 ζ4 q � � � � and σ(E1) = ζqE1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The module V (1, 2) ⊕ V (1, 2) does not lift in characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' There is no way to permute the direct summands so that the eigenvalues of σ are given by ζϵ m, αζϵ m, α2ζϵ m, α3ζϵ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Notice that α = 2 = ζm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The module V (ϵ1, 21)⊕V (221·ϵ1, 23) does not lift in characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The sum 21+24 is divisible by 4, ϵ2 = 221ϵ1 is compatible, but 21+23 = 44 > 25 so the representation of T in the supposed indecomposable module formed ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' 27 by their sum can not have different eigenvalues which should be 25-th roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Alperin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Local representation theory, volume 11 of Cambridge Studies in Advanced Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Modular representations as an introduction to the local representation theory of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' [2] Frauke M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Bleher, Ted Chinburg, and Aristides Kontogeorgis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Galois structure of the holo- morphic differentials of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Translated from the French by Marvin Jay Greenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' [18] Bradley Weaver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' The local lifting problem for D4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Israel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=', 228(2):587–626, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content=' Department of Mathematics, National and Kapodistrian University of Athens Pane- pistimioupolis, 15784 Athens, Greece Email address: kontogar@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='uoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='gr Department of Mathematics, National and Kapodistrian University of Athens, Panepis- timioupolis, 15784 Athens, Greece Email address: aleksistere@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='uoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfGvsw/content/2301.01032v1.pdf'} +page_content='gr' metadata={'source': 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XX, NO. X, FEB. 2019 +1 +Real-time Feedback Based Online Aggregate EV +Power Flexibility Characterization +Dongxiang Yan, Shihan Huang, and Yue Chen, Member, IEEE +Abstract—As an essential measure to combat global warming, +electric vehicles (EVs) have witnessed rapid growth. Meanwhile, +thanks to the flexibility of EV charging, vehicle-to-grid (V2G) +interaction has captured great attention. However, the direct con- +trol of individual EVs is challenging due to their small capacity, +large number, and private information. Hence, it is the aggregator +that interacts with the grid on behalf of EVs by characterizing +their aggregate flexibility. In this paper, we focus on the aggregate +EV power flexibility characterization problem. First, an offline +model is built to obtain the lower and upper bounds of the +aggregate power flexibility region. It ensures that any trajectory +within the region is feasible. Then, considering that parameters +such as real-time electricity prices and EV arrival/departure +times are not known in advance, an online algorithm is developed +based on Lyapunov optimization techniques. We prove that the +charging time delays for EVs always meet the requirement even +if they are not considered explicitly. Furthermore, real-time +feedback is designed and integrated into the proposed online +algorithm to better unlock EV power flexibility. Comprehensive +performance comparisons are carried out to demonstrate the +advantages of the proposed method. +Index Terms—Aggregate flexibility, charging station, electric +vehicle, Lyapunov optimization, online algorithm. +I. INTRODUCTION +T +HANKS to the low carbon emissions, electric vehicles +(EVs) have been considered a promising solution to +climate change and proliferate in recent years [1]. However, +the uncontrolled charging of a large number of EVs can cause +voltage deviation, line overload, and huge transmission loss +[2], threatening the reliability of the power system. Unlike +inelastic loads, the charging power and charging period of +EVs are more flexible [3]. Therefore, unlocking the power +flexibility hidden in EVs is a promising way to lessen the +adverse impact of EVs on the power grid. +There are extensive literature aiming to design coordinated +charging strategies to optimally schedule EV charging. For +example, to promote local renewable generation consumption, +a dynamic charging strategy was proposed to allow the EV +charging power to dynamically track the PV generation [4] +and wind generation [5]. To save the electricity cost, a +deterministic optimal charging strategy was proposed for a +home energy management system based on the time-of-use +tariffs [6]. A model predictive control (MPC) algorithm was +proposed to minimize the operational cost of EV charging +stations [7] relying on short-term forecasts. To address the +uncertainties related to EV charging, reference [8] proposed a +D. Yan, S. Huang, and Y. Chen are with the Department of Me- +chanical and Automation Engineering, the Chinese University of Hong +Kong, +Hong +Kong +SAR, +China +(e-mail: +dongxiangyan@cuhk.edu.hk, +shhuang@link.cuhk.edu.hk, yuechen@mae.cuhk.edu.hk). +stochastic charging strategy based on the probabilistic model +related to EV daily travels. A combined robust and stochastic +MPC method was developed in [9] to handle the uncertain EV +charging behaviors and renewable generations. A multi-stage +energy management strategy including day-ahead and real- +time stages was developed for a charging station integrated +with PV generation and energy storage [10]. In addition, +a pricing mechanism was suggested in [11] to guide EVs +for economical charging. A double-layer optimization model +was built to reduce the voltage violations caused by EV +charging [12]. Despite the efforts mentioned above that intend +to determine the EV charging power, it is challenging to +directly control a large number of individual EVs due to the +high computational complexity. +To get around this problem, some other literature en- +deavored to characterize the EV charging power flexibility. +Reference [13] proposed to model the aggregate EV charging +flexibility region by the lower and upper bounds of power and +cumulative energy. This aggregate EV model was adopted by +[14] to evaluate the achievable vehicle-to-grid capacity of an +EV fleet and by [15] to quantify the value of EV flexibility in +terms of maintaining distribution system reliability. Reference +[16] further considered the spatio-temporal distribution of the +probability that an EV is available for charging during the +aggregation and clustering processes. An EV dispatchable +region was proposed to allow charging stations to participate in +market bidding [17]. In addition, the aggregate flexibility issue +was also studied in the fields of thermostatically controllable +loads (TCLs) [18], distributed energy resources [19], and +virtual power plant [20]. For example, a geometric approach +was utilized to model the aggregate flexibility of TCLs [21]. +An inner box approximation method was proposed to charac- +terize the power flexibility region of various distributed energy +resources [22]. +The above works provide sound techniques for evaluating +EV flexibility in an offline manner. It means that the aggregator +is assumed to have complete information of future uncertainty +realizations, e.g., EV arrival/departure time, and electricity +prices. In practice, those data are usually unavailable or +inaccurate, making the obtained region fail to reflect the actual +EV aggregate flexibility in real-time. Thus, an online algorithm +is desired. A straightforward approach is the greedy algorithm +that decomposes the offline problem into subproblems in each +time slot by neglecting the time-coupling constraints [23]. +Obviously, the result could be far from optimum. Hence, we +resort to another approach, Lyapunov optimization, that can +run in an online manner but with an outcome near to the +offline optimum [24]. Lyapunov optimization has been used in +arXiv:2301.03342v1 [math.OC] 9 Jan 2023 + +JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 +2 +microgrid control [25], energy storage sharing [26], and data +center energy management [27]. For EV charging, a charging +strategy based on Lyapunov optimization was proposed to +minimize the total electricity cost [28]. However, it cannot +guarantee that EVs will depart with desired amount of energy. +To meet the EV charging requirement, virtual delay queues +were introduced to minimize the charging cost under uncertain +renewable generations and electricity prices [29], [30]. +Though the optimal online EV charging strategy has been +widely studied as above, the online aggregate EV power +flexibility characterization problem has not been well explored +yet. The latter problem is more complicated than the former +one, requiring a new model and algorithm design. This paper +proposes a real-time feedback based online aggregate EV +power flexibility characterization method. Our main contribu- +tions are two-fold: +1) Model. We first propose an offline optimization model +to characterize the aggregate EV power flexibility region. +It decomposes the time-coupled flexibility region into each +time slot and gives their lower and upper bounds. We prove +that any trajectory within the region is achievable. Then, +by categorizing the EVs according to their allowable time +delays, we develop a counterpart of the offline model that +enables the further utilization of the Lyapunov optimization +framework. The proposed model has not been reported in +previous literature. +2) Algorithm. A real-time feedback based online algorithm +is developed to derive the aggregate EV power flexibility +region sequentially. First, to fit into the Lyapunov optimization +framework, charging task queues and delay-aware virtual +queues are introduced to reformulate the model. Then, a +drift-plus-penalty term is constructed and by minimizing its +upper bound, an online algorithm is developed. We prove +that the charging time delays for EVs will not exceed their +maximum allowable values even if they are not explicitly +considered. The bound of optimality gap between the offline +and online outcomes is provided theoretically. Furthermore, +real-time dispatch strategy based feedback is designed and +integrated into the online algorithm. The proposed real-time +feedback based online algorithm is prediction free and can +adapt to uncertainties such as random electricity prices and EV +charging behaviors. Furthermore, it can make use of the most +recent information, allowing it to even outperform the offline +model with full knowledge of future uncertainty realizations +but without the updated dispatch information. +The rest of this paper is organized as follows. Section +II formulates the offline model for deriving aggregate EV +charging power flexibility region. Section III and IV introduce +the Lyapunov optimization method and real-time feedback +design, respectively, to generate flexibility region in an online +manner. Simulation results are presented in Section V. Finally, +Section VI concludes this paper. +II. PROBLEM FORMULATION +In this section, we first introduce the concept of aggregate +EV power flexibility and then formulate an offline optimization +problem to approximate it. +A. Aggregate EV Charging Power Flexibility +As shown in Fig. 1, when an EV v ∈ V arrives at the +charging station, it submits its charging task to the aggregator. +The task is described by (ta +v, td +v, ea +v, ed +v), where ta +v is its arrival +time, td +v is its departure time, ea +v is the initial battery energy +level at ta +v, and ed +v is the desired energy level when it leaves. +For the EV v, the maximum allowable charging time delay is +Rv = td +v−ta +v. The EV charging task needs to be finished within +this declared time duration. With the submitted information, +the aggregator can flexibly schedule the EV charging to meet +the charging requirement. Two possible trajectories to meet the +EV charging need are depicted in Fig. 1. Let {pc +v,t, ∀t} be the +charging power of EV v over time. The range that the charging +power can vary within is called the power flexibility of EV +v. If we sum the power flexibility of all EVs in a charging +station up, we can get the aggregate EV power flexibility of +the charging station. +(𝑡1 +𝑎, 𝑡1 +𝑑, +𝑒1 +𝑎, 𝑒1 +𝑑) +(𝑡2 +𝑎, 𝑡2 +𝑑, +𝑒2 +𝑎, 𝑒2 +𝑑) +(𝑡𝑣𝑎, 𝑡𝑣𝑑, +𝑒𝑣𝑎, 𝑒𝑣𝑑) +Aggregator +Distribution System Operator +Aggregate +dispatch power +Aggregate power +flexibility region +Ƽ𝑝𝑑,𝑡, Ƹ𝑝𝑑,𝑡 +𝑝𝑑,𝑡 +𝑑𝑖𝑠𝑝 +𝑝1,𝑡 +𝑑𝑖𝑠𝑝 +𝑝2,𝑡 +𝑑𝑖𝑠𝑝 +𝑝𝑣,𝑡 +𝑑𝑖𝑠𝑝 +EV 1 +EV 2 +EV v +1. Generate EV aggregate power flexibility region +2. Disaggregation +Fig. 1. System diagram and illustration of EV power flexibility. +However, it is difficult to characterize the EV power flexibil- +ity for each time slot due to the temporal-coupled EV charging +constraints. The EV power flexibility in the current time slot is +affected by those in the past time slots and further affects those +in the future time slots. This is different from the traditional +controllable generators whose flexibility can be described by +the minimum and maximum power outputs in each time slot. +In the following, we aim to derive an aggregate EV power +flexibility region that: 1) is time-decoupled so that it can be +used in real-time power system operation; and 2) any trajectory +within it can meet the EV charging requirement. +B. Offline Problem Formulation +Suppose there are T time slots, indexed by t ∈ T += +{1, ..., T}. The desired time-decoupled aggregate EV power +flexibility region can be represented by a series of intervals +[ˇpd,t, ˆpd,t], ∀t ∈ T . The intervals can be specified by a lower +power trajectory {ˇpd,t, ∀t} and an upper power trajectory +{ˆpd,t, ∀t}. To obtain the lower and upper power trajectories, +we formulate the following offline optimization problem: +P1 : +max +ˆpd,t,ˇpd,t,∀t lim +T →∞ +1 +T +T +� +t=1 +E +� +Ft +� +, +(1a) + +Power +Aggregate power +flexibility region +flexibilitJOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 +3 +where +Ft = πt(ˆpd,t − ˇpd,t), ∀t, +(1b) +subject to +ˆpd,t = +� +v∈V +ˆpc +v,t, ∀t, +(1c) +0 ≤ ˆpc +v,t ≤ pmax +v +, ∀v, ∀t, +(1d) +ˆev,t+1 = ˆev,t + ˆpc +v,t∆t, ∀v, ∀t ̸= T, +(1e) +ˆev,tav = eini +v , ˆev,tdv ≥ ed +v, ∀v, +(1f) +emin +v +≤ ˆev,t ≤ emax +v +, ∀v, ∀t, +(1g) +ˇpd,t = +� +n∈V +ˇpc +v,t, ∀t, +(1h) +0 ≤ ˇpc +v,t ≤ pmax +v +, ∀v, ∀t, +(1i) +ˇev,t+1 = ˇev,t + ˇpc +v,t∆t, ∀v, ∀t ̸= T, +(1j) +ˇev,tav = eini +v , ˇev,tdv ≥ ed +v, ∀v, +(1k) +emin +v +≤ ˇev,t ≤ emax +v +, ∀v, ∀t, +(1l) +ˇpd,t ≤ ˆpd,t, ∀t, +(1m) +ˆpc +v,t/ˇpc +v,t = +� ˆpc +v,t/ˇpc +v,t, +if t ∈ [ta +v, td +v] +0, +if t < ta +v ∪ t > td +v +. +(1n) +In the objective function (1a)-(1b), πt, ∀t are the real-time +electricity prices, showing the unit value of power flexibility +in different time slots. Hence, the objective function aims to +maximize the value of total aggregate EV power flexibility. +Constraint (1c) defines the upper bound of aggregate EV +power flexibility region. The charging power of an EV v is +limited by (1d), where pmax +v +is the maximum charging power. +Constraint (1f) defines the EV’s initial energy level and the +charging requirement. (1e) and (1g) describe the EV’s energy +dynamics and battery capacity. Similarly, (1h)-(1l) are the +constraints related to the lower bound of the aggregate EV +power flexibility. (1m) is the joint constraint to ensure that +{ˆpd,t, ∀t} and {ˇpd,t, ∀t} provide the upper and lower bounds, +respectively. (1n) limits that charging only happens during the +EV’s declared parking time. +Proposition 1: Any aggregate EV charging power trajectory +within [ˇpd,1, ˆpd,1] × · · · × [ˇpd,T , ˆpd,T ] is achievable. +The proof of Proposition 1 can be found in Appendix A. +Despite this nice property, the offline optimization problem +above cannot be solved directly since it requires complete +knowledge of the future EV charging tasks and future elec- +tricity prices, which are usually not available in practice. +Therefore, an online algorithm is necessary. To this end, in +the next section, we will first propose a closely related but +more flexible form of the problem studied. Then, we adopt the +Lyapunov optimization framework to reformulate the offline +problem into an online one. We construct charging task queues +and delay-aware virtual queues to ensure the satisfaction of +charging requirements. Furthermore, considering the impact +of real-time dispatch decisions on the future aggregate EV +power flexibility, a real-time feedback based online flexibility +characterization method is developed in Section IV to avoid +the potential underestimate of EV power flexibility. +III. ONLINE ALGORITHM +In this section, we adopt the Lyapunov optimization frame- +work to solve the offline problem P1 in an online manner. +The proposed algorithm can output an aggregate EV power +flexibility value with an economic value close to P1. +A. Problem Modification +As mentioned above, the charging station serves dozens of +EVs every day, and each EV arrives along with a charging task, +i.e., (ta +v, td +v, ea +v, ed +v). Those EV charging tasks can be first stored +in a queue and be served later according to a first-in-first- +out basis. Since different EVs may have different allowable +charging time delays, we use multiple queues to classify and +collect the EV charging tasks. Suppose there are G types +of charging time delays Rgs, each of which is indexed by +g ∈ {1, 2, ..., G}. Correspondingly, we construct G queues to +collect the respective charging tasks, and each queue is denoted +by Qg. For queue Qg, Qg,t refers to its charging task backlog +in time slot t. The queue backlog growth is described by +Qg,t+1 = max[Qg,t − xg,t, 0] + ag,t, +(2) +where xg,t is the charging power for EVs in group g at time +t, and ag,t is the arrival rate of EV charging tasks of group g +at time t. In particular, ag,t sums up the energy demand of all +EVs that arrive at the beginning of time t, i.e., +ag,t = +� +v∈Vg +ag,v,t, +(3) +where ag,v,t is the charging demand of EV v of group g in +time slot t. Vg is the set of EVs in group g. +Recalling that our target in P1 is to derive an upper bound +and a lower bound for the aggregate EV power flexibility +region, we correspondingly define the upper bound queue ˆQg,t +and the lower bound queue ˇQg,t. Similar to (2), we have +ˆQg,t+1 = max[ ˆQg,t − ˆxg,t, 0] + ˆag,t, +(4) +ˇQg,t+1 = max[ ˇQg,t − ˇxg,t, 0] + ˇag,t, +(5) +where ˆxg,t and ˇxg,t are the charging power for upper and +lower bound queues, respectively, i.e., ˆxg,t = � +v∈Vg ˆpc +v,t and +ˇxg,t = � +v∈Vg ˇpc +v,t. +The upper and lower bounds of arriving charging demand, +i.e., ˆag,t and ˇag,t, are determined by +ˆag,t = +� +v∈Vg +ˆag,v,t, ˇag,t = +� +v∈Vg +ˇag,v,t, +(6) +Particularly, the lower bound of arriving charging demand +ˇag,v,t can be determined in the following charging as soon as +possible way, +ˇag,v,t = +� +� +� +pmax +v +, +ta +v ≤ t < ⌊ˇtmin +v +⌋ + ta +v +ˇecha +v +/ηc − ⌊ˇtmin +v +⌋pmax +v +, +t = ⌊ˇtmin +v +⌋ + ta +v +0, +otherwise +, +(7) +where ˇecha +v += ed +v −ea +v, ˇtmin +v +is the minimum required charging +time determined by ˇtmin +v += +ˇecha +v +pmax +v +ηc , and ⌊.⌋ means rounding +down to the nearest integer. + +JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 +4 +Different from ˇag,v,t, the upper bound of arrival charging +demand ˆag,v,t is determined using the maximum charging +demand emax +v +instead of ed +v. Denote ˆecha +v += emax +v +− ea +v, +ˆtmin +v += +ˆecha +v +pmax +v +ηc , and then ˆag,v,t can be determined by +ˆag,v,t = +� +� +� +pmax +v +, +ta +v ≤ t < ⌊ˆtmin +v +⌋ + ta +v +ˆecha +v +/ηc − ⌊ˆtmin +v +⌋pmax +v +, +t = ⌊ˆtmin +v +⌋ + ta +v +0, +otherwise. +(8) +We then formulate the aggregate EV power flexibility char- +acterization problem as follows: +P2 : +min +ˆxg,t,ˇxg,t lim +T →∞ +1 +T +T +� +t=1 +E +� +− Ft +� +, +(9a) +subject to +lim +T →∞ +1 +T +T +� +t=1 +E[ˆag,t − ˆxg,t] ≤ 0, ∀g +(9b) +lim +T →∞ +1 +T +T +� +t=1 +E[ˇag,t − ˇxg,t] ≤ 0, ∀g +(9c) +0 ≤ ˆxg,t ≤ min{xg,max, ˆQg,t}, ∀g +(9d) +0 ≤ ˇxg,t ≤ min{xg,max, ˇQg,t}, ∀g +(9e) +ˆxg,t ≥ ˇxg,t, ∀g +(9f) +where xg,max = � +v∈Vg pmax +v +. Constraint (9b) ensures that if +using the upper bound trajectory {ˆxg,t, ∀t}, the total charg- +ing requirement can be satisfied in the long run. Constraint +(9c) poses a similar requirement for the lower bound trajec- +tory {ˇxg,t, ∀t}. Based on (9b), (9c), and the definitions of +ˆQg,t+1, ˇQg,t+1 in (4)-(5), we can prove that the queues ˆQg,t +and ˇQg,t are mean rate stable. To be specific, +ˆQg,t+1 − ˆag,t ≥ ˆQg,t − ˆxg,t, ∀t +(10) +Summing (10) up over all t and divide both sides by T yields +0 ≤ E[ ˆQg,T ] +T +≤ +�T +t=1 E[ˆag,t − ˆxg,t] +T +(11) +Hence, lim +T →∞ +E[ ˆ +Qg,T ] +T += 0. Similarly, lim +T →∞ +E[ ˇ +Qg,T ] +T += 0. +Constraints (9d) and (9e) give the upper and lower bounds +of the aggregate EV charging power for group g, respectively. +The upper bound is no less than the lower bound, as shown in +(9f). The P2 provides a counterpart problem for P1. Similar +to the proof of Proposition 1, we can prove that any trajectory +between [ˇxg,t, ˆxg,t] is achievable. However, the allowable +charging delay is not considered in P2, which may result in +unfulfilled EV charging tasks upon departure. +B. Construct Virtual Queues +To overcome the aforementioned charging delay issue, we +introduce delay-aware virtual queues, +ˆZg,t+1 = max{ ˆZg,t + ηg +Rg +I ˆ +Qg,t>0 − ˆxg,t, 0}, ∀g, ∀t +(12) +ˇZg,t+1 = max{ ˇZg,t + ηg +Rg +I ˇ +Qg,t>0 − ˇxg,t, 0}, ∀g, ∀t +(13) +where I ˆ +Qg,t>0 and I ˇ +Qg,t>0 are indicator functions of ˆQg,t and +ˇQg,t, respectively. They are equal to 1 if there exists unserved +charging tasks in the queues, i.e., ˆQg,t > 0 and ˇQg,t > 0. +Using ηg +Rg to times it, this whole term constitutes a penalty to +the virtual queue backlog. ηg is a user-defined parameter that +can adjust the growth rate of the virtual queues. For instance, +increasing the value of ηg leads to a fast queue growth and a +larger backlog value, calling for more attention to accelerate +the charging process. We prove that, when Qg,t and Zg,t have +finite upper bounds, with a proper ηg, the charging time delay +for EVs in group g is bounded. +Proposition 2: Suppose ˆQg,t, ˇQg,t, ˆZg,t, and ˇZg,t have finite +upper bounds, e.g., ˆQg,t ≤ ˆQg,max, ˇQg,t ≤ ˇQg,max ˆZg,t ≤ +ˆZg,max, and ˇZg,t ≤ ˇZg,max. The charging time delay of all +EVs in group g is upper bounded by ˆδg,max and ˇδg,max time +slots, where +ˆδg,max := ( ˆQg,max + ˆZg,max)Rg +ηg +, +(14) +ˇδg,max := ( ˇQg,max + ˇZg,max)Rg +ηg +. +(15) +The proof of Proposition 2 can be found in Appendix B. +It ensures that the charging tasks can always be fulfilled +within the available charging periods by properly setting the +parameters ηg, ∀g. +C. Lyapunov Optimization +Based on the charging task queues and delay-aware virtual +queues, the Lyapunov optimization framework is applied as +follows. +1) Lyapunov +Function: +First, +we +define +Θt += +( ˆ +Qt, ˆ +Zt, ˇ +Qt, ˇ +Zt) as the concatenated vector of queues, +where +ˆ +Qt = ( ˆQ1,t, ..., ˆQG,t), +(16a) +ˆ +Zt = ( ˆZ1,t, ..., ˆZG,t), +(16b) +ˇ +Qt = ( ˇQ1,t, ..., ˇQG,t), +(16c) +ˇ +Zt = ( ˇZ1,t, ..., ˇZG,t). +(16d) +The Lyapunov function is then defined as +L(Θt) = 1 +2 +� +g∈G +ˆQ2 +g,t + 1 +2 +� +g∈G +ˆZ2 +g,t + 1 +2 +� +g∈G +ˇQ2 +g,t + 1 +2 +� +g∈G +ˇZ2 +g,t, +(17) +where L(Θt) can be considered as a measure of the queue +size. A smaller L(Θt) is preferred to push (virtual) queues +ˆQg,t, ˆZg,t, ˇQg,t, and ˇZg,t to be less congested. +2) Lyapunov Drift: The conditional one-time slot Lyapunov +drift is defined as follows: +∆(Θt) = E[L(Θt+1) − L(Θt)|Θt], +(18) +where the expectation is taken with respect to the random Θt. +The Lyapunov drift is a measure of the expectation of the +queue size growth given the current state Θt. Intuitively, by +minimizing the Lyapunov drift, virtual queues are expected +to be stabilized. However, only minimizing the Lyapunov +drift may lead to a low aggregate EV power flexibility value. + +JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 +5 +Therefore, we include the expected aggregate flexibility value +(1b) for the time slot t to (18). The drift-plus-penalty term is +obtained, i.e., +∆(Θt) + V E[−Ft|Θt], +(19) +where V is a weight parameter that controls the trade-off +between (virtual) queues stability and aggregate EV power +flexibility maximization. +3) Minimizing the Upper Bound: (19) is still time-coupled +due to the definition of ∆(Θt). To adapt to online imple- +mentation, instead of directly minimizing the drift-plus-penalty +term, we minimize the upper bound to obtain the upper and +lower bounds of aggregate EV power flexibility region. We +first calculate the one-time slot Lyapunov drift: +L(Θt+1) − L(Θt) += 1 +2 +� +g∈G +� � +ˆQ2 +g,t+1 − ˆQ2 +g,t +� ++ +� +ˆZ2 +g,t+1 − ˆZ2 +g,t +� ++ +� ˇQ2 +g,t+1 − ˇQ2 +g,t +� ++ +� ˇZ2 +g,t+1 − ˇZ2 +g,t +� � +. +(20) +Use queue ˆQg,t as an example, based on the queue update +equations (4), we have +ˆQ2 +g,t+1 = {max[ ˆQg,t − ˆxg,t, 0] + ˆag,t}2 +≤ ˆQ2 +g,t + ˆa2 +g,max + ˆx2 +g,max + 2 ˆQg,t(ˆag,t − ˆxg,t). +(21) +Thus, +1 +2 +� +ˆQ2 +g,t+1 − ˆQ2 +g,t +� +≤ 1 +2 +� +ˆx2 +g,max + ˆa2 +g,max +� ++ ˆQg,t (ˆag,t − ˆxg,t) . +(22) +Similarly, for queue ˇQg,t, ˆZg,t, and ˇZg,t, we have +1 +2 +� ˇQ2 +g,t+1 − ˇQ2 +g,t +� +≤ 1 +2 +� +ˇx2 +g,max + ˇa2 +g,max +� ++ ˇQg,t (ˇag,t − ˇxg,t) . +(23) +1 +2[ ˆZ2 +g,t+1 − ˆZ2 +g,t] ≤ 1 +2 max[( ηg +Rg +)2, ˆx2 +g,max] ++ ˆZg,t[ ηg +Rg +− ˆxg,t]. +(24) +1 +2[ ˇZ2 +g,t+1 − ˇZ2 +g,t] ≤ 1 +2 max[( ηg +Rg +)2, ˇx2 +g,max] ++ ˇZg,t[ ηg +Rg +− ˇxg,t]. +(25) +We then substitute inequalities (22),(23), (24) and (25) into +drift-plus-penalty term and yield +∆(Θt) + V E[−Ft|Θt] +≤ A + V E[−Ft|Θt] + +� +g∈G +ˆQg,tE [ˆag,t − ˆxg,t|Θt] ++ +� +g∈G +ˇQg,tE [ˇag,t − ˇxg,t|Θt] + +� +g∈G +ˆZg,tE [−ˆxg,t|Θt] ++ +� +g∈G +ˇZg,tE [−ˇxg,t|Θt] , +(26) +where A is a constant, i.e., +A = 1 +2 +� +g∈G +(ˆx2 +g,max + ˆa2 +g,max) + 1 +2 +� +g∈G +max[( ηg +Rg +)2, ˆx2 +g,max] ++ 1 +2 +� +g∈G +(ˇx2 +g,max + ˇa2 +g,max) + 1 +2 +� +g∈G +max[( ηg +Rg +)2, ˇx2 +g,max] ++ +� +g∈G +[ ˆZg,max +ηg +Rg +] + +� +g∈G +[ ˇZg,max +ηg +Rg +]. +By reorganizing the expression in (26) and ignoring the con- +stant terms, we can obtain the following online optimization +problem, +P3 : +min +ˆxg,t,ˇxg,t,∀g,∀t +� +g∈G +(−V πt − ˆQg,t − ˆZg,t)ˆxg,t ++ +� +g∈G +(V πt − ˇQg,t − ˇZg,t)ˇxg,t, +(27) +s.t. (9d) − (9f), +where ˆQg,t, ˇQg,t, ˇZg,t, and ˇZg,t are first updated based on +(4),(5),(12), and (13) before solving P3 in each time slot. +In each time slot t, given the current system queue state +Θt, the proposed method determines the current upper and +lower aggregate EV power flexibility bounds ˆxg,t and ˇxg,t by +solving problem P3. Hence, the original offline optimization +problem P1 has been decoupled into simple online (real-time) +problems, which can be executed in each time slot without +requiring prior knowledge of future uncertain states. Since +the modified problem P3 is slightly different from the offline +one P1, an important issue we care about is: what’s the gap +between the optimal solutions of the online problem P3 the +and offline problem P1? +Proposition 3: Denote the obtained long-term time-average +aggregate EV power flexibility value of P1 and P3 by F ∗ +and F pro, respectively. We have +0 ≤ −F pro + F ∗ ≤ 1 +V A, +(28) +where A is a constant defined in (26). +The proof of Proposition 3 can be found in Appendix C. +The optimality gap can be controlled by the parameter V . A +bigger V value leads to a smaller optimality gap but increased +queue sizes. In contrast, a smaller V value makes the queues +more stable but results in a larger optimality gap. +IV. DISAGGREGATION +AND REAL-TIME FEEDBACK DESIGN +In each time slot t, given the aggregate EV power flexibility +region [� +g ˇx∗ +g,t, � +g ˆx∗ +g,t], the distribution system operator +(DSO) can determine the optimal aggregate dispatch strategy +for EVs. This aggregate dispatch strategy should be further +disaggregated to obtain the control strategy for each EV, +which is studied in this section. Moreover, considering that +the current dispatch strategy will influence the aggregate EV +power flexibility in future time slots, real-time feedback is +designed and integrated with the proposed online flexibility +characterization method in Section III. + +JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 +6 +A. Disaggregation +Suppose the dispatch strategy in time slot t is pdisp +agg,t ∈ +[� +g ˇx∗ +g,t, � +g ˆx∗ +g,t]. This can be determined by the DSO by +solving an economic dispatch problem based on the up-to- +date information (e.g., the electricity price πt, the grid-side +renewable generation). Since this paper focuses on the online +characterization of aggregate EV power flexibility, the eco- +nomic dispatch problem by the DSO is omitted for simplicity. +Interested readers can refer to [31]. +Let the dispatch ratio αt be +αt = +pdisp +agg,t − � +g ˇx∗ +g,t +� +g ˆx∗ +g,t − � +g ˇx∗ +g,t +. +(29) +Then, the dispatched power pdisp +g,t +for each group g can be +determined according to the ratio, i.e., +pdisp +g,t += (1 − αt)ˇx∗ +g,t + αtˆx∗ +g,t, +(30) +which satisfies +pdisp +agg,t = +� +g +pdisp +g,t . +The next step is to allocate pdisp +g,t +to the EVs in the group +g. All EVs in the group g are sorted according to their arrival +time. Then, we follow the first-in-first-service principle to +allocate the energy; namely, the EV that comes earlier will +be charged with the maximum charging power. We have +pdisp +v,t += min +� +pdisp +g,t , pmax +v +, emax +v +− ev,t +∆t +� +, ∀v ∈ Vg, +(31) +where Vg refers to the set of EVs in group g. The third term +on the right side is used to ensure that the EV will not exceed +its allowable maximum energy level. +After this charging assignment for an earlier EV is com- +pleted, the following update procedures will execute +pdisp +g,t +← (pdisp +g,t +− pdisp +v,t ), +(32) +which means deducting pdisp +v,t +from the total remaining dis- +patched power pdisp +g,t . +Then, the pdisp +g,t +is allocated to the next earlier arrival EVs +until the aggregate EV charging power is completely assigned. +At this time, the disaggregation is finished. +The dispatched power disaggregation algorithm is presented +in Algorithm 1. +B. State Update to Improve Power Flexibility Region +Following the disaggregation procedures in Algorithm 1, we +can get the actual EV dispatched charging power pdisp +v,t , ∀v. +By now, we can move on to the next time slot t + 1 and +evaluate the EV power flexibility by solving problem P3, +determine the dispatch strategy, disaggregate the dispatched +power, and so on. But considering that the current actual +dispatched EV charging power can affect the future aggregate +EV power flexibility, which is ignored in the aforementioned +processes. Therefore, we propose a real-time feedback method +to integrate the current actual EV dispatched charging power +into the future aggregate EV power flexibility characterization. +Algorithm 1 EV Dispatched Power Disaggregation +1: Initialization: aggregate EV dispatched power pdisp +agg,t. +2: Calculate the dispatched aggregate charging power pdisp +g,t +for each group g using (30). +3: for Each group g ∈ G do +4: +for Each EV v in group g do +5: +if the EV is not available for charging then +6: +Let EV v’s charging power pdisp +v,t += 0, ∀v. +7: +else +8: +Calculate pdisp +v,t +according to (31). +9: +Update the remaining aggregate power via (32). +10: +if If the updated pdisp +g,t += 0 then +11: +Break and return to Step 3. +12: +end if +13: +end if +14: +end for +15: end for +Disaggregation +State update +Dispatch +Aggregate power flexibility +region +State +feedback +t=t+1 +Fig. 2. Overall procedure of the proposed method. +The overall procedure is shown in Fig. 2 with the right-hand +side blue box showing the real-time feedback. +To be specific, we change the constraints (9d)-(9e) for time +slot t into +ˆxg,t = pdisp +g,t , ˇxg,t = pdisp +g,t . +(33) +Since pdisp +g,t +∈ [ˇx∗ +g,t, ˆx∗ +g,t], after replacing (9d)-(9e) with (33), +the problem P3 is still solvable and the optimal solution is +ˆxupdate∗ +g,t += pdisp +g,t , ˇxupdate∗ +g,t += pdisp +g,t , ∀g. With these updated +lower and upper bounds, we update the queues ˆQg,t+1, ˇQg,t+1, +ˆZg,t+1, ˇZg,t+1 according to (4), (5), (12) and (13), respectively. +Then, we move on the solve problem P3 for time slot t + 1 +using the updated ˆQg,t+1, ˇQg,t+1, ˆZg,t+1, ˇZg,t+1. +So far, we have developed a real-time feedback based online +aggregate EV power flexibility characterization method as well +as the EV dispatched charging power disaggregation approach. +A completed description of the proposed method is shown in +Algorithm 2. +V. SIMULATION RESULTS AND DISCUSSIONS +In this section, we evaluate the performance of the proposed +online algorithm and compare it with other approaches. +A. System Setup +The time resolution is set as 10 minutes. The entire sim- +ulation duration considered is 24 hours, i.e., 144 time slots + +JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 +7 +Algorithm 2 Real-time Feedback Based Online Aggregate EV +Power Flexibility Characterization and Disaggregation +I. Aggregation +1: Aggregator classifies the arriving EVs and pushes them +into different queues ˆQg, ˇQg, ˆZg and ˇZg according to +their declared charging delay time Rg. +2: Solve problem P3 and obtain the aggregate EV power +flexibility region [ˇx∗ +g,t, ˆx∗ +g,t]. +3: Update queues ˆQg,t+1, ˇQg,t+1, ˆZg,t+1, and ˇZg,t+1 ac- +cording to (4), (5), (12) and (13), respectively. +II. Dispatch and Disaggregation +4: Receive the dispatch decision from the DSO, and decom- +pose it to each group according to (29) and (30). +5: Perform EV dispatched power disaggregation according to +Algorithm 1. +III. Real-Time Feedback and Update +6: Update the lower and upper bounds ˆxupdate∗ +g,t +, ˇxupdate∗ +g,t +, ∀g +of EV aggregate power flexibility region. +7: Update queues ˆQg,t+1, ˇQg,t+1, ˆZg,t+1, and ˇZg,t+1 ac- +cording to (4), (5), (12) and (13), respectively. +8: Move to the next time slot t = t+1, and repeat the above +steps I-III. +each with a time interval of 10 minutes. To reflect the actual +fluctuations in the electricity prices, we use the real-time +electricity price data obtained from the PJM market [32]. The +dynamic electricity price data profile is shown in Fig. 3. For +the setting of EVs, we consider 30 EVs that are divided into +three groups with different allowable charging time delays, i.e., +G = 3. Each group has 10 EVs. The EVs in the same group +have identical allowable time delay, i.e., Rg. Particularlly, +R1 = 8 hours, R2 = 6 hours, and R3 = 7 hours. In addition, +for EV battery parameters, we refer to the Nissan Leaf EV +model with a battery pack of 40 kWh and a maximum charging +power of 6.6 kW [33]. Considering that the EV charging +behavior is uncertain, the EVs’ arriving times are randomly +generated. The initial battery energy level of each EV is +selected from a uniform distribution in [0.3, 0.5] × 40 kWh +randomly [29]. We set the required state-of-charge (SOC) 1 +upon departure as 0.5 and the maximum SOC upon departure +as 0.9. The weight parameter value of V is chosen as 1000, +and the value of ηg is set as 648, 540, and 756 for the three +groups, respectively. +Fig. 3. Real-time electricity price profile. +1The SOC of an EV is the ratio between the battery energy level and the +battery capacity. +B. Effectiveness of the proposed method +We first show how the obtained aggregate EV power +flexibility region looks like. Since the power grid dispatch +determined by the DSO is beyond the scope of this paper, +here the dispatch ratio αt in (29) is assumed to be randomly +generated within the range of [0, 1] in each time slot, as shown +in Fig. 4. Based on the generated dispatch ratio, we apply +the proposed online flexibility characterization method and +real-time feedback in turns (as in Algorithm 2) to obtain the +aggregate EV power flexibility region (grey area) over time for +each group and the charging station as a whole. The results are +shown in Fig. 5. As seen, the power flexibility region varies +over time. This is because EVs dynamically arrive and leave. +Fig. 4. Randomly generated dispatch ratio αt. +Fig. 5. The obtained aggregate EV power flexibility region. +To validate the effectiveness of the proposed algorithm, dis- +aggregation of the dispatched EV charging power is performed +and we check if the SOC curves of EVs satisfy the charging +requirements. Here, if the final EV SOC value can reach +or exceed the EV owner’s requirement (SOC ≥ 0.5) upon +leaving, then it means that the proposed method is effective. +The left of Fig. 6 shows the actual EV charging SOC curves +under the randomly generated dispatch ratio αt in Fig. 4. Each +curve represents an EV. As we can see from the figure, all +EVs’ final SOC is between 0.58 and 0.7, greater than the +required value 0.5 and less than the maximum value 0.9. The +right-hand side of Fig. 6 shows the number of delayed time +slots (NDTS) to reach the requirement SOC=0.5. We can find +that the maximum NDST is 10 for group 1, 7 for group 2, +and 12 for group 3. All of them are within their respective +declared allowable charging delay, i.e. Rg. This validates the +proposed algorithm in providing maximum power flexibility +while meeting the charging requirement. +Furthermore, Fig. 7 shows the queue backlog evolution of +the three groups over time. Taking group 1 for example, the +lower and upper bound queues ˇQ1 and ˆQ1 first increase be- +cause EVs continue to arrive with their charging tasks pushing + +JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 +8 +Rg=8h=48 +time slots +Rg=6h=36 +time slots +Rg=7h=42 +time slots +Fig. 6. +EV charging SOC of each group and the number of delayed time +slots needed to reach SOC=0.5. +into the queues. Then as time moves on, through the EV +charging dispatch pdisp +v,t , ∀v, ∀t determined by disaggregation, +the EV SOC gradually increases and reaches the minimum +charging requirement 0.5. Hence, the lower bound queue ˇQ1 +becomes zero because the ˇag,t, ∀g, ∀t is set using the charging +as soon as possible method to meet the minimum charging +requirement as in (7). The upper bound queue ˆQ1 is still larger +than zero since the EV SOC has not reach the maximum value +0.9 (see Fig. 6), so there is charging flexibility. At the same +time, since ˆQ1 is nonnegative, the delay aware upper bound +queue ˆZ1 keeps growing, aiming to increase the charging +power. The queue evolution in groups 2 and 3 can be analyzed +similarly. +Fig. 7. Queue backlog of each group. +C. Performance Evaluation +To show the advantage of the proposed online algorithm, +two widely used benchmarks in the literature are performed. +• Benchmark 1 (B1): This is a greedy algorithm that EVs +start charging at the maximum charging power upon +arrival. Let us denote the arrival time as t0. When the EV +SOC reaches the minimum charging requirement 0.5, the +lower bound of charging power ˇpd,t turns to be zero (time: +t1), and the upper bound of charging power ˆpd,t remains +the maximum charging power until the EV SOC reaches +the maximum value 0.9 (time: t2). The aggregate power +flexibility region for [t0, t1] is empty and for t ∈ [t1, t2] is +the region between 0 and the maximum charging power. +• Benchmark 2 (B2): This is the offline method. It directly +solves P1 to obtain the aggregate EV power flexibility +regions over the whole time horizon by assuming known +future information. Though not realistic, it provides a +theoretical benchmark to verify the performance of other +methods. But it is worth noting that since it does not +take into account the real-time actual dispatch strategy +when calculating the aggregate flexibility, its performance +may be worse than the proposed real-time feedback based +method even though it is an offline method. +Fig. 8 shows the accumulated flexibility values (�t +τ=1 Fτ) +under the three different methods, and TABLE I summarizes +the total flexibility value (�T +t=1 Ft) under different methods. +The B1, i.e., greedy algorithm, has the worst performance and +the lowest total flexibility value due to the myopic strategy. For +B2, since it has complete future knowledge of EV behaviors +and real-time electricity prices, it outperforms B1. However, +this method is usually impossible in practice since the accurate +future information is hardly available. Though predictions on +future uncertainty realizations may be obtained, the potential +prediction errors limit B2’s performance. In contrast, the +proposed online algorithm achieves the best performance with +the highest total power flexibility value. This is owing to the +fact that it runs in a online manner with real-time feedback +that allows it to utilize the most recent dispatch information +to update its state. In addition, compared to the offline method +B2, it does not require prior knowledge of future information +or forecasts, which is more practical. +Fig. 8. Accumulated flexibility value under different methods. +TABLE I +TOTAL FLEXIBILITY VALUE COMPARISON BETWEEN B1, B2, AND THE +PROPOSED ALGORITHM (UNIT: USD). +Methods +B1 +B2 +Proposed +Value +517.69 +586 +647.21 +Improvement +- +13.2% +25% +The above result is obtained under the random dispatch ratio +αt (see Fig. 4). In fact, the dispatch ratio can affect the actual +charging power of each EV and further affect their aggregate +power flexibility. Therefore, it is interesting to investigate the + +JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 +9 +impact of αt on the aggregate EV power flexibility (or the total +power flexibility value (9)). Here, we use a uniform α over +time, i.e., αt = α, ∀t. We change α from 0 to 1 and record the +total power flexibility value in Fig. 9. As seen, the total power +flexibility value depends on the dispatch ratio α. Generally, +a larger α leads to a larger power flexibility value. However, +this increase is nonlinear. When the dispatch ratio α exceeds +0.5, the total power flexibility value no longer increases. In the +extreme case when α = 0, the total power flexibility value is +520 USD, which is still larger than the greedy algorithm B1. +In addition, it can be concluded that if the average dispatch +ratio α is greater than 0.2, then the proposed online algorithm +more likely outperforms the offline method. This demonstrates +the advantage of the proposed algorithm. We also present the +aggregate EV power flexibility region under different α in Fig. +10. As α decreases, the aggregate EV power flexibility region +gradually narrows. This is because under a low dispatch ratio, +the EVs are charged at a low charging rate and more likely to +fail to meet the charging requirement; hence, the lower bound +of aggregate EV power flexibility region is raised to ensure +the EVs can meet the charging requirement in the remaining +time. +Fig. 9. The impact of α on total power flexibility value. +0 +50 +Power +[kW] +ub +lb +=0 +0 +50 +Power +[kW] +ub +lb +=0.1 +0 +50 +Power +[kW] +ub +lb +=0.2 +0 +50 +Power +[kW] +ub +lb +=0.3 +0 +50 +Power +[kW] +ub +lb +=0.4 +0 +20 +40 +60 +80 +100 +120 +140 +Time [10 min] +0 +50 +Power +[kW] +ub +lb +=0.5 +0 +50 +Power +[kW] +ub +lb +=0 +0 +50 +Power +[kW] +ub +lb +=0.1 +0 +50 +Power +[kW] +ub +lb +=0.2 +0 +50 +Power +[kW] +ub +lb +=0.3 +0 +50 +Power +[kW] +ub +lb +=0.4 +0 +20 +40 +60 +80 +100 +120 +140 +Time [10 min] +0 +50 +Power +[kW] +ub +lb +=0.5 +Fig. 10. The aggregate EV power flexibility under different α. +D. Impact of Parameters +According to (19), the parameter V controls the trade- +off between stabilizing the queues and maximizing the total +power flexibility value in the objective function (27). Here, we +change the value of V to investigate its impact on the total +power flexibility value. As shown in Fig. 11, the total power +flexibility value becomes larger with an increasing V . +Fig. 11. The impact of V on the total power flexibility value. +Fig. 12. The impact of V on the average time delay. +Fig. 12 depicts the impact of V on the number of time slots +needed for EVs to meet their required battery energy level ed +v. +We calculate the maximum/minimum/average number of time +slots for the EVs in each group. As seen, with the growth +of V , the number of delayed time slots slightly increases. +This is because a larger V means putting more emphasis on +maximizing the total power flexibility, which may result in a +reduced lower bound ˇxg,tof the aggregate EV power flexibility +region. Consequently, the charging time needed to reach the +required energy level becomes longer. Comparing the three +groups, we can find that the time delays of groups 1 and 3 are +generally longer than that of group 2, which is owing to the +shorter allowable time delay Rg. +Fig. 13 shows the impact of ηg on the total flexibility value +and the number of time slots needed for EVs to meet their +required battery energy level ed +v. We can find that a larger ηg +results in a lower total power flexibility value and less number +of delayed time slots. This is because a larger ηg forces the +virtual delay-aware queue ˇZg to grow rapidly, allowing EVs +to get charged quickly. Meanwhile, the power flexibility is +sacrificed. +VI. CONCLUSION +With the proliferation of EVs, it is necessary to better +utilize their charging power flexibility, making them valuable +resources rather than burdens on the power grid. This paper +proposes a real-time feedback based online aggregate EV +power flexibility characterization method. It can output the +aggregate flexibility region for each time slot in an online +manner, with a total flexibility value over time similar to the +offline counterpart. We prove that by choosing an aggregate + +JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 +10 +Fig. 13. +The impact of ηg on the total flexibility value and average time +delay. +dispatch strategy within the obtained flexibility region for +each time slot, the corresponding disaggregated EV control +strategies allow all EVs to satisfy their charging requirements. +Simulations demonstrate the effectiveness and benefits of the +proposed method. 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Abusorrah, +“Optimal transactive energy trading of electric vehicle charging stations +with on-site PV generation in constrained power distribution networks,” +IEEE Trans. Smart Grid, vol. 13, no. 2, pp. 1427–1440, 2022. +APPENDIX A +PROOF OF PROPOSITION 1 +Let {pd,t, ∀t} be the aggregate power trajectory. For each +time slot t ∈ T , since pd,t ∈ [ˇp∗ +d,t, ˆp∗ +d,t], we can define an + +JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 +11 +auxiliary coefficient: +βt := +ˆp∗ +d,t − pd,t +ˆp∗ +d,t − ˇp∗ +d,t +∈ [0, 1] +(A.1) +so that pd,t = βtˇp∗ +d,t + (1 − βt)ˆp∗ +d,t. Then, we can construct a +feasible EV dispatch strategy by letting +pc +v,t = βtˇpc∗ +v,t + (1 − βt)ˆpc∗ +v,t, +(A.2a) +ev,t = βtˇec∗ +v,t + (1 − βt)ˆec∗ +v,t. +(A.2b) +for all time slots t ∈ T . +We prove that it is a feasible EV dispatch strategy as +follows, +pd,t = βtˇp∗ +d,t + (1 − βt)ˆp∗ +d,t += βt +� +v∈V +ˇpc∗ +v,t + (1 − βt) +� +v∈V +ˆpc∗ +v,t += +� +v∈V +� +βtˇpc∗ +v,t + (1 − βt)ˆpc∗ +v,t +� += +� +v∈V +pc +v,t +(A.3) +Hence, constraint (1c) holds for pd,t and pc +v,t, ∀v. Similarly, +we can prove that constraints (1d)-(1g) are met. Therefore, +we have constructed a feasible EV dispatch strategy, which +completes the proof. +■ +APPENDIX B +PROOF OF PROPOSITION 2 +Here, we use the contradiction. If a charging request ˆag,t +arrives in time slot t cannot be fulfilled on or before time +slot t + ˆδg,max. Then, queue ˆQg,τ > 0always holds for τ ∈ +[t + 1, ..., t + ˆδg,max]. Thus, we have I ˆ +Qg,t>0 = 1. According +to delay virtual queue dynamics (12), for all τ ∈ [t+1, ..., t+ +ˆδg,max], we have +ˆZg,τ+1 ≥ ˆZg,τ + ηg +Rg +− ˆxg,τ, ∀g, ∀t. +(B.1) +By summing the above inequalities over τ ∈ [t + 1, ..., t + +ˆδg,max], we have +ˆZg,t+ˆδg,max+1 − ˆZg,t+1 ≥ ηg +Rg +ˆδg,max + +t+ˆδg,max +� +τ=t+1 +(−ˆxg,τ). +(B.2) +Since ˆZg,t+ˆδg,max+1 ≤ ˆZg,max and ˆZg,t+1 ≥ 0, we have +ˆZg,max ≥ ηg +Rg +ˆδg,max + +t+ˆδg,max +� +τ=t+1 +(−ˆxg,τ). +(B.3) +Since the charging tasks are processed in a first-in-first-out +manner, and the charging request is not fulfilled by t+ˆδg,max, +we have +t+ˆδg,max +� +τ=t+1 +(ˆxg,τ) < ˆQg,max +(B.4) +Combining the above two inequalities, we obtain +ˆZg,max > ηg +Rg +ˆδg,max − ˆQg,max, +(B.5) +which implies +ˆδg,max < ( ˆQg,max + ˆZg,max)Rg +ηg +. +(B.6) +However, this result contradicts the definition of ˆδg,max in +(14). Therefore, the worst case delay should be less than or +equal to ˆδg,max as defined in (14). +The proof of (15) follows a similar procedure, and we omit +it here for brevity. +■ +APPENDIX C +PROOF OF PROPOSITION 3 +Denote the solution of P3 by the proposed algorithm by +ˆxpro +g,t and ˇxpro +g,t , and the optimal solution of P1 by ˆx∗ +g,t and +ˇx∗ +g,t. According to (26), we have +∆(Θt) + V E[−F pro +t +|Θt] +≤ A + V E[−F pro +t +|Θt] + +� +g∈G +ˆQg,tE +� +ˆag,t − ˆxpro +g,t |Θt +� ++ +� +g∈G +ˇQg,tE +� +ˇag,t − ˇxpro +g,t |Θt +� ++ +� +g∈G +ˆZg,tE +� +−ˆxpro +g,t |Θt +� ++ +� +g∈G +ˇZg,tE +� +−ˇxpro +g,t |Θt +� +, +≤ A + V E[−F ∗ +t |Θt] + +� +g∈G +ˆQg,tE +� +ˆag,t − ˆx∗ +g,t|Θt +� ++ +� +g∈G +ˇQg,tE +� +ˇag,t − ˇx∗ +g,t|Θt +� ++ +� +g∈G +ˆZg,tE +� +−ˆx∗ +g,t|Θt +� ++ +� +g∈G +ˇZg,tE +� +−ˇx∗ +g,t|Θt +� +, +≤ A + V E[−F ∗ +t |Θt] +(C.1) +The result is based on the fact that +lim +T →∞ +1 +T +T +� +t=1 +E [ˆag,t − ˆxg,t|Θt] ≤ 0 +(C.2) +lim +T →∞ +1 +T +T +� +t=1 +E [ˇag,t − ˇxg,t|Θt] ≤ 0 +(C.3) +lim +T →∞ +1 +T +T +� +t=1 +E [−ˆxg,t|Θt] ≤ 0 +(C.4) +lim +T →∞ +1 +T +T +� +t=1 +E [−ˇxg,t|Θt] ≤ 0 +(C.5) +which is due to constraints (9b)-(9e). +By summing the above inequality (C.1) over time slots t ∈ +{1, 2, . . . , T}, we have +T +� +t=1 +V E[−F pro +t +] +≤ AT + V +T +� +t=1 +E[−F ∗ +t ] − E[L(ΘT +1)] + E[L(Θ1)]. + +JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 +12 +Based on the fact that L(ΘT +1) and L(Θ1) are finite, we +divide both sides of the above inequalities by V T and let +T → ∞, then we have +lim +T →∞ +1 +T +T +� +t=1 +E(−F pro +t +) ≤ A +V + lim +T →∞ +1 +T +T +� +t=1 +E(−F ∗ +t ), +which completes the proof. +■ + diff --git a/EdE1T4oBgHgl3EQfqQXv/content/tmp_files/load_file.txt b/EdE1T4oBgHgl3EQfqQXv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1e152b732268a0646d2d25bccec4a68daa444f3 --- /dev/null +++ b/EdE1T4oBgHgl3EQfqQXv/content/tmp_files/load_file.txt @@ -0,0 +1,976 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf,len=975 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 1 Real-time Feedback Based Online Aggregate EV Power Flexibility Characterization Dongxiang Yan, Shihan Huang, and Yue Chen, Member, IEEE Abstract—As an essential measure to combat global warming, electric vehicles (EVs) have witnessed rapid growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Meanwhile, thanks to the flexibility of EV charging, vehicle-to-grid (V2G) interaction has captured great attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, the direct con- trol of individual EVs is challenging due to their small capacity, large number, and private information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Hence, it is the aggregator that interacts with the grid on behalf of EVs by characterizing their aggregate flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In this paper, we focus on the aggregate EV power flexibility characterization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' First, an offline model is built to obtain the lower and upper bounds of the aggregate power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' It ensures that any trajectory within the region is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, considering that parameters such as real-time electricity prices and EV arrival/departure times are not known in advance, an online algorithm is developed based on Lyapunov optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We prove that the charging time delays for EVs always meet the requirement even if they are not considered explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Furthermore, real-time feedback is designed and integrated into the proposed online algorithm to better unlock EV power flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Comprehensive performance comparisons are carried out to demonstrate the advantages of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Index Terms—Aggregate flexibility, charging station, electric vehicle, Lyapunov optimization, online algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' INTRODUCTION T HANKS to the low carbon emissions, electric vehicles (EVs) have been considered a promising solution to climate change and proliferate in recent years [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, the uncontrolled charging of a large number of EVs can cause voltage deviation, line overload, and huge transmission loss [2], threatening the reliability of the power system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Unlike inelastic loads, the charging power and charging period of EVs are more flexible [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Therefore, unlocking the power flexibility hidden in EVs is a promising way to lessen the adverse impact of EVs on the power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' There are extensive literature aiming to design coordinated charging strategies to optimally schedule EV charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For example, to promote local renewable generation consumption, a dynamic charging strategy was proposed to allow the EV charging power to dynamically track the PV generation [4] and wind generation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To save the electricity cost, a deterministic optimal charging strategy was proposed for a home energy management system based on the time-of-use tariffs [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A model predictive control (MPC) algorithm was proposed to minimize the operational cost of EV charging stations [7] relying on short-term forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To address the uncertainties related to EV charging, reference [8] proposed a D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Yan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Huang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Chen are with the Department of Me- chanical and Automation Engineering, the Chinese University of Hong Kong, Hong Kong SAR, China (e-mail: dongxiangyan@cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='hk, shhuang@link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='hk, yuechen@mae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' stochastic charging strategy based on the probabilistic model related to EV daily travels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A combined robust and stochastic MPC method was developed in [9] to handle the uncertain EV charging behaviors and renewable generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A multi-stage energy management strategy including day-ahead and real- time stages was developed for a charging station integrated with PV generation and energy storage [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In addition, a pricing mechanism was suggested in [11] to guide EVs for economical charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A double-layer optimization model was built to reduce the voltage violations caused by EV charging [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Despite the efforts mentioned above that intend to determine the EV charging power, it is challenging to directly control a large number of individual EVs due to the high computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To get around this problem, some other literature en- deavored to characterize the EV charging power flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Reference [13] proposed to model the aggregate EV charging flexibility region by the lower and upper bounds of power and cumulative energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This aggregate EV model was adopted by [14] to evaluate the achievable vehicle-to-grid capacity of an EV fleet and by [15] to quantify the value of EV flexibility in terms of maintaining distribution system reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Reference [16] further considered the spatio-temporal distribution of the probability that an EV is available for charging during the aggregation and clustering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' An EV dispatchable region was proposed to allow charging stations to participate in market bidding [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In addition, the aggregate flexibility issue was also studied in the fields of thermostatically controllable loads (TCLs) [18], distributed energy resources [19], and virtual power plant [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For example, a geometric approach was utilized to model the aggregate flexibility of TCLs [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' An inner box approximation method was proposed to charac- terize the power flexibility region of various distributed energy resources [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The above works provide sound techniques for evaluating EV flexibility in an offline manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' It means that the aggregator is assumed to have complete information of future uncertainty realizations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', EV arrival/departure time, and electricity prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In practice, those data are usually unavailable or inaccurate, making the obtained region fail to reflect the actual EV aggregate flexibility in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Thus, an online algorithm is desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A straightforward approach is the greedy algorithm that decomposes the offline problem into subproblems in each time slot by neglecting the time-coupling constraints [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Obviously, the result could be far from optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Hence, we resort to another approach, Lyapunov optimization, that can run in an online manner but with an outcome near to the offline optimum [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Lyapunov optimization has been used in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='03342v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='OC] 9 Jan 2023 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 2 microgrid control [25], energy storage sharing [26], and data center energy management [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For EV charging, a charging strategy based on Lyapunov optimization was proposed to minimize the total electricity cost [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, it cannot guarantee that EVs will depart with desired amount of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To meet the EV charging requirement, virtual delay queues were introduced to minimize the charging cost under uncertain renewable generations and electricity prices [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Though the optimal online EV charging strategy has been widely studied as above, the online aggregate EV power flexibility characterization problem has not been well explored yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The latter problem is more complicated than the former one, requiring a new model and algorithm design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This paper proposes a real-time feedback based online aggregate EV power flexibility characterization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Our main contribu- tions are two-fold: 1) Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We first propose an offline optimization model to characterize the aggregate EV power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' It decomposes the time-coupled flexibility region into each time slot and gives their lower and upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We prove that any trajectory within the region is achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, by categorizing the EVs according to their allowable time delays, we develop a counterpart of the offline model that enables the further utilization of the Lyapunov optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The proposed model has not been reported in previous literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2) Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A real-time feedback based online algorithm is developed to derive the aggregate EV power flexibility region sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' First, to fit into the Lyapunov optimization framework, charging task queues and delay-aware virtual queues are introduced to reformulate the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, a drift-plus-penalty term is constructed and by minimizing its upper bound, an online algorithm is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We prove that the charging time delays for EVs will not exceed their maximum allowable values even if they are not explicitly considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The bound of optimality gap between the offline and online outcomes is provided theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Furthermore, real-time dispatch strategy based feedback is designed and integrated into the online algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The proposed real-time feedback based online algorithm is prediction free and can adapt to uncertainties such as random electricity prices and EV charging behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Furthermore, it can make use of the most recent information, allowing it to even outperform the offline model with full knowledge of future uncertainty realizations but without the updated dispatch information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Section II formulates the offline model for deriving aggregate EV charging power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Section III and IV introduce the Lyapunov optimization method and real-time feedback design, respectively, to generate flexibility region in an online manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Simulation results are presented in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Finally, Section VI concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' PROBLEM FORMULATION In this section, we first introduce the concept of aggregate EV power flexibility and then formulate an offline optimization problem to approximate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Aggregate EV Charging Power Flexibility As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 1, when an EV v ∈ V arrives at the charging station, it submits its charging task to the aggregator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The task is described by (ta v, td v, ea v, ed v), where ta v is its arrival time, td v is its departure time, ea v is the initial battery energy level at ta v, and ed v is the desired energy level when it leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For the EV v, the maximum allowable charging time delay is Rv = td v−ta v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The EV charging task needs to be finished within this declared time duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' With the submitted information, the aggregator can flexibly schedule the EV charging to meet the charging requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Two possible trajectories to meet the EV charging need are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Let {pc v,t, ∀t} be the charging power of EV v over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The range that the charging power can vary within is called the power flexibility of EV v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' If we sum the power flexibility of all EVs in a charging station up, we can get the aggregate EV power flexibility of the charging station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (𝑡1 𝑎, 𝑡1 𝑑, 𝑒1 𝑎, 𝑒1 𝑑) (𝑡2 𝑎, 𝑡2 𝑑, 𝑒2 𝑎, 𝑒2 𝑑) (𝑡𝑣𝑎, 𝑡𝑣𝑑, 𝑒𝑣𝑎, 𝑒𝑣𝑑) Aggregator Distribution System Operator Aggregate dispatch power Aggregate power flexibility region Ƽ𝑝𝑑,𝑡, Ƹ𝑝𝑑,𝑡 𝑝𝑑,𝑡 𝑑𝑖𝑠𝑝 𝑝1,𝑡 𝑑𝑖𝑠𝑝 𝑝2,𝑡 𝑑𝑖𝑠𝑝 𝑝𝑣,𝑡 𝑑𝑖𝑠𝑝 EV 1 EV 2 EV v 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Generate EV aggregate power flexibility region 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Disaggregation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' System diagram and illustration of EV power flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, it is difficult to characterize the EV power flexibil- ity for each time slot due to the temporal-coupled EV charging constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The EV power flexibility in the current time slot is affected by those in the past time slots and further affects those in the future time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This is different from the traditional controllable generators whose flexibility can be described by the minimum and maximum power outputs in each time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In the following, we aim to derive an aggregate EV power flexibility region that: 1) is time-decoupled so that it can be used in real-time power system operation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' and 2) any trajectory within it can meet the EV charging requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Offline Problem Formulation Suppose there are T time slots, indexed by t ∈ T = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The desired time-decoupled aggregate EV power flexibility region can be represented by a series of intervals [ˇpd,t, ˆpd,t], ∀t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The intervals can be specified by a lower power trajectory {ˇpd,t, ∀t} and an upper power trajectory {ˆpd,t, ∀t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To obtain the lower and upper power trajectories, we formulate the following offline optimization problem: P1 : max ˆpd,t,ˇpd,t,∀t lim T →∞ 1 T T � t=1 E � Ft � , (1a) Power Aggregate power flexibility region flexibilitJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 3 where Ft = πt(ˆpd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t − ˇpd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1b) subject to ˆpd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t = � v∈V ˆpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1c) 0 ≤ ˆpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t ≤ pmax v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1d) ˆev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t+1 = ˆev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t + ˆpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t∆t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t ̸= T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1e) ˆev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='tav = eini v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ˆev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='tdv ≥ ed v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1f) emin v ≤ ˆev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t ≤ emax v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1g) ˇpd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t = � n∈V ˇpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1h) 0 ≤ ˇpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t ≤ pmax v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1i) ˇev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t+1 = ˇev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t + ˇpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t∆t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t ̸= T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1j) ˇev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='tav = eini v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ˇev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='tdv ≥ ed v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1k) emin v ≤ ˇev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t ≤ emax v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1l) ˇpd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t ≤ ˆpd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1m) ˆpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t/ˇpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t = � ˆpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t/ˇpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' if t ∈ [ta v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' td v] 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' if t < ta v ∪ t > td v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1n) In the objective function (1a)-(1b), πt, ∀t are the real-time electricity prices, showing the unit value of power flexibility in different time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Hence, the objective function aims to maximize the value of total aggregate EV power flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Constraint (1c) defines the upper bound of aggregate EV power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The charging power of an EV v is limited by (1d), where pmax v is the maximum charging power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Constraint (1f) defines the EV’s initial energy level and the charging requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1e) and (1g) describe the EV’s energy dynamics and battery capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Similarly, (1h)-(1l) are the constraints related to the lower bound of the aggregate EV power flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1m) is the joint constraint to ensure that {ˆpd,t, ∀t} and {ˇpd,t, ∀t} provide the upper and lower bounds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1n) limits that charging only happens during the EV’s declared parking time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Proposition 1: Any aggregate EV charging power trajectory within [ˇpd,1, ˆpd,1] × · · · × [ˇpd,T , ˆpd,T ] is achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The proof of Proposition 1 can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Despite this nice property, the offline optimization problem above cannot be solved directly since it requires complete knowledge of the future EV charging tasks and future elec- tricity prices, which are usually not available in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Therefore, an online algorithm is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To this end, in the next section, we will first propose a closely related but more flexible form of the problem studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, we adopt the Lyapunov optimization framework to reformulate the offline problem into an online one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We construct charging task queues and delay-aware virtual queues to ensure the satisfaction of charging requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Furthermore, considering the impact of real-time dispatch decisions on the future aggregate EV power flexibility, a real-time feedback based online flexibility characterization method is developed in Section IV to avoid the potential underestimate of EV power flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ONLINE ALGORITHM In this section, we adopt the Lyapunov optimization frame- work to solve the offline problem P1 in an online manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The proposed algorithm can output an aggregate EV power flexibility value with an economic value close to P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Problem Modification As mentioned above, the charging station serves dozens of EVs every day, and each EV arrives along with a charging task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', (ta v, td v, ea v, ed v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Those EV charging tasks can be first stored in a queue and be served later according to a first-in-first- out basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Since different EVs may have different allowable charging time delays, we use multiple queues to classify and collect the EV charging tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Suppose there are G types of charging time delays Rgs, each of which is indexed by g ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Correspondingly, we construct G queues to collect the respective charging tasks, and each queue is denoted by Qg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For queue Qg, Qg,t refers to its charging task backlog in time slot t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The queue backlog growth is described by Qg,t+1 = max[Qg,t − xg,t, 0] + ag,t, (2) where xg,t is the charging power for EVs in group g at time t, and ag,t is the arrival rate of EV charging tasks of group g at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In particular, ag,t sums up the energy demand of all EVs that arrive at the beginning of time t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ag,t = � v∈Vg ag,v,t, (3) where ag,v,t is the charging demand of EV v of group g in time slot t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Vg is the set of EVs in group g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Recalling that our target in P1 is to derive an upper bound and a lower bound for the aggregate EV power flexibility region, we correspondingly define the upper bound queue ˆQg,t and the lower bound queue ˇQg,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Similar to (2), we have ˆQg,t+1 = max[ ˆQg,t − ˆxg,t, 0] + ˆag,t, (4) ˇQg,t+1 = max[ ˇQg,t − ˇxg,t, 0] + ˇag,t, (5) where ˆxg,t and ˇxg,t are the charging power for upper and lower bound queues, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ˆxg,t = � v∈Vg ˆpc v,t and ˇxg,t = � v∈Vg ˇpc v,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The upper and lower bounds of arriving charging demand, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ˆag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t and ˇag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' are determined by ˆag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t = � v∈Vg ˆag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ˇag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t = � v∈Vg ˇag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (6) Particularly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' the lower bound of arriving charging demand ˇag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t can be determined in the following charging as soon as possible way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ˇag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t = � � � pmax v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ta v ≤ t < ⌊ˇtmin v ⌋ + ta v ˇecha v /ηc − ⌊ˇtmin v ⌋pmax v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' t = ⌊ˇtmin v ⌋ + ta v 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' otherwise ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (7) where ˇecha v = ed v −ea v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ˇtmin v is the minimum required charging time determined by ˇtmin v = ˇecha v pmax v ηc ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' and ⌊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='⌋ means rounding down to the nearest integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 4 Different from ˇag,v,t, the upper bound of arrival charging demand ˆag,v,t is determined using the maximum charging demand emax v instead of ed v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Denote ˆecha v = emax v − ea v, ˆtmin v = ˆecha v pmax v ηc , and then ˆag,v,t can be determined by ˆag,v,t = � � � pmax v , ta v ≤ t < ⌊ˆtmin v ⌋ + ta v ˆecha v /ηc − ⌊ˆtmin v ⌋pmax v , t = ⌊ˆtmin v ⌋ + ta v 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (8) We then formulate the aggregate EV power flexibility char- acterization problem as follows: P2 : min ˆxg,t,ˇxg,t lim T →∞ 1 T T � t=1 E � − Ft � , (9a) subject to lim T →∞ 1 T T � t=1 E[ˆag,t − ˆxg,t] ≤ 0, ∀g (9b) lim T →∞ 1 T T � t=1 E[ˇag,t − ˇxg,t] ≤ 0, ∀g (9c) 0 ≤ ˆxg,t ≤ min{xg,max, ˆQg,t}, ∀g (9d) 0 ≤ ˇxg,t ≤ min{xg,max, ˇQg,t}, ∀g (9e) ˆxg,t ≥ ˇxg,t, ∀g (9f) where xg,max = � v∈Vg pmax v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Constraint (9b) ensures that if using the upper bound trajectory {ˆxg,t, ∀t}, the total charg- ing requirement can be satisfied in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Constraint (9c) poses a similar requirement for the lower bound trajec- tory {ˇxg,t, ∀t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Based on (9b), (9c), and the definitions of ˆQg,t+1, ˇQg,t+1 in (4)-(5), we can prove that the queues ˆQg,t and ˇQg,t are mean rate stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To be specific, ˆQg,t+1 − ˆag,t ≥ ˆQg,t − ˆxg,t, ∀t (10) Summing (10) up over all t and divide both sides by T yields 0 ≤ E[ ˆQg,T ] T ≤ �T t=1 E[ˆag,t − ˆxg,t] T (11) Hence, lim T →∞ E[ ˆ Qg,T ] T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Similarly, lim T →∞ E[ ˇ Qg,T ] T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Constraints (9d) and (9e) give the upper and lower bounds of the aggregate EV charging power for group g, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The upper bound is no less than the lower bound, as shown in (9f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The P2 provides a counterpart problem for P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Similar to the proof of Proposition 1, we can prove that any trajectory between [ˇxg,t, ˆxg,t] is achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, the allowable charging delay is not considered in P2, which may result in unfulfilled EV charging tasks upon departure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Construct Virtual Queues To overcome the aforementioned charging delay issue, we introduce delay-aware virtual queues, ˆZg,t+1 = max{ ˆZg,t + ηg Rg I ˆ Qg,t>0 − ˆxg,t, 0}, ∀g, ∀t (12) ˇZg,t+1 = max{ ˇZg,t + ηg Rg I ˇ Qg,t>0 − ˇxg,t, 0}, ∀g, ∀t (13) where I ˆ Qg,t>0 and I ˇ Qg,t>0 are indicator functions of ˆQg,t and ˇQg,t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' They are equal to 1 if there exists unserved charging tasks in the queues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ˆQg,t > 0 and ˇQg,t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Using ηg Rg to times it, this whole term constitutes a penalty to the virtual queue backlog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ηg is a user-defined parameter that can adjust the growth rate of the virtual queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For instance, increasing the value of ηg leads to a fast queue growth and a larger backlog value, calling for more attention to accelerate the charging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We prove that, when Qg,t and Zg,t have finite upper bounds, with a proper ηg, the charging time delay for EVs in group g is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Proposition 2: Suppose ˆQg,t, ˇQg,t, ˆZg,t, and ˇZg,t have finite upper bounds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ˆQg,t ≤ ˆQg,max, ˇQg,t ≤ ˇQg,max ˆZg,t ≤ ˆZg,max, and ˇZg,t ≤ ˇZg,max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The charging time delay of all EVs in group g is upper bounded by ˆδg,max and ˇδg,max time slots, where ˆδg,max := ( ˆQg,max + ˆZg,max)Rg ηg , (14) ˇδg,max := ( ˇQg,max + ˇZg,max)Rg ηg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (15) The proof of Proposition 2 can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' It ensures that the charging tasks can always be fulfilled within the available charging periods by properly setting the parameters ηg, ∀g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Lyapunov Optimization Based on the charging task queues and delay-aware virtual queues, the Lyapunov optimization framework is applied as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 1) Lyapunov Function: First, we define Θt = ( ˆ Qt, ˆ Zt, ˇ Qt, ˇ Zt) as the concatenated vector of queues, where ˆ Qt = ( ˆQ1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ˆQG,t), (16a) ˆ Zt = ( ˆZ1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ˆZG,t), (16b) ˇ Qt = ( ˇQ1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ˇQG,t), (16c) ˇ Zt = ( ˇZ1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ˇZG,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (16d) The Lyapunov function is then defined as L(Θt) = 1 2 � g∈G ˆQ2 g,t + 1 2 � g∈G ˆZ2 g,t + 1 2 � g∈G ˇQ2 g,t + 1 2 � g∈G ˇZ2 g,t, (17) where L(Θt) can be considered as a measure of the queue size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A smaller L(Θt) is preferred to push (virtual) queues ˆQg,t, ˆZg,t, ˇQg,t, and ˇZg,t to be less congested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2) Lyapunov Drift: The conditional one-time slot Lyapunov drift is defined as follows: ∆(Θt) = E[L(Θt+1) − L(Θt)|Θt], (18) where the expectation is taken with respect to the random Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The Lyapunov drift is a measure of the expectation of the queue size growth given the current state Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Intuitively, by minimizing the Lyapunov drift, virtual queues are expected to be stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, only minimizing the Lyapunov drift may lead to a low aggregate EV power flexibility value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 5 Therefore, we include the expected aggregate flexibility value (1b) for the time slot t to (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The drift-plus-penalty term is obtained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ∆(Θt) + V E[−Ft|Θt], (19) where V is a weight parameter that controls the trade-off between (virtual) queues stability and aggregate EV power flexibility maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 3) Minimizing the Upper Bound: (19) is still time-coupled due to the definition of ∆(Θt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To adapt to online imple- mentation, instead of directly minimizing the drift-plus-penalty term, we minimize the upper bound to obtain the upper and lower bounds of aggregate EV power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We first calculate the one-time slot Lyapunov drift: L(Θt+1) − L(Θt) = 1 2 � g∈G � � ˆQ2 g,t+1 − ˆQ2 g,t � + � ˆZ2 g,t+1 − ˆZ2 g,t � + � ˇQ2 g,t+1 − ˇQ2 g,t � + � ˇZ2 g,t+1 − ˇZ2 g,t � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (20) Use queue ˆQg,t as an example, based on the queue update equations (4), we have ˆQ2 g,t+1 = {max[ ˆQg,t − ˆxg,t, 0] + ˆag,t}2 ≤ ˆQ2 g,t + ˆa2 g,max + ˆx2 g,max + 2 ˆQg,t(ˆag,t − ˆxg,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (21) Thus, 1 2 � ˆQ2 g,t+1 − ˆQ2 g,t � ≤ 1 2 � ˆx2 g,max + ˆa2 g,max � + ˆQg,t (ˆag,t − ˆxg,t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (22) Similarly, for queue ˇQg,t, ˆZg,t, and ˇZg,t, we have 1 2 � ˇQ2 g,t+1 − ˇQ2 g,t � ≤ 1 2 � ˇx2 g,max + ˇa2 g,max � + ˇQg,t (ˇag,t − ˇxg,t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (23) 1 2[ ˆZ2 g,t+1 − ˆZ2 g,t] ≤ 1 2 max[( ηg Rg )2, ˆx2 g,max] + ˆZg,t[ ηg Rg − ˆxg,t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (24) 1 2[ ˇZ2 g,t+1 − ˇZ2 g,t] ≤ 1 2 max[( ηg Rg )2, ˇx2 g,max] + ˇZg,t[ ηg Rg − ˇxg,t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (25) We then substitute inequalities (22),(23), (24) and (25) into drift-plus-penalty term and yield ∆(Θt) + V E[−Ft|Θt] ≤ A + V E[−Ft|Θt] + � g∈G ˆQg,tE [ˆag,t − ˆxg,t|Θt] + � g∈G ˇQg,tE [ˇag,t − ˇxg,t|Θt] + � g∈G ˆZg,tE [−ˆxg,t|Θt] + � g∈G ˇZg,tE [−ˇxg,t|Θt] , (26) where A is a constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', A = 1 2 � g∈G (ˆx2 g,max + ˆa2 g,max) + 1 2 � g∈G max[( ηg Rg )2, ˆx2 g,max] + 1 2 � g∈G (ˇx2 g,max + ˇa2 g,max) + 1 2 � g∈G max[( ηg Rg )2, ˇx2 g,max] + � g∈G [ ˆZg,max ηg Rg ] + � g∈G [ ˇZg,max ηg Rg ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' By reorganizing the expression in (26) and ignoring the con- stant terms, we can obtain the following online optimization problem, P3 : min ˆxg,t,ˇxg,t,∀g,∀t � g∈G (−V πt − ˆQg,t − ˆZg,t)ˆxg,t + � g∈G (V πt − ˇQg,t − ˇZg,t)ˇxg,t, (27) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (9d) − (9f), where ˆQg,t, ˇQg,t, ˇZg,t, and ˇZg,t are first updated based on (4),(5),(12), and (13) before solving P3 in each time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In each time slot t, given the current system queue state Θt, the proposed method determines the current upper and lower aggregate EV power flexibility bounds ˆxg,t and ˇxg,t by solving problem P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Hence, the original offline optimization problem P1 has been decoupled into simple online (real-time) problems, which can be executed in each time slot without requiring prior knowledge of future uncertain states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Since the modified problem P3 is slightly different from the offline one P1, an important issue we care about is: what’s the gap between the optimal solutions of the online problem P3 the and offline problem P1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Proposition 3: Denote the obtained long-term time-average aggregate EV power flexibility value of P1 and P3 by F ∗ and F pro, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We have 0 ≤ −F pro + F ∗ ≤ 1 V A, (28) where A is a constant defined in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The proof of Proposition 3 can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The optimality gap can be controlled by the parameter V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A bigger V value leads to a smaller optimality gap but increased queue sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In contrast, a smaller V value makes the queues more stable but results in a larger optimality gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' DISAGGREGATION AND REAL-TIME FEEDBACK DESIGN In each time slot t, given the aggregate EV power flexibility region [� g ˇx∗ g,t, � g ˆx∗ g,t], the distribution system operator (DSO) can determine the optimal aggregate dispatch strategy for EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This aggregate dispatch strategy should be further disaggregated to obtain the control strategy for each EV, which is studied in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Moreover, considering that the current dispatch strategy will influence the aggregate EV power flexibility in future time slots, real-time feedback is designed and integrated with the proposed online flexibility characterization method in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Disaggregation Suppose the dispatch strategy in time slot t is pdisp agg,t ∈ [� g ˇx∗ g,t, � g ˆx∗ g,t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This can be determined by the DSO by solving an economic dispatch problem based on the up-to- date information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', the electricity price πt, the grid-side renewable generation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Since this paper focuses on the online characterization of aggregate EV power flexibility, the eco- nomic dispatch problem by the DSO is omitted for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Interested readers can refer to [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Let the dispatch ratio αt be αt = pdisp agg,t − � g ˇx∗ g,t � g ˆx∗ g,t − � g ˇx∗ g,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (29) Then, the dispatched power pdisp g,t for each group g can be determined according to the ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', pdisp g,t = (1 − αt)ˇx∗ g,t + αtˆx∗ g,t, (30) which satisfies pdisp agg,t = � g pdisp g,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The next step is to allocate pdisp g,t to the EVs in the group g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' All EVs in the group g are sorted according to their arrival time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, we follow the first-in-first-service principle to allocate the energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' namely, the EV that comes earlier will be charged with the maximum charging power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We have pdisp v,t = min � pdisp g,t , pmax v , emax v − ev,t ∆t � , ∀v ∈ Vg, (31) where Vg refers to the set of EVs in group g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The third term on the right side is used to ensure that the EV will not exceed its allowable maximum energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' After this charging assignment for an earlier EV is com- pleted, the following update procedures will execute pdisp g,t ← (pdisp g,t − pdisp v,t ), (32) which means deducting pdisp v,t from the total remaining dis- patched power pdisp g,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, the pdisp g,t is allocated to the next earlier arrival EVs until the aggregate EV charging power is completely assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' At this time, the disaggregation is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The dispatched power disaggregation algorithm is presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' State Update to Improve Power Flexibility Region Following the disaggregation procedures in Algorithm 1, we can get the actual EV dispatched charging power pdisp v,t , ∀v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' By now, we can move on to the next time slot t + 1 and evaluate the EV power flexibility by solving problem P3, determine the dispatch strategy, disaggregate the dispatched power, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' But considering that the current actual dispatched EV charging power can affect the future aggregate EV power flexibility, which is ignored in the aforementioned processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Therefore, we propose a real-time feedback method to integrate the current actual EV dispatched charging power into the future aggregate EV power flexibility characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Algorithm 1 EV Dispatched Power Disaggregation 1: Initialization: aggregate EV dispatched power pdisp agg,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2: Calculate the dispatched aggregate charging power pdisp g,t for each group g using (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 3: for Each group g ∈ G do 4: for Each EV v in group g do 5: if the EV is not available for charging then 6: Let EV v’s charging power pdisp v,t = 0, ∀v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 7: else 8: Calculate pdisp v,t according to (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 9: Update the remaining aggregate power via (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 10: if If the updated pdisp g,t = 0 then 11: Break and return to Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 12: end if 13: end if 14: end for 15: end for Disaggregation State update Dispatch Aggregate power flexibility region State feedback t=t+1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Overall procedure of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The overall procedure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2 with the right-hand side blue box showing the real-time feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To be specific, we change the constraints (9d)-(9e) for time slot t into ˆxg,t = pdisp g,t , ˇxg,t = pdisp g,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (33) Since pdisp g,t ∈ [ˇx∗ g,t, ˆx∗ g,t], after replacing (9d)-(9e) with (33), the problem P3 is still solvable and the optimal solution is ˆxupdate∗ g,t = pdisp g,t , ˇxupdate∗ g,t = pdisp g,t , ∀g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' With these updated lower and upper bounds, we update the queues ˆQg,t+1, ˇQg,t+1, ˆZg,t+1, ˇZg,t+1 according to (4), (5), (12) and (13), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, we move on the solve problem P3 for time slot t + 1 using the updated ˆQg,t+1, ˇQg,t+1, ˆZg,t+1, ˇZg,t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' So far, we have developed a real-time feedback based online aggregate EV power flexibility characterization method as well as the EV dispatched charging power disaggregation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A completed description of the proposed method is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' SIMULATION RESULTS AND DISCUSSIONS In this section, we evaluate the performance of the proposed online algorithm and compare it with other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' System Setup The time resolution is set as 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The entire sim- ulation duration considered is 24 hours, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', 144 time slots JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 7 Algorithm 2 Real-time Feedback Based Online Aggregate EV Power Flexibility Characterization and Disaggregation I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Aggregation 1: Aggregator classifies the arriving EVs and pushes them into different queues ˆQg, ˇQg, ˆZg and ˇZg according to their declared charging delay time Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2: Solve problem P3 and obtain the aggregate EV power flexibility region [ˇx∗ g,t, ˆx∗ g,t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 3: Update queues ˆQg,t+1, ˇQg,t+1, ˆZg,t+1, and ˇZg,t+1 ac- cording to (4), (5), (12) and (13), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Dispatch and Disaggregation 4: Receive the dispatch decision from the DSO, and decom- pose it to each group according to (29) and (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 5: Perform EV dispatched power disaggregation according to Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Real-Time Feedback and Update 6: Update the lower and upper bounds ˆxupdate∗ g,t , ˇxupdate∗ g,t , ∀g of EV aggregate power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 7: Update queues ˆQg,t+1, ˇQg,t+1, ˆZg,t+1, and ˇZg,t+1 ac- cording to (4), (5), (12) and (13), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 8: Move to the next time slot t = t+1, and repeat the above steps I-III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' each with a time interval of 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To reflect the actual fluctuations in the electricity prices, we use the real-time electricity price data obtained from the PJM market [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The dynamic electricity price data profile is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For the setting of EVs, we consider 30 EVs that are divided into three groups with different allowable charging time delays, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', G = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Each group has 10 EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The EVs in the same group have identical allowable time delay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Particularlly, R1 = 8 hours, R2 = 6 hours, and R3 = 7 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In addition, for EV battery parameters, we refer to the Nissan Leaf EV model with a battery pack of 40 kWh and a maximum charging power of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='6 kW [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Considering that the EV charging behavior is uncertain, the EVs’ arriving times are randomly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The initial battery energy level of each EV is selected from a uniform distribution in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5] × 40 kWh randomly [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We set the required state-of-charge (SOC) 1 upon departure as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5 and the maximum SOC upon departure as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The weight parameter value of V is chosen as 1000, and the value of ηg is set as 648, 540, and 756 for the three groups, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Real-time electricity price profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 1The SOC of an EV is the ratio between the battery energy level and the battery capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Effectiveness of the proposed method We first show how the obtained aggregate EV power flexibility region looks like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Since the power grid dispatch determined by the DSO is beyond the scope of this paper, here the dispatch ratio αt in (29) is assumed to be randomly generated within the range of [0, 1] in each time slot, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Based on the generated dispatch ratio, we apply the proposed online flexibility characterization method and real-time feedback in turns (as in Algorithm 2) to obtain the aggregate EV power flexibility region (grey area) over time for each group and the charging station as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' As seen, the power flexibility region varies over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This is because EVs dynamically arrive and leave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Randomly generated dispatch ratio αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The obtained aggregate EV power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To validate the effectiveness of the proposed algorithm, dis- aggregation of the dispatched EV charging power is performed and we check if the SOC curves of EVs satisfy the charging requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Here, if the final EV SOC value can reach or exceed the EV owner’s requirement (SOC ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5) upon leaving, then it means that the proposed method is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 6 shows the actual EV charging SOC curves under the randomly generated dispatch ratio αt in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Each curve represents an EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' As we can see from the figure, all EVs’ final SOC is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='58 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='7, greater than the required value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5 and less than the maximum value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The right-hand side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 6 shows the number of delayed time slots (NDTS) to reach the requirement SOC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We can find that the maximum NDST is 10 for group 1, 7 for group 2, and 12 for group 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' All of them are within their respective declared allowable charging delay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This validates the proposed algorithm in providing maximum power flexibility while meeting the charging requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 7 shows the queue backlog evolution of the three groups over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Taking group 1 for example, the lower and upper bound queues ˇQ1 and ˆQ1 first increase be- cause EVs continue to arrive with their charging tasks pushing JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 8 Rg=8h=48 time slots Rg=6h=36 time slots Rg=7h=42 time slots Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' EV charging SOC of each group and the number of delayed time slots needed to reach SOC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' into the queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then as time moves on, through the EV charging dispatch pdisp v,t , ∀v, ∀t determined by disaggregation, the EV SOC gradually increases and reaches the minimum charging requirement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Hence, the lower bound queue ˇQ1 becomes zero because the ˇag,t, ∀g, ∀t is set using the charging as soon as possible method to meet the minimum charging requirement as in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The upper bound queue ˆQ1 is still larger than zero since the EV SOC has not reach the maximum value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='9 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 6), so there is charging flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' At the same time, since ˆQ1 is nonnegative, the delay aware upper bound queue ˆZ1 keeps growing, aiming to increase the charging power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The queue evolution in groups 2 and 3 can be analyzed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Queue backlog of each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Performance Evaluation To show the advantage of the proposed online algorithm, two widely used benchmarks in the literature are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Benchmark 1 (B1): This is a greedy algorithm that EVs start charging at the maximum charging power upon arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Let us denote the arrival time as t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' When the EV SOC reaches the minimum charging requirement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5, the lower bound of charging power ˇpd,t turns to be zero (time: t1), and the upper bound of charging power ˆpd,t remains the maximum charging power until the EV SOC reaches the maximum value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='9 (time: t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The aggregate power flexibility region for [t0, t1] is empty and for t ∈ [t1, t2] is the region between 0 and the maximum charging power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Benchmark 2 (B2): This is the offline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' It directly solves P1 to obtain the aggregate EV power flexibility regions over the whole time horizon by assuming known future information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Though not realistic, it provides a theoretical benchmark to verify the performance of other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' But it is worth noting that since it does not take into account the real-time actual dispatch strategy when calculating the aggregate flexibility, its performance may be worse than the proposed real-time feedback based method even though it is an offline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 8 shows the accumulated flexibility values (�t τ=1 Fτ) under the three different methods, and TABLE I summarizes the total flexibility value (�T t=1 Ft) under different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The B1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', greedy algorithm, has the worst performance and the lowest total flexibility value due to the myopic strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For B2, since it has complete future knowledge of EV behaviors and real-time electricity prices, it outperforms B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, this method is usually impossible in practice since the accurate future information is hardly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Though predictions on future uncertainty realizations may be obtained, the potential prediction errors limit B2’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In contrast, the proposed online algorithm achieves the best performance with the highest total power flexibility value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This is owing to the fact that it runs in a online manner with real-time feedback that allows it to utilize the most recent dispatch information to update its state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In addition, compared to the offline method B2, it does not require prior knowledge of future information or forecasts, which is more practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Accumulated flexibility value under different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' TABLE I TOTAL FLEXIBILITY VALUE COMPARISON BETWEEN B1, B2, AND THE PROPOSED ALGORITHM (UNIT: USD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Methods B1 B2 Proposed Value 517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='69 586 647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='21 Improvement 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2% 25% The above result is obtained under the random dispatch ratio αt (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In fact, the dispatch ratio can affect the actual charging power of each EV and further affect their aggregate power flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Therefore, it is interesting to investigate the JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 9 impact of αt on the aggregate EV power flexibility (or the total power flexibility value (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Here, we use a uniform α over time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', αt = α, ∀t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We change α from 0 to 1 and record the total power flexibility value in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' As seen, the total power flexibility value depends on the dispatch ratio α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Generally, a larger α leads to a larger power flexibility value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, this increase is nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' When the dispatch ratio α exceeds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5, the total power flexibility value no longer increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In the extreme case when α = 0, the total power flexibility value is 520 USD, which is still larger than the greedy algorithm B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In addition, it can be concluded that if the average dispatch ratio α is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2, then the proposed online algorithm more likely outperforms the offline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This demonstrates the advantage of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We also present the aggregate EV power flexibility region under different α in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' As α decreases, the aggregate EV power flexibility region gradually narrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This is because under a low dispatch ratio, the EVs are charged at a low charging rate and more likely to fail to meet the charging requirement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' hence, the lower bound of aggregate EV power flexibility region is raised to ensure the EVs can meet the charging requirement in the remaining time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The impact of α on total power flexibility value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 0 50 Power [kW] ub lb =0 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='1 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='3 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='4 0 20 40 60 80 100 120 140 Time [10 min] 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5 0 50 Power [kW] ub lb =0 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='1 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='3 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='4 0 20 40 60 80 100 120 140 Time [10 min] 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The aggregate EV power flexibility under different α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Impact of Parameters According to (19), the parameter V controls the trade- off between stabilizing the queues and maximizing the total power flexibility value in the objective function (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Here, we change the value of V to investigate its impact on the total power flexibility value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 11, the total power flexibility value becomes larger with an increasing V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The impact of V on the total power flexibility value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The impact of V on the average time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 12 depicts the impact of V on the number of time slots needed for EVs to meet their required battery energy level ed v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We calculate the maximum/minimum/average number of time slots for the EVs in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' As seen, with the growth of V , the number of delayed time slots slightly increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This is because a larger V means putting more emphasis on maximizing the total power flexibility, which may result in a reduced lower bound ˇxg,tof the aggregate EV power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Consequently, the charging time needed to reach the required energy level becomes longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Comparing the three groups, we can find that the time delays of groups 1 and 3 are generally longer than that of group 2, which is owing to the shorter allowable time delay Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 13 shows the impact of ηg on the total flexibility value and the number of time slots needed for EVs to meet their required battery energy level ed v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We can find that a larger ηg results in a lower total power flexibility value and less number of delayed time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This is because a larger ηg forces the virtual delay-aware queue ˇZg to grow rapidly, allowing EVs to get charged quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Meanwhile, the power flexibility is sacrificed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' CONCLUSION With the proliferation of EVs, it is necessary to better utilize their charging power flexibility, making them valuable resources rather than burdens on the power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This paper proposes a real-time feedback based online aggregate EV power flexibility characterization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' It can output the aggregate flexibility region for each time slot in an online manner, with a total flexibility value over time similar to the offline counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We prove that by choosing an aggregate JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The impact of ηg on the total flexibility value and average time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' dispatch strategy within the obtained flexibility region for each time slot, the corresponding disaggregated EV control strategies allow all EVs to satisfy their charging requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Simulations demonstrate the effectiveness and benefits of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' It is worth noting that the proposed method can even outperform the offline method in some cases since it can utilize up-to-date dispatch information via real-time feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Future research may take into account the conflicting interests between the operator, aggregator, and EVs when deriving the flexibility region.' metadata={'source': 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“Optimal transactive energy trading of electric vehicle charging stations with on-site PV generation in constrained power distribution networks,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Smart Grid, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 1427–1440, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' APPENDIX A PROOF OF PROPOSITION 1 Let {pd,t, ∀t} be the aggregate power trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For each time slot t ∈ T , since pd,t ∈ [ˇp∗ d,t, ˆp∗ d,t], we can define an JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 11 auxiliary coefficient: βt := ˆp∗ d,t − pd,t ˆp∗ d,t − ˇp∗ d,t ∈ [0, 1] (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='1) so that pd,t = βtˇp∗ d,t + (1 − βt)ˆp∗ d,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, we can construct a feasible EV dispatch strategy by letting pc v,t = βtˇpc∗ v,t + (1 − βt)ˆpc∗ v,t, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2a) ev,t = βtˇec∗ v,t + (1 − βt)ˆec∗ v,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2b) for all time slots t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We prove that it is a feasible EV dispatch strategy as follows, pd,t = βtˇp∗ d,t + (1 − βt)ˆp∗ d,t = βt � v∈V ˇpc∗ v,t + (1 − βt) � v∈V ˆpc∗ v,t = � v∈V � βtˇpc∗ v,t + (1 − βt)ˆpc∗ v,t � = � v∈V pc v,t (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='3) Hence, constraint (1c) holds for pd,t and pc v,t, ∀v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Similarly, we can prove that constraints (1d)-(1g) are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Therefore, we have constructed a feasible EV dispatch strategy, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ■ APPENDIX B PROOF OF PROPOSITION 2 Here, we use the contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' If a charging request ˆag,t arrives in time slot t cannot be fulfilled on or before time slot t + ˆδg,max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, queue ˆQg,τ > 0always holds for τ ∈ [t + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', t + ˆδg,max].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Thus, we have I ˆ Qg,t>0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' According to delay virtual queue dynamics (12), for all τ ∈ [t+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', t+ ˆδg,max], we have ˆZg,τ+1 ≥ ˆZg,τ + ηg Rg − ˆxg,τ, ∀g, ∀t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='1) By summing the above inequalities over τ ∈ [t + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', t + ˆδg,max], we have ˆZg,t+ˆδg,max+1 − ˆZg,t+1 ≥ ηg Rg ˆδg,max + t+ˆδg,max � τ=t+1 (−ˆxg,τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2) Since ˆZg,t+ˆδg,max+1 ≤ ˆZg,max and ˆZg,t+1 ≥ 0, we have ˆZg,max ≥ ηg Rg ˆδg,max + t+ˆδg,max � τ=t+1 (−ˆxg,τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='3) Since the charging tasks are processed in a first-in-first-out manner, and the charging request is not fulfilled by t+ˆδg,max, we have t+ˆδg,max � τ=t+1 (ˆxg,τ) < ˆQg,max (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='4) Combining the above two inequalities, we obtain ˆZg,max > ηg Rg ˆδg,max − ˆQg,max, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5) which implies ˆδg,max < ( ˆQg,max + ˆZg,max)Rg ηg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='6) However, this result contradicts the definition of ˆδg,max in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Therefore, the worst case delay should be less than or equal to ˆδg,max as defined in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The proof of (15) follows a similar procedure, and we omit it here for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ■ APPENDIX C PROOF OF PROPOSITION 3 Denote the solution of P3 by the proposed algorithm by ˆxpro g,t and ˇxpro g,t , and the optimal solution of P1 by ˆx∗ g,t and ˇx∗ g,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' According to (26), we have ∆(Θt) + V E[−F pro t |Θt] ≤ A + V E[−F pro t |Θt] + � g∈G ˆQg,tE � ˆag,t − ˆxpro g,t |Θt � + � g∈G ˇQg,tE � ˇag,t − ˇxpro g,t |Θt � + � g∈G ˆZg,tE � −ˆxpro g,t |Θt � + � g∈G ˇZg,tE � −ˇxpro g,t |Θt � , ≤ A + V E[−F ∗ t |Θt] + � g∈G ˆQg,tE � ˆag,t − ˆx∗ g,t|Θt � + � g∈G ˇQg,tE � ˇag,t − ˇx∗ g,t|Θt � + � g∈G ˆZg,tE � −ˆx∗ g,t|Θt � + � g∈G ˇZg,tE � −ˇx∗ g,t|Θt � , ≤ A + V E[−F ∗ t |Θt] (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='1) The result is based on the fact that lim T →∞ 1 T T � t=1 E [ˆag,t − ˆxg,t|Θt] ≤ 0 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2) lim T →∞ 1 T T � t=1 E [ˇag,t − ˇxg,t|Θt] ≤ 0 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='3) lim T →∞ 1 T T � t=1 E [−ˆxg,t|Θt] ≤ 0 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='4) lim T →∞ 1 T T � t=1 E [−ˇxg,t|Θt] ≤ 0 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5) which is due to constraints (9b)-(9e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' By summing the above inequality (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='1) over time slots t ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' , T}, we have T � t=1 V E[−F pro t ] ≤ AT + V T � t=1 E[−F ∗ t ] − E[L(ΘT +1)] + E[L(Θ1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 12 Based on the fact that L(ΘT +1) and L(Θ1) are finite, we divide both sides of the above inequalities by V T and let T → ∞, then we have lim T →∞ 1 T T � t=1 E(−F pro t ) ≤ A V + lim T →∞ 1 T T � t=1 E(−F ∗ t ), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ■' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} diff --git a/F9AyT4oBgHgl3EQfSveK/vector_store/index.faiss 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Kamenshchik,† Alessandro Tronconi,‡ and Giovanni Venturi§ +Dipartimento di Fisica e Astronomia, Università di Bologna, via Irnerio 46, 40126 Bologna, Italy +I.N.F.N., Sezione di Bologna, I.S. FLAG, viale B. Pichat 6/2, 40127 Bologna, Italy +We analyze the consequences of different evolutions of the Hubble parameter on the spectrum of +scalar inflationary perturbations. The analysis is restricted to inflationary phases described by a +transient evolution, when uncommon features arise in the inflationary spectra which may lead to +an amplitude enhancement. We then discuss how the spectrum is, respectively, amplified or blue- +tilted in the presence or absence of a growing solution of the Mukhanov-Sasaki equation. The cases +of general relativity with a minimally coupled inflaton and that of induced gravity are considered +explicitly. Finally, some remarks on constant roll inflation are discussed. +I. +INTRODUCTION +The possibility that a large amount of the Dark Matter (DM) content in our Universe is made of (primordial) +black holes (PBHs) has been seriously considered in the last few years [1]. This idea seems compelling because it +could improve our understanding of cosmological evolution and, in particular, of inflation [2]. Moreover, the PBH +hypothesis is also intriguing due to the increasing amount of direct and indirect observations of black holes (BHs) +out of the astrophysical range, as well as the current lack of evidence for particle models of DM that go beyond the +Standard Model of particle physics. +According to the present observational bounds [3], it is possible that even the whole DM content of the Universe +today is comprised of PBHs originated from the collapse of matter overdensities in a certain wavelength interval +of inflationary perturbations. In this scenario, the abundance of PBHs is related to the amplitude of the inflaton +fluctuations, the enhancement of which must be by several orders of magnitude with respect to (w.r.t.) the amplitude +probed by Cosmic Microwave Background (CMB) radiation. Nonetheless, the microscopic physics that originate such +a mechanism of amplification is still debated. For example, the amplification needed can be generated by a phase of +ultra slow roll (USR) inflation in the presence of an inflection point of the inflaton potential [4]. This USR phase +is the consequence of a transient period of inflatonary evolution, when slow-roll conditions are violated, and the +inflaton then relaxes towards a de Sitter attractor. In contrast to the case of the fluctuations imprinted in the CMB, +the perturbations [5] do not freeze at horizon exit in this case, as a growing solution of the Mukhanov-Sasaki (MS) +equation is present, and it is responsible for the amplification of the modes. Other possibilities have been considered +in the literature, such as an inflationary model able to generate a blue-tilted spectrum without the presence of the +growing solution [6]. +In this article, these two mechanisms of amplification are considered. Instead of analyzing the possible consequences +of different inflationary models obtained by varying the form of the inflaton-gravity action, we shall here consider +different evolutions of the Hubble parameter and correspondingly obtain the inflaton action. Within this approach, +even if the inflaton potential cannot be exactly reconstructed, the features of the resulting spectra can still be +calculated, and one may verify whether their amplitude is amplified. For simplicity, our starting point is the case +of a minimally coupled inflaton, then some non-minimally coupled models are also investigated. Moreover, different +techniques for the reconstruction are adopted. +The article is organised as follows. In Section II, we review the general formalism of the dynamics of the inflationary +perturbations adopting a slightly unconventional formalism, and we derive the conditions for the existence of a growing +solution in the MS equation or simply a blue-tilted spectrum in the absence of this solution. Furthermore, a useful +relation between the odd and even slow-roll (SR) parameters in a certain hierarchy is obtained. This relation is valid +for transient periods described by a certain time evolution, and it will be employed across the entire article. In Section +III, different models are analysed, and the procedure for reconstruction is illustrated. In Section IV, the application +of the formalism to constant roll inflation is studied. Finally, the conclusions are drawn in Section V. +∗ leonardo.chataignier@unibo.it +† kamenshchik@bo.infn.it +‡ tronconi@bo.infn.it +§ giovanni.venturi@bo.infn.it + +2 +II. +INFLATIONARY PERTURBATIONS +Let us first review the formalism of the inflationary perturbations. On adopting a slightly unconventional approach, +we find the conditions which must hold in order to have an amplification of the inflationary spectrum either as the +wavenumber k grows or as time evolves. In a realistic inflationary scenario, wherein amplification starts at some +given time, both mechanisms essentially lead to an enhancement of the shortest wavelength part of the spectrum +(k > kCMB). +The conditions are then expressed in a model-independent form, which is valid provided the SR +parameters are “constant”, and we use it in what follows to discuss different scenarios. +In general, after some manipulations, the Mukhanov-Sasaki (MS) equation takes the following form +v′′ +k + +� +k2 − z′′ +z +� +vk = 0 , +(1) +where the prime denotes the derivative w.r.t. conformal time η and z is a time-dependent function that depends on +the specific model of inflation. For example, in the case of general relativity (GR) with a minimally coupled inflaton, +one has z = a√ǫ1, which leads to (see e.g. [7]) +z′′ +z = a2H2 +� +2 − ǫ1 + ǫ2 +�3 +2 + ǫ2 +4 − ǫ1 +2 + ǫ3 +2 +�� +≡ a2H2fMS(ǫi) , +(2) +with ǫ1 = − ˙H/H2, ǫi+1 = ǫ−1 +i dǫi/dN for i > 0 and N = ln a. The infinite set of ǫi’s form the so-called hierarchy of +“Hubble flow functions” of SR parameters. It is important to note that, depending on the model of inflation, other +hierarchies are commonly used, and they are associated with the evolution of different (homogeneous) degrees of +freedom. +In general, one has +z′′ +z ≡ a2H2fMS , +(3) +where fMS is a dimensionless quantity that takes a different form depending on the inflationary model. It can then +be expressed as a function of the SR parameters ǫi’s. +It is now convenient to define the new independent variable ξ = k/(aH), where dξ/dη = −aH(1 − ǫ1)ξ < 0 during +inflation. Due to +d +dη = −aH(1 − ǫ1)ξ d +dξ +(4) +and +d2 +dη2 = a2H2(1 − ǫ1)2 +� +ξ2 d2 +dξ2 + +ǫ1ǫ2 +(1 − ǫ1)2 ξ d +dξ +� +, +(5) +we are led to +� +ξ2 d2vk +dξ2 + +ǫ1ǫ2 +(1 − ǫ1)2 ξ dvk +dξ +� ++ ξ2 − fMS(ǫi) +(1 − ǫ1)2 +vk = 0 . +(6) +On rewriting the MS equation in terms of ξ one eliminates its explicit dependence on aH. +In the regime where the SR parameters are constant, and in the long wavelength limit (ξ → 0) Eq. (6) can be +algebraically solved and the features of the primordial spectra can be derived in a straightforward manner. Indeed, +in this limit, the two independent solutions of Eq. (6) have the form vk = ξα, where α satisfies the algebraic equation +α2 + +� +ǫ1ǫ2 +(1 − ǫ1)2 − 1 +� +α − fMS(ǫi) +(1 − ǫ1)2 = 0 , +(7) +with +α1,2 = +− +� +ǫ1ǫ2 +(1−ǫ1)2 − 1 +� +± +�� +ǫ1ǫ2 +(1−ǫ1)2 − 1 +�2 ++ 4 fMS(ǫi) +(1−ǫ1)2 +2 +. +(8) + +3 +For instance, when fMS is defined by Eq. (2), and in the pure de Sitter case (ǫi = 0), we obtain +α1,2 = 1 ± 3 +2 +. +(9) +For this case, the positive solution, α1 = 2, decreases in time, while the negative solution, α2 = −1, increases, and it +remains nontrivial in the limit ξ → 0, which leads to +vk,dS ∼ k−1/2 +� k +aH +�−1 +, Rk,dS ∼ k−3/2 aH +z += k−3/2H , +(10) +where Rk ≡ vk/z is the curvature perturbation (z = a in the de Sitter case), and the prefactor k−1/2 is essentially +fixed by the initial (Bunch-Davies) conditions. The quantity Rk is independent of time, and the spectral index can +be straightforwardly computed to be +ns − 1 = d ln ∆2 +s +d ln k +, +(11) +with ∆2 +s ≡ |Rk,dS|2k3/(2π2), which leads to the well known de Sitter result (ns − 1)dS = 0. +In the SR case (|ǫi| ≪ 1), the SR parameters can be approximated by constants and the expressions (8) are still +valid but must be expanded to first order for consistency. One then obtains +α1,2 = 1 ± √9 + 12ǫ1 + 6ǫ2 +2 +≃ 1 ± (3 + 2ǫ1 + ǫ2) +2 +, +(12) +which implies (ns − 1)SR = −2ǫ1 − ǫ2. +We note that there is a caveat one must take into account for USR. In this case, one finds the same solutions for +the α’s as the de Sitter case, but the definition of the curvature perturbations is different, since zUSR ∝ a√ǫ1 → 0. +Then, the amplitude of primordial curvature perturbations depends on time and is amplified. In the USR case, the +spectral index cannot be calculated analytically with the same procedure as illustrated for de Sitter and SR. +One can better illustrate the differences among the three cases just mentioned by solving the equation for Rk, +R′′ +k + 2z′ +z Rk + k2Rk = 0 . +(13) +In terms of ξ, Eq. (13) can be conveniently rewritten as +ξ2 d2Rk +dξ2 ++ +� +ǫ1ǫ2 − 2 (1 − ǫ1) d ln z +dN +(1 − ǫ1)2 +� +ξ dRk +dξ ++ +ξ2 +(1 − ǫ1)2 Rk = 0 . +(14) +In GR with a minimally coupled inflaton, we have d ln z/dN = 1 + ǫ2/2. Then, for constant SR parameters and in +the long wavelength limit, the last term is negligible, and the equation admits a constant solution and a solution +proportional to ξβ, where +β = 3 − 4ǫ1 + ǫ2 + ǫ1 (ǫ1 − 2ǫ2) +(1 − ǫ1)2 +. +(15) +If ξβ decreases in time, the constant solution dominates in the ξ → 0 limit. This is what happens for de Sitter and +SR. In contrast, if ξβ increases in time, it dominates in the ξ → 0 limit. This is what occurs for USR leading to +results that are very different from de Sitter and SR, namely, an amplitude of the spectrum that increases in time. +The non-constant solution is +Rk ∝ +� k +aH +�β +∼ e−β(1−ǫ1) N , +(16) +and it increases or decreases depending on the sign of +Φ ≡ β (1 − ǫ1) = 3 − 4ǫ1 + ǫ2 + ǫ1 (ǫ1 − 2ǫ2) +(1 − ǫ1) +, +(17) + +4 +increasing if Φ < 0 and decreasing if Φ > 0. Only in the latter case the spectral index of primordial spectrum can be +analytically calculated by using the definition (11). For a general inflationary model, one finds +∆2 +s ∝ k2+2α2 = k +2− +� +ǫ1ǫ2 +(1−ǫ1)2 −1 +� +− +�� +ǫ1ǫ2 +(1−ǫ1)2 −1 +�2+4 fMS(ǫi) +(1−ǫ1)2 , +(18) +and +ns − 1 = 2 − +� +ǫ1ǫ2 +(1 − ǫ1)2 − 1 +� +− +�� +ǫ1ǫ2 +(1 − ǫ1)2 − 1 +�2 ++ 4 fMS(ǫi) +(1 − ǫ1)2 . +(19) +A. +Evolutions with “Constant” SR Parameters +Let us now illustrate an important point. +The results obtained above are exact when the SR parameters are +constant. However, given the recursive definition of the SR parameters (ǫi+1 = ǫ−1 +i dǫi/dN), a constant set of ǫi’s +corresponds either to H = const and ǫi = 0 (de Sitter case), or ǫ1 = const and ǫi = 0 for i > 1 (power law inflation). +It may thus seem redundant to present the general formalism for such a restricted range of applications. Nevertheless, +we note that the above results can be applied to a wider set of problems. First, as we already mentioned, the general +results for Φ and ns − 1 can be applied to the SR case, in which the expressions must be expanded to the first order +for consistency, since the SR parameters are approximately constant when they are small. Furthermore, the large a +limit of some transient phase (such as the USR phase) leads to non-trivial sequences of “constant” SR parameters. In +these cases, one obtains a hierarchy of, for example, ǫi’s with constant, non-zero SR parameters for either even or odd +values of i, while the remaining SR parameters are zero. For instance, let ǫi +N→∞ += +li +Li(N) with limN→∞ Li(N) = 0. +Then, due to their recursive definition, one obtains +ǫiǫi+1 ≡ dǫi +dN +a→∞ += +Li,N(N) , +(20) +which leads to limN→∞ ǫi+1 = 0, provided limN→∞ Li,N(N) = 0, and, in particular, +ǫi+1 +N→∞ += +Li,N(N) +li + Li(N) . +(21) +Moreover, +ǫi+2 ≡ dǫi+1/dN +ǫi+1 +N→∞ += +Li,NN(N) +Li,N(N) + ǫi+1 . +(22) +Let us now suppose Li(N) ∝ e−γN ∼ a−γ, with γ > 0. In this case: +ǫi+2 +N→∞ += +−γ + ǫi+1 , +(23) +and the subsequent terms of the hierarchy take values equal to zero and −γ: +lim +N→∞ ǫi = li, +lim +N→∞ ǫi+1+2n = 0, +lim +N→∞ ǫi+2n = −γ . +(24) +Therefore, due to their definition, an infinite sequence of SR parameters may take alternate “constant” values in the +large a limit. This property is crucial in the analysis that follows, and it depends on the form of Li(N). Indeed, +exponential forms lead to the result (24) but, in contrast, if Li ∝ N −γ, then the sequence obtained is limN→∞ ǫj = 0 +for j > i. +It is also worthwhile to mention that similar results can be generalized to other hierarchies of SR parameters because +they only depend on the recursive definition of the SR parameters [analogously to Eq. (20)] and on the form of Li. +For example, the same results can be extended to the hierarchy of “scalar field flow functions” that is defined by +δ0 = φ/φ0 and δiδi+1 = dδi/dN. In general, the ǫi’s and the δi’s are related through the homogeneous Friedmann +and Klein-Gordon equations, and, in some scenarios, it is useful to use one or both hierarchies. + +5 +III. +MODEL RECONSTRUCTION +We are interested in reconstructing scalar field potentials that describe transient inflationary solutions, which are +associated with varying SR parameters with a “constant” behaviour in the future (and necessarily ǫ1 < 1). Therefore, +the results illustrated in the previous section can be adopted to study such models and to verify whether they can +generate an amplification of the primordial spectrum. Finding the entire evolution of the scalar field is not necessary +for this purpose, and we will only calculate the potential and the asymptotic behaviour of the homogeneous quantities +in terms of the corresponding SR parameters. The potentials that lead to an amplification can then be used to build +an inflationary model that fits the CMB observations and which produces a large amount of DM in the form of PBHs +at the end of inflation. +A. +GR with a Minimally Coupled Inflaton +In order to proceed with the reconstruction, let us first briefly review the homogeneous Einstein equation, +H2 = +1 +3MP +2 +�1 +2 +˙φ2 + V (φ) +� +, +(25) +˙H = − +˙φ2 +2MP +2 , +(26) +which leads to +MP +2H2 (3 − ǫ1) = V . +(27) +This last equation can be used to reconstruct the potential. The Eqs. (26) and (27) can be conveniently used for the +reconstructions starting from some ansatz for H = H(a). In this case, Eq. (26) becomes +ǫ1 = +1 +2MP +2 +� dφ +d ln a +�2 +, +(28) +which can be integrated to obtain, when possible, a = a(φ). +Let us first consider the following evolution of the Hubble constant: +H = H0 +� +α + A +an +�m +, +(29) +where A, α, n > 0. Similarly to USR, the evolution described by Eq. (29) has a de Sitter attractor in the future, +and, indeed, H(a) is that of USR when n = 6 and m = 1/2. (It is interesting to note that this evolution represents +a general solution in the model with a minimally coupled scalar field and a constant potential, or, in other words, in +a universe driven by a mixture of two fluids: a cosmological constant and stiff matter. It is curious that n = 6 and +m = 1/4 yield the general solution for the universe driven by the Chaplygin gas [8].) We also note that the transient +is described by A/an ∼ e−nN and that a result similar to Eq. (24) is then expected. This is easily verified if we +explicitly calculate the hierarchy of SR parameters: +ǫ1 = m · n +A +αan + A = m ǫ3 = m ǫ5 = . . . +a→+∞ +−→ 0 , +(30) +and +ǫ2 = −n +αan +αan + A = ǫ4 = ǫ6 = . . . +a→+∞ +−→ −n , +(31) +where a > [(m · n − 1)A/α]1/n is necessary for inflation to occur. We can integrate and invert Eq. (28) to obtain +exp +�φ − φ0 +MP +� n +2m +� += x + 1 +x − 1 +x0 − 1 +x0 + 1 , +(32) + +6 +with x ≡ A−1/2√ +αan + A and x, x0 > 1. Notice that φ = φ0 when x = x0. Conversely, φ = φ∞, with +φ∞ ≡ φ0 + MP +� +2m +n ln B0 , +(33) +for x → ∞. Eq. (32) can be solved for x, which yields +x = e +φ−φ0 +MP +√ n +2m + B0 +e +φ−φ0 +MP +√ n +2m − B0 +, +(34) +where B0 = (x0 − 1)/(x0 + 1), and the reconstructed potential is finally +V = H2 +0 +� αx2 +x2 − 1 +�2m � +3 − n · m +x2 +� +. +(35) +For n = 6 and m = 1/2, one recovers a constant potential and the USR evolution, as expected. For other choices of the +parameters n and m, the expression for the potential in terms of φ is a complicated function with exponentials that +need not be written here explicitly. However, this cumbersome expression is exact. Since the asymptotic behaviour +of the potential at φ ∼ φ∞ determines the limiting values of the SR parameters, we simply give the form of V around +φ∞, which is +V ≃ 3H2 +0α2m +� +1 + n +4 +� +1 − n +2 +� �φ − φ∞ +MP +�2� +. +(36) +Finally, let us calculate the consequences of the background evolution given by Eq; (29) on the inflationary spectrum. +The value of Φ is +Φ = 3 − 4ǫ1 + ǫ2 + ǫ1 (ǫ1 − 2ǫ2) +(1 − ǫ1) +a→+∞ +−→ 3 − n , +(37) +and for n > 3 the curvature perturbations Rk are amplified, after their horizon exit, as time passes. In contrast, if +0 < n < 3, from the constant solution for Rk, one finds +ns − 1 = n > 0 , +(38) +which implies the amplitude is that of a blue-tilted spectrum, which grows as the wavenumber k increases. +We +conclude that for GR with a minimally coupled inflaton, the inflationary evolution described by Eq. (29), with a +transient phase and a de Sitter attractor in the future, leads to an inflationary enhancement. The corresponding +inflaton dynamics is driven by the potential (35) and similar behaviours can be obtained from potentials of the form +(36) with the field close to φ∞. +B. +Power Law solutions +In this section, we generalise the results obtained from Eq. (29), and study the transient phase with a Power Law +inflation attractor. For this case, in contrast to de Sitter, it is only possible to reconstruct the inflaton potential +exactly for particular choices of the parameters. Close to the attractor, an approximate reconstruction can always +be obtained, and that is enough for the purposes of model building. The amplification of the primordial spectrum +can still be studied in full generality, as it depends on the asymptotic values of the SR parameters, which can be +calculated exactly. In this case, and in the large a limit, one obtains +ǫ1 → const + L(a) , +(39) +with L(a) → 0. In analogy to the previous case, we consider +ǫ1 = +� +β + B +an +�m +→ βm , +(40) +with β, B, n > 0 and βm < 1 (so as to have acceleration close to the attractor). Notice that, when β = 0, one finds +a transient phase with a de Sitter attractor, but ǫ1 in Eq. (40) is different from that in the set (30). This case is + +7 +expected to generate a hierarchy of the form (24) in the large a limit. Indeed, the ansatz (40) leads to the following +hierarchy of SR parameters: +ǫ2 = −mǫ4 = −mǫ6 = · · · = − n m B +B + β an → 0 , +(41) +and +ǫ3 = ǫ5 = · · · = − +n βan +B + β an → −n , +(42) +where, in contrast to the de Sitter case examined in the previous section, now the even SR parameters tend to zero. +By proceeding with reconstruction and integrating Eqs. (40) and (28), one finds, respectively, +H = H0 exp +� +− +� +β + B +an +�1+m +(1 + m)nβ +2F1 +� +1, 1 + m, 2 + m, 1 + B +βan +�� +, +(43) +and +φ − φ0 = f(a) − f(a0) , +(44) +where +f(a) = +2 +√ +2MP +� +β + B +an +� 2+m +2 +2F1 +� +1, 1 + m +2 , 2 + m +2 , 1 + +B +βan +� +(2 + m)nβ +. +(45) +In this case, the exact reconstruction of the potential is rather complicated unless one adopts simplifying assumptions. +For example, let m = −1 and 0 < β < 1. Then, +H = +H0 +[n (B + βan)] +1 +nβ , +(46) +and +φ − φ0 = MP +ln +� +1+ +� +ǫ1(a) +β +1− +� +ǫ1(a) +β +1− +� +ǫ1(a0) +β +1+ +� +ǫ1(a0) +β +� +n√β +. +(47) +In the a → ∞ limit, one obtains +φ∞ = φ0 + MP +ln +� +β+1 +β−1A0 +� +n√β +, +(48) +with A0 ≡ +� +1 − +� +ǫ1(a0)/β +� +/ +� +1 + +� +ǫ1(a0)/β +� +. The relation (47) can be inverted to obtain an = an(φ): +an = an +0 +�� +1 − +� +ǫ1(a0) +β +� ++ +� +1 + +� +ǫ1(a0) +β +� +en√β(φ−φ0)/MP +�2 +4en√β(φ−φ0)/MP +, +(49) +and finally the potential can be reconstructed, provided we substitute Eq. (49) into Eq. (27). In terms of an, it then +takes the following form: +V = +H2 +0 +[n (B + βan)] +2 +nβ +� +3 − +� +β + B +an +�m� +. +(50) +The expression in terms of φ is cumbersome and it will not be needed. It is also worthwhile to note that such a +potential depends on the homogeneous inflaton through the exponential function exp +� +n√βφ/MP +� +. This functional +dependence is expected as it is the generalization of the standard Power-Law inflation potential, which only contains +one exponential function of the inflaton. Moreover, various approximate reconstruction methods can be used to obtain + +8 +the shape of the potential close to the attractor but we omit this discussion here. Whereas the exact reconstruction can +be obtained for certain values of the parameters, the behaviour of the resulting inflationary spectra can be calculated +exactly from Eqs. (41) and (42). For generic values of m, one may compute +Φ = β2m − 4βm + 3 +(1 − βm) += 3 − βm > 0 . +(51) +This shows the absence of the growing solution for Eq. (13). The spectral index is then simply given by +ns − 1 = − 2βm +1 − βm < 0 . +(52) +This is the same result as the one obtained for the Power Law attractor solution. In contrast to the de Sitter case, +the resulting primordial spectrum, if evaluated on the trajectory which approaches the attractor (and close to it), +coincides with the spectrum calculated on the attractor itself, and no amplification or peculiar features emerge. It is +also noteworthy that this result is not restricted to the evolution given by Eq. (40), as it only depends on the limits +(41) and (42), which are not partcular to Eq. (40). For instance, starting from the ansatz +H(a) = H0 +�a0 +a +�βm � +1 + A0 +an +�m +, +(53) +where n, A0 > 0, the resulting spectra are the same. +The observed absence of amplification in the cases of Power-Law inflation considered here is relevant because Power- +Law is the exactly solvable inflationary model that is most akin to SR. One may then conjecture that similar results +(and, in particular, the lack of amplification) hold for SR inflation when the inflaton approaches the attractor solution, +and the enhancement is a peculiarity of de Sitter. +C. +Non-Minimally Coupled Inflaton +Let us now consider the different scenario of a non-minimally coupled inflaton. In order to perform the reconstruc- +tion, we first review the basic homogeneous equations for this model: +H2 = +1 +3F(φ) +� ˙φ2 +2 + V − 3HF,φ ˙φ +� +, +(54) +and +˙H = − +1 +2F(φ) +� +(1 + F,φφ) ˙φ2 + F,φ +� +¨φ − H ˙φ +�� +, +(55) +where F(φ) represents a general non-minimal coupling and F = MP reproduces the minimally coupled case. In contrast +to the previous cases, the homogeneous equations and the reconstruction procedure now become more involved. Then, +for simplicity, we shall henceforth limit our study to the induced gravity (IG) case, where F(φ) = ξφ2 [9, 10]. This +simplifying choice is also justified by the fact that both Higgs inflation and Starobinsky inflation (in the Einstein +Frame) occur in a regime that is very close to pure IG. +Reconstructing the inflaton potential for a given H(a) is not as straightforward as for GR with a minimally coupled +inflaton, and we found exact potentials only for certain values of the parameters and for the de Sitter attractor case +[cf. Eq. (29)]. Nevertheless, we can still predict the shape of the inflationary spectra or, at least, the possibility of an +amplification in the large a limit. +D. +De Sitter limit +Let us consider H(a) given by Eq. (29). In IG, the following exact relations hold between some SR parameters: +ǫ1 = +δ1 +1 + δ1 +� δ1 +2ξ + 2δ1 + δ2 − 1 +� +, +(56) + +9 +ǫ1 = +1 +2ξ(1 + 6ξ) +� +(1 + 2ξ)δ2 +1 − 8ξδ1 − 6ξ2 +� +1 + 2δ1 − δ2 +1 +6ξ +� �d ln V +d ln φ − 4 +�� +. +(57) +Before discussing the reconstruction of the inflaton potential, we must first calculate the asymptotic values of the SR +parameters, which are pivotal in the analysis of the amplification of the spectrum. From Eq. (54), the potential can +then be obtained as +V = 3ξφ2H2 +� +1 + 2δ1 − 1 +6ξ δ2 +1 +� +, +(58) +provided H = H(φ) and δ1 = δ1(φ) are known [indeed, Eq. (58) is the IG counterpart of Eq. (27) in GR]. +Let us first calculate the SR parameters in the large a limit. Since lima→∞ ǫ1 = 0, one either has lima→∞ δ1 = 0 +and lima→∞ δ2 ̸= 0, or lima→∞ δ2 = 0 and lima→∞ δ1 ̸= 0 but satisfying the relation +δ1,∞ = +2ξ +1 + 4ξ . +(59) +These results follow from the functional dependence of H(a) on a inherited by ǫ1 and δi’s and by the general result +given in Eq. (24), which is applied here to the SR hierarchy δi. Notice that, in contrast to GR, two different de Sitter +trajectories are present in IG, and they are associated with two different evolutions of the inflaton field. Using Eq. +(57) in the same limit for a, one obtains that the potential, on the attractor, must satisfy +d ln V∞ +d ln φ − 4 = 0 ⇒ V∞ ∝ φ4 +(60) +in the former case and +d ln V∞ +d ln φ − 4 = 0 ⇒ V∞ ∝ φ2 +(61) +in the latter case. +We can now proceed to evaluate the full hierarchy of δi’s. Starting from Eq. (56) and differentiating, we find +ǫ2 = δ2 +� +(1 + 4ξ)δ2 +1 + 2ξ (δ2 + δ3 − 1) + 2δ1 (1 + 4ξ + ξδ3) +� +(1 + δ1) [(1 + 4ξ)δ1 + 2ξ(δ2 − 1)] +, +(62) +and, by further differentiating, the ǫi’s with arbitrary large i can be obtained. In the large-a limit, we have already +calculated ǫ2i = −n and ǫ2i+1 = 0 [cf. Eqs. (30) and (31)], and one then obtains two possible hierarchies for the δi’s: +δ2i+1,∞ = 0, δ2i,∞ = ǫ2,∞ = −n , +(63) +and +δ1,∞ = +2ξ +1 + 4ξ , δ2i+1,∞ = −n, δ2i,∞ = 0 . +(64) +This latter statement cannot be simply verified by substitution because the limits involved do not commute. For +example, on substituting first δ2 = 0 in Eq. (62), one obtains ǫ2 = 0, which is not correct. One must solve (at least +perturbatively in the large-a limit) Eq. (56) and then evaluate the limits with the help of the solution found. The +above results are correctly reproduced only if we proceed in this manner. +The exact reconstruction of the inflaton potential is not possible in general. Nonetheless, in specific cases, the +potential may be derived exactly as follows. Consider the following ansatz for δ1: +δ1 = n0 + n1a−n +d0 + d1a−n , +(65) +which is suggested by the expression for ǫ1 and Eq. (56). If n0 = 0 and n1 ̸= 0, then Eq. (65) can be integrated, and +the resulting φ(a) is inverted as follows +φ(a) = φ0 +� +d0 + d1a−n�− n1 +n d1 ⇒ a−n = +� +φ(a) +φ0 +�− n d1 +n1 − d0 +d1 +. +(66) + +10 +The coefficients n1, d0, d1 and ξ can finally be fixed by the requirement that Eq. (65) be a solution of Eq. (56). Two +nontrivial solutions can be found: +d0 = −(1 + n) n1α +A m n +, d1 = −(1 + n)n1 +m n +, ξ = +m +2(1 − 3m + n + m n) , +(67) +or +d0 = −(1 + n) n1α +A m n +, d1 = −(1 + n + m n)n1 +m n +, ξ = +m +2(1 − 2m + n + m n) . +(68) +Notice that more exact solutions for δ1 can be found if we start from the ansatz (65) and n0 ̸= 0. However, by further +integrating these solutions to obtain φ(a), one is led to non-invertible functions, and the reconstruction cannot be +completed. For both Eqs. (67) and (68) one has +δ1,∞ = 0 , +(69) +and one can explicitly verify that the hierarchies belong to the set (63). Notice that n1 in Eqs. (67) and (68) can be +arbitrarily chosen, as should be due to the form of the ansatz (65). Let us, for simplicity, complete the reconstruction +choosing n and m to reproduce USR in the IG context (n = 6, m = 1/2). In this case, Eqs. (67) and (68) take the +following form: +n0 = 0, d0 = − 7α +3An1, d1 = −7 +3n1, ξ = 1/10 ⇒ δ1 = − +3A +7 (A + αa6) , +(70) +n0 = 0, d0 = − 7α +3An1, d1 = −10 +3 n1, ξ = 1/36 ⇒ δ1 = − +3A +10A + 7αa6 . +(71) +From Eq. (29), a(φ) in (66) and Eq. (70), one finds +δ1 = 3 +7 +� +α +�φ0 +φ +�14 +− 1 +� +and +H2 = H2 +0 +� φ +φ0 +�14 +, +(72) +with φ/φ0 +a→∞ +−→ α1/14 and φ > φ0, while for Eq. (71) one finds +δ1 = 3 +10 +� +α +�φ0 +φ +�20 +− 1 +� +and +H2 = H2 +0 +�� 7 +10 +φ +φ0 +�20 ++ 3α +10 +� +, +(73) +with φ/φ0 +a→∞ +−→ α1/20 and φ > φ0. Finally, by using Eq. (58), one obtains +V = − +3H2 +0 +490φ12φ14 +0 +� +8φ28 − 72αφ14 +0 φ14 + 15α2φ28 +0 +� +(74) +for the first exact solution, and +V = − +H2 +0 +6000φ38φ20 +0 +� +7φ20 + 3αφ20 +0 +� � +7φ40 − 84αφ20 +0 φ20 + 27α2φ40 +0 +� +(75) +for the second. In the a → ∞ limit, the potentials (74) and (75) satisfy the condition d ln V/d ln φ = 4. The potential +can have negative values but, in the vicinity of φ ≃ φ∞, the potential is positive and V∞ > 0. +We discuss at last the behaviour of the primordial scalar curvature spectrum. The general formulae illustrated in +the Sec. II can be easily generalised to the IG case wherein +zIG = aφδ1 +� +1 + 6ξ +1 + δ1 +, +(76) +and Φ is given by +Φ = +� +1 − ǫ1 − +ǫ1ǫ2 +(1 − ǫ1) + +� +2 + 2δ1 + 2δ2 − δ1δ2 +1 + δ1 +�� +. +(77) + +11 +If we evaluate Φ w.r.t. to the hierarchies (63) and (64), one observes that only constants and terms linear in the SR +parameters remain. Moreover ǫ1,∞ = 0 and Φ then simplifies to +Φ = 3 + 2δ1 + 2δ2 , +(78) +which can be negative only for the hierarchy (63) but is strictly positive for the hierarchy (64), provided we restrict +ourselves to positive values of the non-minimal coupling ξ. In the former case, Φ = 3 − 2n, which implies that the +growing solution exists for n > 3/2. +If no growing solution exists [as is the case for (64) or (63) with n < 3/2], an amplification of the spectrum is only +possible if the spectrum is blue-tilted. Let us then evaluate ns − 1. In the IG case, fMS(ǫi) in the MS equation is +given by +fMS = δ2 +1 + δ2 +2 + (3 − ǫ1) (1 + δ1 + δ2) + δ2δ3 + +δ1δ2 +� +ǫ1 + δ1 − 3δ2 − δ3 + 2δ1δ2 +1+δ1 − 2 +� +1 + δ1 +− 1 , +(79) +and, as usual, it can be simplified to obtain the following expression for the scalar spectral index: +ns − 1 = 3 − +� +1 + 4 (δ2 +1 + δ2 +2 + 3 (1 + δ1 + δ2) − 1) . +(80) +Then, for the hierarchy (63) and n < 3/2, we obtain +ns − 1 = 3 − |3 − 2n| = 2n , +(81) +which is indeed blue-tilted, while for the hierarchy (64), we find +ns − 1 = − +4ξ +1 + 4ξ , +(82) +which is red-tilted. +We therefore conclude that solutions having H of the form given in Eq. (29), in the IG context, may lead to a +spectrum enhancement for evolutions asymptotically described by the hierarchy (63) and either for n > 3/2 (due to +the presence of the growing solution) or 0 < n < 3/2 (in absence of the growing solution but with the blue-tilted +spectrum). +IV. +APPLICATIONS +We have so far studied the consequences of cosmological evolutions with a transient phase, which is crucial to +potentially obtain the amplification required by the formation of PBHs. Indeed, the presence of the transient generates, +in the large-a limit, a sequence of values for the SR parameters that is otherwise not obtained. We then reconstructed, +when possible, the potentials that led to the desired evolution. In this section, our approach will be slightly different, +as we shall study the presence of the transient solutions in the particular dynamical regime of constant roll (CR) +inflation [11], which is the natural generalisation of USR. +CR solutions satisfy the equation +¨φ + BH ˙φ = 0 , +(83) +where B > 0, and one recovers the USR solution for B = 3, while the case of |B| ≪ 1 reproduces standard SR. We +observe that the CR condition (83) can be rewritten, in terms of the SR parameters, as +δ2 + δ1 − ǫ1 + B = 0 . +(84) +Eq. (84) is model independent, since it only depends on the definitions of ǫi’s and δi’s, and can be easily integrated +to obtain +dφ +d ln aH +� a +a0 +�B += C3 ⇒ ˙φ = C3 +�a0 +a +�B +, +(85) +where C3 is an integration constant. +In the minimally coupled case, CR can generate an amplification of the primordial scalar spectrum. In what follows, +after a revision of this result (which was analysed in [7]) we shall consider CR in IG and study its consequences. + +12 +A. +Constant Roll in GR with a Minimally Coupled Inflaton +In GR with a minimally coupled inflaton, on imposing CR conditions and adopting the Hamilton-Jacobi (HJ) +formalism, it is possible to reconstruct the evolution of the Hubble parameter and the corresponding potential [7]. In +particular, one finds that H(φ) is the following superposition of two exponential functions: +H(φ) = C1 exp +�� +B +2 +φ +MP +� ++ C2 exp +� +− +� +B +2 +φ +MP +� +. +(86) +In [7], the solution (86) with one exponential (C1 = 0 or C2 = 0), as well as the cosh and sinh cases, are analysed +with the aim of finding the exact solutions compatible with CMB observations [12] (and thus not amplified). +Here, in a slightly different approach, we consider the general case, and we study the power enhancement of the +spectrum. Eq. (26) can be rewritten in terms of the SR parameters +ǫ1 = +φ2 +2MP +2 δ2 +1 , +(87) +from which, using the chain rule, we obtain +ǫ1 = −δ1 +d ln H +d ln φ . +(88) +Eq. (87) then becomes +ǫ1 = 2MP +2 +φ2 +�d ln H +d ln φ +� +. +(89) +The potential can subsequently be reconstructed by substituting Eqs. (86) and (89) into Eq. (27): +V (φ) = MP +2H(φ)2 +� +3 − 2MP +2 +φ2 +�d ln H +d ln φ +�2� +. +(90) +In order to obtain the corresponding evolution, one must integrate and invert the equation +δ1 = −2MP +2 +φ2 +d ln H +d ln φ , +(91) +which can be easily derived from Eq. (87) by using (88). One finds +� a +a0 +�B += +x +B (C2 − C1x2) , +(92) +where x = exp +�� +B +2 +φ +MP +� +. It is straightforward to invert Eq. (92) so as to obtain x = x(a). Correspondingly, one has +H(a) = ± +4C1C2 + +� a0 +a +�2B ∓ +� a0 +a +�B � +4C1C2 + +� a0 +a +�2B +∓ +� a0 +a +�B + +� +4C1C2 + +� a0 +a +�2B +a→∞ +−→ ±8C1C2 + +� a0 +a +�2B +4√C1C2 +. +(93) +Notice that the same result can be obtained if one uses the CR definition (84) instead of Eq. (26). +The last, approximate, equality in Eq. (93) is the large-a limit of H(a), and this shows that the CR evolution is +asymptotically equivalent to the evolution given in Eq. (29) with m = 1 and n = 2B. The results obtained in the +Sec. II for large a are therefore inherited by CR. Thus, one obtains ǫ2i+1,∞ = 0 and ǫ2i,∞ = 2B. Correspondingly, +Φ = 3 − 2B , +(94) +which shows that the curvature perturbations are amplified for B > 3/2 due to the presence of a growing solution. In +contrast, if 0 < B < 3/2, one finds a blue-tilted spectrum +ns − 1 = 3 − +� +(3 − 2B)2 = 2B > 0 , +(95) +i.e., a spectrum enhancement in the absence of growing solutions. Therefore, CR inflation admits transient solutions +that always lead to an amplification. Finally, it is worthwhile to mention that the solutions with C1 = 0 or C2 = 0 +simply correspond to the attractor solutions for power-law inflation, and thus they are not associated with any +amplification effect. + +13 +B. +Constant Roll with a Non-Minimally Coupled Inflaton +Let us now consider CR in the IG context. For this case, the HJ formalism leads to [13] +H(φ) = C1φ(B+p)/2 + +C2 +φ(p−B)/2 , +(96) +where p = +� +(B + 2)2 + 2B(2 + ξ−1), and (B + p)/2 and (p − B)/2 are both positive with (p − B)/2 < (B + p)/2. +For simplicity, we shall take C1,2 > 0 and we restrict the analysis to the φ > 0 interval. Studying the spectrum +enhancement for CR in the IG case is more complicated than for GR. This is essentially a consequence of the +complicated form of Eq. (56) in comparison to Eq. (26) in the GR case. However, the simple relation (84) holds, and +it can be used to simplify the equations. First, with Eq. (84), one may eliminate δ2 from Eq. (56) and obtain +ǫ1 = 1 + 2ξ +2ξ +δ2 +1 − (B + 1)δ1 . +(97) +Subsequently, by using Eq. (88), one finds +δ1 = +2ξ +1 + 2ξ +� +B + 1 − d ln H +d ln φ +� +, +(98) +and the potential can be reconstructed by substituting Eqs. (96) and (98) into Eq. (58). +The evolution could be obtained by integrating Eq. (98) and inverting the result. However, analytically inverting +the resulting equation for arbitrary values of the parameters is impossible. As we are only interested in the asymptotic +form of H(a), one can employ a perturbative approach. Integration of Eq. (98) yields +�a0 +a +�B += φ +2+B +2 +� +(B + p + 2) C1φ +p +2 + (B − p + 2) C2 +φ +p +2 +� +, +(99) +where B + p + 2 > 0 and B − p + 2 < 0. Therefore, in the large-a limit, the inversion of Eq. (99) leads to +φ(a) = φ∞ + +� +i>1 +φi +�a0 +a +�i B +∼ φ∞ + φ1 +�a0 +a +�B +, +(100) +where φ∞ is positive. By substituting Eq. (100) into Eq. (96) and expanding for large a (properly accounting for the +next-to-leading-order contributions), one finally obtains the asymptotic form of H(a), which reads +H ∼ H∞ + H1 +�a0 +a +�B +. +(101) +Comparison to Eq. (29) shows that m = 1 and n = B, and the corresponding hierarchy of δi’s is given by Eq.(63) +since +δ1,∞ = lima→∞ ˙φ +H∞φ∞ += 0 , +(102) +where ˙φ is given by (85). One then obtains +Φ = 3 − 2B, ns − 1 = 2B , +(103) +and, when 0 < B < 3/2, +ns − 1 = 2B , +(104) +which are the same results as GR with a minimally coupled inflaton. Indeed, in the a → ∞ limit, the homogeneous +inflaton is frozen at a certain value and one essentially recovers the evolution of the minimally coupled case, where +“Newton’s constant” is now reproduced by the (constant) asymptotic value of the inflaton. Furthermore the a depen- +dence of the solution is a consequence of the fact that the CR condition (84) is independent of the specific inflationary +model, provided H∞ and φ∞ are found to be (finite) constants. + +14 +C. +Jordan and Einstein frame mapping +In the previous section, we found the same asymptotic behaviour for the spectra in the minimally coupled case and +in the IG case. This result was obtained in spite of the fact that CR condition is not frame invariant; i.e., the CR +condition in the Einstein Frame (EF) is not mapped, in general, into a CR condition in the Jordan Frame (JF). In +this section, we briefly review this statement and discuss its consequences. +It is well known that, by a suitable conformal transformation and a redefinition of the scalar field (inflaton), one +can map a minimally coupled theory (defined in the so-called EF) into a non-minimally coupled one (in the JF). In +partcular, for IG, the mapping is given by the following transformation rules (see e.g. [14]): +a(t) = MP +√ξσ ˜a(t), N(t) = MP +√ξσ +˜ +N(t) , +(105) +and +φ = MP +� +1 + 6ξ +ξ +ln σ +σ0 +, ˜V (φ(σ)) = MP +2 +ξ2σ4 V (σ) , +(106) +where the tilde refers to the Einstein frame, φ is the scalar field in the EF, σ is that in the JF. The mapping induces +the following transformations of the Hubble parameter: +˜H = d˜a/dt +˜N˜a += (1 + δ1) MP +√ξσ H , +(107) +where +H(t) = da(t)/dt +a(t)N(t), ǫi+1 = +dǫi/dt +ǫiN(t)H(t), δi+1 = +dδi/dt +δiN(t)H(t) +(108) +are the Hubble and SR parameters in the JF. From the relation (105), one also finds that +d +d ln ˜a = (1 + δ1)−1 +d +d ln a . +(109) +It is now straightforward to obtain the relations between SR parameters in the two frames: +˜ǫ1 ≡ −d ln ˜H +d ln ˜a = − (1 + δ1)−1 +d +d ln a ln +� +(1 + δ1) MP +√ξσ H +� += +δ1 + ǫ1 − δ1δ2 +1+δ1 +1 + δ1 +. +(110) +Given the relation (56), one then finds +˜ǫ1 = (1 + 6ξ)δ2 +1 +2ξ(1 + δ1)2 . +(111) +From Eqs. +(56) and (111), given that CR for a minimally coupled inflaton has ˜ǫ1,∞ = 0, one concludes that, +correspondingly, in the JF one has δ1,∞ = 0 and ǫ1,∞ = 0. If we differentiate Eq. (111), we obtain the following +relations among other SR parameters in the two frames +˜ǫ2 = +2δ2 +(1 + δ1)2 , +(112) +˜ǫ3 = δ3 − 2δ1δ2 + δ1δ3 +(1 + δ1)2 +, +(113) +˜ǫ4 = +δ1δ2 +� +2δ2 − 2δ1δ2 + 3δ3 + 3δ1δ3 − (1 + δ1)2 δ3δ4 +� +(1 + δ1)2 (2δ1δ2 − δ1δ3 − δ3) +, +(114) +and further ˜ǫi’s can be found by iterating the procedure but are useless for what follows. + +15 +Similarly, one can directly calculate the relations of the ˜δi’s with the dynamical variables in the JF: +˜δ1 ≡ +˙φ +˜N ˜Hφ += +� +1 + 6ξ +ξ +MP +φ +δ1 +1 + δ1 +, +(115) +and +˜δ2 ≡ d˜δ1/dt +˜N ˜H˜δ1 += −˜δ1 + +δ2 +(1 + δ1)2 . +(116) +From the last relation and Eq. (56), one has that the CR condition in the EF, +˜δ2 + ˜δ1 − ˜ǫ1 + B = 0 , +(117) +is mapped into the following condition in the JF +δ2 + (B − 1) δ1 − ǫ1 + B = 0 . +(118) +Notice that only for B = 2 the CR condition is frame invariant. Nonetheless, both equations reduce to δ2,∞ = ˜δ2,∞ = +−B at late times, and the evolution is indistinguishable, at least as far as the homogeneous degrees of freedom and +the inflationary spectra are concerned. +We conclude that, whereas the scalar spectral index ns − 1 is frame invariant, Φ is generally not frame invariant. +This can be checked directly by substitution. However, assuming CR holds in the EF, one verifies that Φ and ns − 1 +are both frame invariant in the asymptotic regime. +TABLE I. Results summary +Inflation Asymptotic Growing +Blue-Tilted +Model +Svolution +Solution Spectral Index +GR +dS +n > 3 +0 < n < 3 +GR +PL +− +− +IG +dS, δ1,∞ = 0 +n > 3/2 +0 < n < 3/2 +IG +dS, δ1,∞ ̸= 0 +- +- +CR+GR +dS +B > 3/2 +0 < B < 3/2 +CR+IG +dS, δ1,∞ = 0 +B > 3/2 +0 < B < 3/2 +V. +CONCLUSIONS +In this article, we have analyzed the effects of different transient phases, which may occur during inflation due to a +particularity of the inflaton potential, on the primordial inflationary spectrum of scalar perturbations. These transients +have been studied in the last few years as sources of amplification of the amplitude of the curvature spectrum. It +is important to notice that if the amplitude of scalar perturbations grows large enough, it may induce gravitational +collapse and consequently seed the formation of primordial black holes after inflation ends. In the literature, several +mechanisms for such an amplification during inflation have been proposed. In particular, the presence of an ultra +slow-roll or, more generally, a constant-roll phase has been studied. Whereas in the former case the amplification is +due to the existence of a growing solution to the equation of motion of the curvature perturbations, in the latter case +the amplification can also be generated by a blue-tilted spectrum in absence of the growing solution. +The purpose of this paper was precisely to examine general features of the aforementioned models starting from +a rather generic ansatz for the Hubble parameter as a function of the scale factor. +This general description of +the transient phase is model independent, and many results obtained can be readily applied to several modified +gravity models. The matter-gravity dynamics is described in terms of the hierarchies of SR parameters, both at the +homogeneous level and at the level of perturbations. These hierarchies, when the transient phase that describes the +approach to some inflationary attractor is considered, have been shown to take a peculiar form wherein either odd or +even terms of the hierarchy are null and the remaining ones are different for zero. This general feature is a peculiarity +of the asymptotic form of the SR parameters close to the attractor, and it is then used as a simplifying assumption + +16 +throughout the entire article. The resulting hierarchies, in the large-a limit and for the cases considered, were used +to calculate the behavior of the primordial curvature spectrum as the parametrisation of H(a) was varied. Then, +when possible, the corresponding inflaton potential was fully reconstructed. An overview of the spectra enhancement +results was presented in Table (I). +For simplicity, only the induced gravity case has been considered here as a generalisation of general relativity with +a minimally coupled inflaton. Induced gravity is particularly relevant since both Higgs and Starobinsky inflationary +models (which are in good agreement with observations) take place in the ‘induced gravity phase’. We note that while +transient evolutions that have the de Sitter universe as a limit (such as USR) can lead to an amplification, the results +differ when power-law inflation is considered as the limit of a transitory dynamics and, for the cases we were able to +solve explicitly, no modification of the scalar spectrum was obtained. Finally, the constant-roll case was discussed in +more detail as an application of the preceding results, and the issue of the transition from the Einstein frame to the +Jordan frame was also scrutinized. +[1] G. F. Chapline, Nature 253 (1975) no.5489, 251-252 doi:10.1038/253251a0; P. Meszaros, Astron. Astrophys. 38 (1975), +5-13 +[2] A.A. Starobinsky. Springer. in H.J. De Vega and N. Sanchez (eds.) Current trends in field theory quantum gravity and +strings, Lecture Notes in Physics 246 Verlag, Heidelberg, 1986), pp. 107-126. A.D. Linde. Academic. Particle Physics and +Inflationary Cosmology (Harwood New York, 1990). +[3] B. Carr and F. Kuhnel, Ann. Rev. Nucl. Part. Sci. 70 (2020), 355-394 doi:10.1146/annurev-nucl-050520-125911 +[4] J. Garcia-Bellido and E. Ruiz Morales, Phys. Dark Univ. 18 (2017) 47 doi:10.1016/j.dark.2017.09.007 [arXiv:1702.03901 +[astro-ph.CO]]. G. Ballesteros and M. Taoso, Phys. Rev. D 97 (2018) no.2, 023501 doi:10.1103/PhysRevD.97.023501 +[arXiv:1709.05565 [hep-ph]]. M. Cicoli, V. A. Diaz and F. G. Pedro, JCAP 1806 (2018) no.06, 034 doi:10.1088/1475- +7516/2018/06/034 [arXiv:1803.02837 [hep-th]]. H. Motohashi and W. Hu, Phys. Rev. D 96 (2017) no.6, 063503 +doi:10.1103/PhysRevD.96.063503 [arXiv:1706.06784 [astro-ph.CO]]. C. Germani and T. Prokopec, Phys. Dark Univ. 18 +(2017) 6 doi:10.1016/j.dark.2017.09.001 [arXiv:1706.04226 [astro-ph.CO]]. +[5] V.F. Mukhanov, Sov. Phys. JETP 68, 1297 (1988); J. M. Maldacena, JHEP 0305 (2003) 013; V. F. Mukhanov, +H. A. Feldman and R. H. Brandenberger, Phys. Rept. 215 (1992) 203. V.F. Mukhanov, Phys. Lett. B 218, 17 (1989); +J. M. Bardeen, Phys. Rev. D 22, 1882 (1980). doi:10.1103/PhysRevD.22.1882; M. Sasaki, Prog. Theor. Phys. 70 (1983) +394. doi:10.1143/PTP.70.394 +[6] A. Y. Kamenshchik, A. Tronconi and G. Venturi, JCAP 01 (2022) no.01, 051 doi:10.1088/1475-7516/2022/01/051 +[arXiv:2110.08112 [gr-qc]] A. Y. Kamenshchik, A. Tronconi, T. Vardanyan and G. Venturi, Phys. Lett. B 791 (2019), +201-205 doi:10.1016/j.physletb.2019.02.036 [arXiv:1812.02547 [gr-qc]] H. Motohashi, S. Mukohyama and M. Oliosi, JCAP +03 (2020), 002 doi:10.1088/1475-7516/2020/03/002 [arXiv:1910.13235 [gr-qc]] +[7] H. Motohashi, +A. A. Starobinsky and J. Yokoyama, +JCAP 09 (2015), +018 doi:10.1088/1475-7516/2015/09/018 +[arXiv:1411.5021 [astro-ph.CO]] +[8] A.Yu. Kamenshchik, U. Moschella and V. Pasquier, Phys. Lett. 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Grav. 31 (2014), 105003 +doi:10.1088/0264-9381/31/10/105003 [arXiv:1312.3540 [hep-th]] + diff --git a/JNE3T4oBgHgl3EQfXQpr/content/tmp_files/load_file.txt b/JNE3T4oBgHgl3EQfXQpr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e9f0a8f3335b4546b1079937fe7480519f3be9f8 --- /dev/null +++ b/JNE3T4oBgHgl3EQfXQpr/content/tmp_files/load_file.txt @@ -0,0 +1,636 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf,len=635 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='04477v1 [gr-qc] 11 Jan 2023 Reconstruction Methods and the Amplification of the Inflationary Spectrum Leonardo Chataignier,∗ Alexander Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Kamenshchik,† Alessandro Tronconi,‡ and Giovanni Venturi§ Dipartimento di Fisica e Astronomia, Università di Bologna, via Irnerio 46, 40126 Bologna, Italy I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=', Sezione di Bologna, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' FLAG, viale B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Pichat 6/2, 40127 Bologna, Italy We analyze the consequences of different evolutions of the Hubble parameter on the spectrum of scalar inflationary perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The analysis is restricted to inflationary phases described by a transient evolution, when uncommon features arise in the inflationary spectra which may lead to an amplitude enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' We then discuss how the spectrum is, respectively, amplified or blue- tilted in the presence or absence of a growing solution of the Mukhanov-Sasaki equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The cases of general relativity with a minimally coupled inflaton and that of induced gravity are considered explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Finally, some remarks on constant roll inflation are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' INTRODUCTION The possibility that a large amount of the Dark Matter (DM) content in our Universe is made of (primordial) black holes (PBHs) has been seriously considered in the last few years [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This idea seems compelling because it could improve our understanding of cosmological evolution and, in particular, of inflation [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Moreover, the PBH hypothesis is also intriguing due to the increasing amount of direct and indirect observations of black holes (BHs) out of the astrophysical range, as well as the current lack of evidence for particle models of DM that go beyond the Standard Model of particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' According to the present observational bounds [3], it is possible that even the whole DM content of the Universe today is comprised of PBHs originated from the collapse of matter overdensities in a certain wavelength interval of inflationary perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In this scenario, the abundance of PBHs is related to the amplitude of the inflaton fluctuations, the enhancement of which must be by several orders of magnitude with respect to (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=') the amplitude probed by Cosmic Microwave Background (CMB) radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Nonetheless, the microscopic physics that originate such a mechanism of amplification is still debated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For example, the amplification needed can be generated by a phase of ultra slow roll (USR) inflation in the presence of an inflection point of the inflaton potential [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This USR phase is the consequence of a transient period of inflatonary evolution, when slow-roll conditions are violated, and the inflaton then relaxes towards a de Sitter attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In contrast to the case of the fluctuations imprinted in the CMB, the perturbations [5] do not freeze at horizon exit in this case, as a growing solution of the Mukhanov-Sasaki (MS) equation is present, and it is responsible for the amplification of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Other possibilities have been considered in the literature, such as an inflationary model able to generate a blue-tilted spectrum without the presence of the growing solution [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In this article, these two mechanisms of amplification are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Instead of analyzing the possible consequences of different inflationary models obtained by varying the form of the inflaton-gravity action, we shall here consider different evolutions of the Hubble parameter and correspondingly obtain the inflaton action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Within this approach, even if the inflaton potential cannot be exactly reconstructed, the features of the resulting spectra can still be calculated, and one may verify whether their amplitude is amplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For simplicity, our starting point is the case of a minimally coupled inflaton, then some non-minimally coupled models are also investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Moreover, different techniques for the reconstruction are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The article is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In Section II, we review the general formalism of the dynamics of the inflationary perturbations adopting a slightly unconventional formalism, and we derive the conditions for the existence of a growing solution in the MS equation or simply a blue-tilted spectrum in the absence of this solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Furthermore, a useful relation between the odd and even slow-roll (SR) parameters in a certain hierarchy is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This relation is valid for transient periods described by a certain time evolution, and it will be employed across the entire article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In Section III, different models are analysed, and the procedure for reconstruction is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In Section IV, the application of the formalism to constant roll inflation is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Finally, the conclusions are drawn in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' ∗ leonardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='chataignier@unibo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='it † kamenshchik@bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='it ‡ tronconi@bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='it § giovanni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='venturi@bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='it 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' INFLATIONARY PERTURBATIONS Let us first review the formalism of the inflationary perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' On adopting a slightly unconventional approach, we find the conditions which must hold in order to have an amplification of the inflationary spectrum either as the wavenumber k grows or as time evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In a realistic inflationary scenario, wherein amplification starts at some given time, both mechanisms essentially lead to an enhancement of the shortest wavelength part of the spectrum (k > kCMB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The conditions are then expressed in a model-independent form, which is valid provided the SR parameters are “constant”, and we use it in what follows to discuss different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In general, after some manipulations, the Mukhanov-Sasaki (MS) equation takes the following form v′′ k + � k2 − z′′ z � vk = 0 , (1) where the prime denotes the derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' conformal time η and z is a time-dependent function that depends on the specific model of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For example, in the case of general relativity (GR) with a minimally coupled inflaton, one has z = a√ǫ1, which leads to (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' [7]) z′′ z = a2H2 � 2 − ǫ1 + ǫ2 �3 2 + ǫ2 4 − ǫ1 2 + ǫ3 2 �� ≡ a2H2fMS(ǫi) , (2) with ǫ1 = − ˙H/H2, ǫi+1 = ǫ−1 i dǫi/dN for i > 0 and N = ln a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The infinite set of ǫi’s form the so-called hierarchy of “Hubble flow functions” of SR parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' It is important to note that, depending on the model of inflation, other hierarchies are commonly used, and they are associated with the evolution of different (homogeneous) degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In general, one has z′′ z ≡ a2H2fMS , (3) where fMS is a dimensionless quantity that takes a different form depending on the inflationary model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' It can then be expressed as a function of the SR parameters ǫi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' It is now convenient to define the new independent variable ξ = k/(aH), where dξ/dη = −aH(1 − ǫ1)ξ < 0 during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Due to d dη = −aH(1 − ǫ1)ξ d dξ (4) and d2 dη2 = a2H2(1 − ǫ1)2 � ξ2 d2 dξ2 + ǫ1ǫ2 (1 − ǫ1)2 ξ d dξ � , (5) we are led to � ξ2 d2vk dξ2 + ǫ1ǫ2 (1 − ǫ1)2 ξ dvk dξ � + ξ2 − fMS(ǫi) (1 − ǫ1)2 vk = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (6) On rewriting the MS equation in terms of ξ one eliminates its explicit dependence on aH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In the regime where the SR parameters are constant, and in the long wavelength limit (ξ → 0) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (6) can be algebraically solved and the features of the primordial spectra can be derived in a straightforward manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Indeed, in this limit, the two independent solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (6) have the form vk = ξα, where α satisfies the algebraic equation α2 + � ǫ1ǫ2 (1 − ǫ1)2 − 1 � α − fMS(ǫi) (1 − ǫ1)2 = 0 , (7) with α1,2 = − � ǫ1ǫ2 (1−ǫ1)2 − 1 � ± �� ǫ1ǫ2 (1−ǫ1)2 − 1 �2 + 4 fMS(ǫi) (1−ǫ1)2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (8) 3 For instance, when fMS is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (2), and in the pure de Sitter case (ǫi = 0), we obtain α1,2 = 1 ± 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (9) For this case, the positive solution, α1 = 2, decreases in time, while the negative solution, α2 = −1, increases, and it remains nontrivial in the limit ξ → 0, which leads to vk,dS ∼ k−1/2 � k aH �−1 , Rk,dS ∼ k−3/2 aH z = k−3/2H , (10) where Rk ≡ vk/z is the curvature perturbation (z = a in the de Sitter case), and the prefactor k−1/2 is essentially fixed by the initial (Bunch-Davies) conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The quantity Rk is independent of time, and the spectral index can be straightforwardly computed to be ns − 1 = d ln ∆2 s d ln k , (11) with ∆2 s ≡ |Rk,dS|2k3/(2π2), which leads to the well known de Sitter result (ns − 1)dS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In the SR case (|ǫi| ≪ 1), the SR parameters can be approximated by constants and the expressions (8) are still valid but must be expanded to first order for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' One then obtains α1,2 = 1 ± √9 + 12ǫ1 + 6ǫ2 2 ≃ 1 ± (3 + 2ǫ1 + ǫ2) 2 , (12) which implies (ns − 1)SR = −2ǫ1 − ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' We note that there is a caveat one must take into account for USR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In this case, one finds the same solutions for the α’s as the de Sitter case, but the definition of the curvature perturbations is different, since zUSR ∝ a√ǫ1 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Then, the amplitude of primordial curvature perturbations depends on time and is amplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In the USR case, the spectral index cannot be calculated analytically with the same procedure as illustrated for de Sitter and SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' One can better illustrate the differences among the three cases just mentioned by solving the equation for Rk, R′′ k + 2z′ z Rk + k2Rk = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (13) In terms of ξ, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (13) can be conveniently rewritten as ξ2 d2Rk dξ2 + � ǫ1ǫ2 − 2 (1 − ǫ1) d ln z dN (1 − ǫ1)2 � ξ dRk dξ + ξ2 (1 − ǫ1)2 Rk = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (14) In GR with a minimally coupled inflaton, we have d ln z/dN = 1 + ǫ2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Then, for constant SR parameters and in the long wavelength limit, the last term is negligible, and the equation admits a constant solution and a solution proportional to ξβ, where β = 3 − 4ǫ1 + ǫ2 + ǫ1 (ǫ1 − 2ǫ2) (1 − ǫ1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (15) If ξβ decreases in time, the constant solution dominates in the ξ → 0 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This is what happens for de Sitter and SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In contrast, if ξβ increases in time, it dominates in the ξ → 0 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This is what occurs for USR leading to results that are very different from de Sitter and SR, namely, an amplitude of the spectrum that increases in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The non-constant solution is Rk ∝ � k aH �β ∼ e−β(1−ǫ1) N , (16) and it increases or decreases depending on the sign of Φ ≡ β (1 − ǫ1) = 3 − 4ǫ1 + ǫ2 + ǫ1 (ǫ1 − 2ǫ2) (1 − ǫ1) , (17) 4 increasing if Φ < 0 and decreasing if Φ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Only in the latter case the spectral index of primordial spectrum can be analytically calculated by using the definition (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For a general inflationary model, one finds ∆2 s ∝ k2+2α2 = k 2− � ǫ1ǫ2 (1−ǫ1)2 −1 � − �� ǫ1ǫ2 (1−ǫ1)2 −1 �2+4 fMS(ǫi) (1−ǫ1)2 , (18) and ns − 1 = 2 − � ǫ1ǫ2 (1 − ǫ1)2 − 1 � − �� ǫ1ǫ2 (1 − ǫ1)2 − 1 �2 + 4 fMS(ǫi) (1 − ǫ1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (19) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Evolutions with “Constant” SR Parameters Let us now illustrate an important point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The results obtained above are exact when the SR parameters are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' However, given the recursive definition of the SR parameters (ǫi+1 = ǫ−1 i dǫi/dN), a constant set of ǫi’s corresponds either to H = const and ǫi = 0 (de Sitter case), or ǫ1 = const and ǫi = 0 for i > 1 (power law inflation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' It may thus seem redundant to present the general formalism for such a restricted range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Nevertheless, we note that the above results can be applied to a wider set of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' First, as we already mentioned, the general results for Φ and ns − 1 can be applied to the SR case, in which the expressions must be expanded to the first order for consistency, since the SR parameters are approximately constant when they are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Furthermore, the large a limit of some transient phase (such as the USR phase) leads to non-trivial sequences of “constant” SR parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In these cases, one obtains a hierarchy of, for example, ǫi’s with constant, non-zero SR parameters for either even or odd values of i, while the remaining SR parameters are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For instance, let ǫi N→∞ = li +Li(N) with limN→∞ Li(N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Then, due to their recursive definition, one obtains ǫiǫi+1 ≡ dǫi dN a→∞ = Li,N(N) , (20) which leads to limN→∞ ǫi+1 = 0, provided limN→∞ Li,N(N) = 0, and, in particular, ǫi+1 N→∞ = Li,N(N) li + Li(N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (21) Moreover, ǫi+2 ≡ dǫi+1/dN ǫi+1 N→∞ = Li,NN(N) Li,N(N) + ǫi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (22) Let us now suppose Li(N) ∝ e−γN ∼ a−γ, with γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In this case: ǫi+2 N→∞ = −γ + ǫi+1 , (23) and the subsequent terms of the hierarchy take values equal to zero and −γ: lim N→∞ ǫi = li, lim N→∞ ǫi+1+2n = 0, lim N→∞ ǫi+2n = −γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (24) Therefore, due to their definition, an infinite sequence of SR parameters may take alternate “constant” values in the large a limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This property is crucial in the analysis that follows, and it depends on the form of Li(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Indeed, exponential forms lead to the result (24) but, in contrast, if Li ∝ N −γ, then the sequence obtained is limN→∞ ǫj = 0 for j > i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' It is also worthwhile to mention that similar results can be generalized to other hierarchies of SR parameters because they only depend on the recursive definition of the SR parameters [analogously to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (20)] and on the form of Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For example, the same results can be extended to the hierarchy of “scalar field flow functions” that is defined by δ0 = φ/φ0 and δiδi+1 = dδi/dN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In general, the ǫi’s and the δi’s are related through the homogeneous Friedmann and Klein-Gordon equations, and, in some scenarios, it is useful to use one or both hierarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' 5 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' MODEL RECONSTRUCTION We are interested in reconstructing scalar field potentials that describe transient inflationary solutions, which are associated with varying SR parameters with a “constant” behaviour in the future (and necessarily ǫ1 < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Therefore, the results illustrated in the previous section can be adopted to study such models and to verify whether they can generate an amplification of the primordial spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Finding the entire evolution of the scalar field is not necessary for this purpose, and we will only calculate the potential and the asymptotic behaviour of the homogeneous quantities in terms of the corresponding SR parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The potentials that lead to an amplification can then be used to build an inflationary model that fits the CMB observations and which produces a large amount of DM in the form of PBHs at the end of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' GR with a Minimally Coupled Inflaton In order to proceed with the reconstruction, let us first briefly review the homogeneous Einstein equation, H2 = 1 3MP 2 �1 2 ˙φ2 + V (φ) � , (25) ˙H = − ˙φ2 2MP 2 , (26) which leads to MP 2H2 (3 − ǫ1) = V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (27) This last equation can be used to reconstruct the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (26) and (27) can be conveniently used for the reconstructions starting from some ansatz for H = H(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In this case, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (26) becomes ǫ1 = 1 2MP 2 � dφ d ln a �2 , (28) which can be integrated to obtain, when possible, a = a(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Let us first consider the following evolution of the Hubble constant: H = H0 � α + A an �m , (29) where A, α, n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Similarly to USR, the evolution described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (29) has a de Sitter attractor in the future, and, indeed, H(a) is that of USR when n = 6 and m = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (It is interesting to note that this evolution represents a general solution in the model with a minimally coupled scalar field and a constant potential, or, in other words, in a universe driven by a mixture of two fluids: a cosmological constant and stiff matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' It is curious that n = 6 and m = 1/4 yield the general solution for the universe driven by the Chaplygin gas [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=') We also note that the transient is described by A/an ∼ e−nN and that a result similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (24) is then expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This is easily verified if we explicitly calculate the hierarchy of SR parameters: ǫ1 = m · n A αan + A = m ǫ3 = m ǫ5 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' a→+∞ −→ 0 , (30) and ǫ2 = −n αan αan + A = ǫ4 = ǫ6 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' a→+∞ −→ −n , (31) where a > [(m · n − 1)A/α]1/n is necessary for inflation to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' We can integrate and invert Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (28) to obtain exp �φ − φ0 MP � n 2m � = x + 1 x − 1 x0 − 1 x0 + 1 , (32) 6 with x ≡ A−1/2√ αan + A and x, x0 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Notice that φ = φ0 when x = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Conversely, φ = φ∞, with φ∞ ≡ φ0 + MP � 2m n ln B0 , (33) for x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (32) can be solved for x, which yields x = e φ−φ0 MP √ n 2m + B0 e φ−φ0 MP √ n 2m − B0 , (34) where B0 = (x0 − 1)/(x0 + 1), and the reconstructed potential is finally V = H2 0 � αx2 x2 − 1 �2m � 3 − n · m x2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (35) For n = 6 and m = 1/2, one recovers a constant potential and the USR evolution, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For other choices of the parameters n and m, the expression for the potential in terms of φ is a complicated function with exponentials that need not be written here explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' However, this cumbersome expression is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Since the asymptotic behaviour of the potential at φ ∼ φ∞ determines the limiting values of the SR parameters, we simply give the form of V around φ∞, which is V ≃ 3H2 0α2m � 1 + n 4 � 1 − n 2 � �φ − φ∞ MP �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (36) Finally, let us calculate the consequences of the background evolution given by Eq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (29) on the inflationary spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The value of Φ is Φ = 3 − 4ǫ1 + ǫ2 + ǫ1 (ǫ1 − 2ǫ2) (1 − ǫ1) a→+∞ −→ 3 − n , (37) and for n > 3 the curvature perturbations Rk are amplified, after their horizon exit, as time passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In contrast, if 0 < n < 3, from the constant solution for Rk, one finds ns − 1 = n > 0 , (38) which implies the amplitude is that of a blue-tilted spectrum, which grows as the wavenumber k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' We conclude that for GR with a minimally coupled inflaton, the inflationary evolution described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (29), with a transient phase and a de Sitter attractor in the future, leads to an inflationary enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The corresponding inflaton dynamics is driven by the potential (35) and similar behaviours can be obtained from potentials of the form (36) with the field close to φ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Power Law solutions In this section, we generalise the results obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (29), and study the transient phase with a Power Law inflation attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For this case, in contrast to de Sitter, it is only possible to reconstruct the inflaton potential exactly for particular choices of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Close to the attractor, an approximate reconstruction can always be obtained, and that is enough for the purposes of model building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The amplification of the primordial spectrum can still be studied in full generality, as it depends on the asymptotic values of the SR parameters, which can be calculated exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In this case, and in the large a limit, one obtains ǫ1 → const + L(a) , (39) with L(a) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In analogy to the previous case, we consider ǫ1 = � β + B an �m → βm , (40) with β, B, n > 0 and βm < 1 (so as to have acceleration close to the attractor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Notice that, when β = 0, one finds a transient phase with a de Sitter attractor, but ǫ1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (40) is different from that in the set (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This case is 7 expected to generate a hierarchy of the form (24) in the large a limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Indeed, the ansatz (40) leads to the following hierarchy of SR parameters: ǫ2 = −mǫ4 = −mǫ6 = · · · = − n m B B + β an → 0 , (41) and ǫ3 = ǫ5 = · · · = − n βan B + β an → −n , (42) where, in contrast to the de Sitter case examined in the previous section, now the even SR parameters tend to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' By proceeding with reconstruction and integrating Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (40) and (28), one finds, respectively, H = H0 exp � − � β + B an �1+m (1 + m)nβ 2F1 � 1, 1 + m, 2 + m, 1 + B βan �� , (43) and φ − φ0 = f(a) − f(a0) , (44) where f(a) = 2 √ 2MP � β + B an � 2+m 2 2F1 � 1, 1 + m 2 , 2 + m 2 , 1 + B βan � (2 + m)nβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (45) In this case, the exact reconstruction of the potential is rather complicated unless one adopts simplifying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For example, let m = −1 and 0 < β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Then, H = H0 [n (B + βan)] 1 nβ , (46) and φ − φ0 = MP ln � 1+ � ǫ1(a) β 1− � ǫ1(a) β 1− � ǫ1(a0) β 1+ � ǫ1(a0) β � n√β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (47) In the a → ∞ limit, one obtains φ∞ = φ0 + MP ln � β+1 β−1A0 � n√β , (48) with A0 ≡ � 1 − � ǫ1(a0)/β � / � 1 + � ǫ1(a0)/β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The relation (47) can be inverted to obtain an = an(φ): an = an 0 �� 1 − � ǫ1(a0) β � + � 1 + � ǫ1(a0) β � en√β(φ−φ0)/MP �2 4en√β(φ−φ0)/MP , (49) and finally the potential can be reconstructed, provided we substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (49) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In terms of an, it then takes the following form: V = H2 0 [n (B + βan)] 2 nβ � 3 − � β + B an �m� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (50) The expression in terms of φ is cumbersome and it will not be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' It is also worthwhile to note that such a potential depends on the homogeneous inflaton through the exponential function exp � n√βφ/MP � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This functional dependence is expected as it is the generalization of the standard Power-Law inflation potential, which only contains one exponential function of the inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Moreover, various approximate reconstruction methods can be used to obtain 8 the shape of the potential close to the attractor but we omit this discussion here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Whereas the exact reconstruction can be obtained for certain values of the parameters, the behaviour of the resulting inflationary spectra can be calculated exactly from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (41) and (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For generic values of m, one may compute Φ = β2m − 4βm + 3 (1 − βm) = 3 − βm > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (51) This shows the absence of the growing solution for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The spectral index is then simply given by ns − 1 = − 2βm 1 − βm < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (52) This is the same result as the one obtained for the Power Law attractor solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In contrast to the de Sitter case, the resulting primordial spectrum, if evaluated on the trajectory which approaches the attractor (and close to it), coincides with the spectrum calculated on the attractor itself, and no amplification or peculiar features emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' It is also noteworthy that this result is not restricted to the evolution given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (40), as it only depends on the limits (41) and (42), which are not partcular to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For instance, starting from the ansatz H(a) = H0 �a0 a �βm � 1 + A0 an �m , (53) where n, A0 > 0, the resulting spectra are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The observed absence of amplification in the cases of Power-Law inflation considered here is relevant because Power- Law is the exactly solvable inflationary model that is most akin to SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' One may then conjecture that similar results (and, in particular, the lack of amplification) hold for SR inflation when the inflaton approaches the attractor solution, and the enhancement is a peculiarity of de Sitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Non-Minimally Coupled Inflaton Let us now consider the different scenario of a non-minimally coupled inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In order to perform the reconstruc- tion, we first review the basic homogeneous equations for this model: H2 = 1 3F(φ) � ˙φ2 2 + V − 3HF,φ ˙φ � , (54) and ˙H = − 1 2F(φ) � (1 + F,φφ) ˙φ2 + F,φ � ¨φ − H ˙φ �� , (55) where F(φ) represents a general non-minimal coupling and F = MP reproduces the minimally coupled case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In contrast to the previous cases, the homogeneous equations and the reconstruction procedure now become more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Then, for simplicity, we shall henceforth limit our study to the induced gravity (IG) case, where F(φ) = ξφ2 [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This simplifying choice is also justified by the fact that both Higgs inflation and Starobinsky inflation (in the Einstein Frame) occur in a regime that is very close to pure IG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Reconstructing the inflaton potential for a given H(a) is not as straightforward as for GR with a minimally coupled inflaton, and we found exact potentials only for certain values of the parameters and for the de Sitter attractor case [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (29)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Nevertheless, we can still predict the shape of the inflationary spectra or, at least, the possibility of an amplification in the large a limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' De Sitter limit Let us consider H(a) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In IG, the following exact relations hold between some SR parameters: ǫ1 = δ1 1 + δ1 � δ1 2ξ + 2δ1 + δ2 − 1 � , (56) 9 ǫ1 = 1 2ξ(1 + 6ξ) � (1 + 2ξ)δ2 1 − 8ξδ1 − 6ξ2 � 1 + 2δ1 − δ2 1 6ξ � �d ln V d ln φ − 4 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (57) Before discussing the reconstruction of the inflaton potential, we must first calculate the asymptotic values of the SR parameters, which are pivotal in the analysis of the amplification of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (54), the potential can then be obtained as V = 3ξφ2H2 � 1 + 2δ1 − 1 6ξ δ2 1 � , (58) provided H = H(φ) and δ1 = δ1(φ) are known [indeed, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (58) is the IG counterpart of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (27) in GR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Let us first calculate the SR parameters in the large a limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Since lima→∞ ǫ1 = 0, one either has lima→∞ δ1 = 0 and lima→∞ δ2 ̸= 0, or lima→∞ δ2 = 0 and lima→∞ δ1 ̸= 0 but satisfying the relation δ1,∞ = 2ξ 1 + 4ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (59) These results follow from the functional dependence of H(a) on a inherited by ǫ1 and δi’s and by the general result given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (24), which is applied here to the SR hierarchy δi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Notice that, in contrast to GR, two different de Sitter trajectories are present in IG, and they are associated with two different evolutions of the inflaton field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (57) in the same limit for a, one obtains that the potential, on the attractor, must satisfy d ln V∞ d ln φ − 4 = 0 ⇒ V∞ ∝ φ4 (60) in the former case and d ln V∞ d ln φ − 4 = 0 ⇒ V∞ ∝ φ2 (61) in the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' We can now proceed to evaluate the full hierarchy of δi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (56) and differentiating, we find ǫ2 = δ2 � (1 + 4ξ)δ2 1 + 2ξ (δ2 + δ3 − 1) + 2δ1 (1 + 4ξ + ξδ3) � (1 + δ1) [(1 + 4ξ)δ1 + 2ξ(δ2 − 1)] , (62) and, by further differentiating, the ǫi’s with arbitrary large i can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In the large-a limit, we have already calculated ǫ2i = −n and ǫ2i+1 = 0 [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (30) and (31)], and one then obtains two possible hierarchies for the δi’s: δ2i+1,∞ = 0, δ2i,∞ = ǫ2,∞ = −n , (63) and δ1,∞ = 2ξ 1 + 4ξ , δ2i+1,∞ = −n, δ2i,∞ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (64) This latter statement cannot be simply verified by substitution because the limits involved do not commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For example, on substituting first δ2 = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (62), one obtains ǫ2 = 0, which is not correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' One must solve (at least perturbatively in the large-a limit) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (56) and then evaluate the limits with the help of the solution found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The above results are correctly reproduced only if we proceed in this manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The exact reconstruction of the inflaton potential is not possible in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Nonetheless, in specific cases, the potential may be derived exactly as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Consider the following ansatz for δ1: δ1 = n0 + n1a−n d0 + d1a−n , (65) which is suggested by the expression for ǫ1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' If n0 = 0 and n1 ̸= 0, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (65) can be integrated, and the resulting φ(a) is inverted as follows φ(a) = φ0 � d0 + d1a−n�− n1 n d1 ⇒ a−n = � φ(a) φ0 �− n d1 n1 − d0 d1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (66) 10 The coefficients n1, d0, d1 and ξ can finally be fixed by the requirement that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (65) be a solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Two nontrivial solutions can be found: d0 = −(1 + n) n1α A m n , d1 = −(1 + n)n1 m n , ξ = m 2(1 − 3m + n + m n) , (67) or d0 = −(1 + n) n1α A m n , d1 = −(1 + n + m n)n1 m n , ξ = m 2(1 − 2m + n + m n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (68) Notice that more exact solutions for δ1 can be found if we start from the ansatz (65) and n0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' However, by further integrating these solutions to obtain φ(a), one is led to non-invertible functions, and the reconstruction cannot be completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For both Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (67) and (68) one has δ1,∞ = 0 , (69) and one can explicitly verify that the hierarchies belong to the set (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Notice that n1 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (67) and (68) can be arbitrarily chosen, as should be due to the form of the ansatz (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Let us, for simplicity, complete the reconstruction choosing n and m to reproduce USR in the IG context (n = 6, m = 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In this case, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (67) and (68) take the following form: n0 = 0, d0 = − 7α 3An1, d1 = −7 3n1, ξ = 1/10 ⇒ δ1 = − 3A 7 (A + αa6) , (70) n0 = 0, d0 = − 7α 3An1, d1 = −10 3 n1, ξ = 1/36 ⇒ δ1 = − 3A 10A + 7αa6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (71) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (29), a(φ) in (66) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (70), one finds δ1 = 3 7 � α �φ0 φ �14 − 1 � and H2 = H2 0 � φ φ0 �14 , (72) with φ/φ0 a→∞ −→ α1/14 and φ > φ0, while for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (71) one finds δ1 = 3 10 � α �φ0 φ �20 − 1 � and H2 = H2 0 �� 7 10 φ φ0 �20 + 3α 10 � , (73) with φ/φ0 a→∞ −→ α1/20 and φ > φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Finally, by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (58), one obtains V = − 3H2 0 490φ12φ14 0 � 8φ28 − 72αφ14 0 φ14 + 15α2φ28 0 � (74) for the first exact solution, and V = − H2 0 6000φ38φ20 0 � 7φ20 + 3αφ20 0 � � 7φ40 − 84αφ20 0 φ20 + 27α2φ40 0 � (75) for the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In the a → ∞ limit, the potentials (74) and (75) satisfy the condition d ln V/d ln φ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The potential can have negative values but, in the vicinity of φ ≃ φ∞, the potential is positive and V∞ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' We discuss at last the behaviour of the primordial scalar curvature spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The general formulae illustrated in the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' II can be easily generalised to the IG case wherein zIG = aφδ1 � 1 + 6ξ 1 + δ1 , (76) and Φ is given by Φ = � 1 − ǫ1 − ǫ1ǫ2 (1 − ǫ1) + � 2 + 2δ1 + 2δ2 − δ1δ2 1 + δ1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (77) 11 If we evaluate Φ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' to the hierarchies (63) and (64), one observes that only constants and terms linear in the SR parameters remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Moreover ǫ1,∞ = 0 and Φ then simplifies to Φ = 3 + 2δ1 + 2δ2 , (78) which can be negative only for the hierarchy (63) but is strictly positive for the hierarchy (64), provided we restrict ourselves to positive values of the non-minimal coupling ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In the former case, Φ = 3 − 2n, which implies that the growing solution exists for n > 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' If no growing solution exists [as is the case for (64) or (63) with n < 3/2], an amplification of the spectrum is only possible if the spectrum is blue-tilted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Let us then evaluate ns − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In the IG case, fMS(ǫi) in the MS equation is given by fMS = δ2 1 + δ2 2 + (3 − ǫ1) (1 + δ1 + δ2) + δ2δ3 + δ1δ2 � ǫ1 + δ1 − 3δ2 − δ3 + 2δ1δ2 1+δ1 − 2 � 1 + δ1 − 1 , (79) and, as usual, it can be simplified to obtain the following expression for the scalar spectral index: ns − 1 = 3 − � 1 + 4 (δ2 1 + δ2 2 + 3 (1 + δ1 + δ2) − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (80) Then, for the hierarchy (63) and n < 3/2, we obtain ns − 1 = 3 − |3 − 2n| = 2n , (81) which is indeed blue-tilted, while for the hierarchy (64), we find ns − 1 = − 4ξ 1 + 4ξ , (82) which is red-tilted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' We therefore conclude that solutions having H of the form given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (29), in the IG context, may lead to a spectrum enhancement for evolutions asymptotically described by the hierarchy (63) and either for n > 3/2 (due to the presence of the growing solution) or 0 < n < 3/2 (in absence of the growing solution but with the blue-tilted spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' APPLICATIONS We have so far studied the consequences of cosmological evolutions with a transient phase, which is crucial to potentially obtain the amplification required by the formation of PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Indeed, the presence of the transient generates, in the large-a limit, a sequence of values for the SR parameters that is otherwise not obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' We then reconstructed, when possible, the potentials that led to the desired evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In this section, our approach will be slightly different, as we shall study the presence of the transient solutions in the particular dynamical regime of constant roll (CR) inflation [11], which is the natural generalisation of USR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' CR solutions satisfy the equation ¨φ + BH ˙φ = 0 , (83) where B > 0, and one recovers the USR solution for B = 3, while the case of |B| ≪ 1 reproduces standard SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' We observe that the CR condition (83) can be rewritten, in terms of the SR parameters, as δ2 + δ1 − ǫ1 + B = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (84) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (84) is model independent, since it only depends on the definitions of ǫi’s and δi’s, and can be easily integrated to obtain dφ d ln aH � a a0 �B = C3 ⇒ ˙φ = C3 �a0 a �B , (85) where C3 is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In the minimally coupled case, CR can generate an amplification of the primordial scalar spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In what follows, after a revision of this result (which was analysed in [7]) we shall consider CR in IG and study its consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Constant Roll in GR with a Minimally Coupled Inflaton In GR with a minimally coupled inflaton, on imposing CR conditions and adopting the Hamilton-Jacobi (HJ) formalism, it is possible to reconstruct the evolution of the Hubble parameter and the corresponding potential [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In particular, one finds that H(φ) is the following superposition of two exponential functions: H(φ) = C1 exp �� B 2 φ MP � + C2 exp � − � B 2 φ MP � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (86) In [7], the solution (86) with one exponential (C1 = 0 or C2 = 0), as well as the cosh and sinh cases, are analysed with the aim of finding the exact solutions compatible with CMB observations [12] (and thus not amplified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Here, in a slightly different approach, we consider the general case, and we study the power enhancement of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (26) can be rewritten in terms of the SR parameters ǫ1 = φ2 2MP 2 δ2 1 , (87) from which, using the chain rule, we obtain ǫ1 = −δ1 d ln H d ln φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (88) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (87) then becomes ǫ1 = 2MP 2 φ2 �d ln H d ln φ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (89) The potential can subsequently be reconstructed by substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (86) and (89) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (27): V (φ) = MP 2H(φ)2 � 3 − 2MP 2 φ2 �d ln H d ln φ �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (90) In order to obtain the corresponding evolution, one must integrate and invert the equation δ1 = −2MP 2 φ2 d ln H d ln φ , (91) which can be easily derived from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (87) by using (88).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' One finds � a a0 �B = x B (C2 − C1x2) , (92) where x = exp �� B 2 φ MP � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' It is straightforward to invert Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (92) so as to obtain x = x(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Correspondingly, one has H(a) = ± 4C1C2 + � a0 a �2B ∓ � a0 a �B � 4C1C2 + � a0 a �2B ∓ � a0 a �B + � 4C1C2 + � a0 a �2B a→∞ −→ ±8C1C2 + � a0 a �2B 4√C1C2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (93) Notice that the same result can be obtained if one uses the CR definition (84) instead of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The last, approximate, equality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (93) is the large-a limit of H(a), and this shows that the CR evolution is asymptotically equivalent to the evolution given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (29) with m = 1 and n = 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The results obtained in the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' II for large a are therefore inherited by CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Thus, one obtains ǫ2i+1,∞ = 0 and ǫ2i,∞ = 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Correspondingly, Φ = 3 − 2B , (94) which shows that the curvature perturbations are amplified for B > 3/2 due to the presence of a growing solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In contrast, if 0 < B < 3/2, one finds a blue-tilted spectrum ns − 1 = 3 − � (3 − 2B)2 = 2B > 0 , (95) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=', a spectrum enhancement in the absence of growing solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Therefore, CR inflation admits transient solutions that always lead to an amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Finally, it is worthwhile to mention that the solutions with C1 = 0 or C2 = 0 simply correspond to the attractor solutions for power-law inflation, and thus they are not associated with any amplification effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' 13 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Constant Roll with a Non-Minimally Coupled Inflaton Let us now consider CR in the IG context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For this case, the HJ formalism leads to [13] H(φ) = C1φ(B+p)/2 + C2 φ(p−B)/2 , (96) where p = � (B + 2)2 + 2B(2 + ξ−1), and (B + p)/2 and (p − B)/2 are both positive with (p − B)/2 < (B + p)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For simplicity, we shall take C1,2 > 0 and we restrict the analysis to the φ > 0 interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Studying the spectrum enhancement for CR in the IG case is more complicated than for GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This is essentially a consequence of the complicated form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (56) in comparison to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (26) in the GR case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' However, the simple relation (84) holds, and it can be used to simplify the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' First, with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (84), one may eliminate δ2 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (56) and obtain ǫ1 = 1 + 2ξ 2ξ δ2 1 − (B + 1)δ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (97) Subsequently, by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (88), one finds δ1 = 2ξ 1 + 2ξ � B + 1 − d ln H d ln φ � , (98) and the potential can be reconstructed by substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (96) and (98) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The evolution could be obtained by integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (98) and inverting the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' However, analytically inverting the resulting equation for arbitrary values of the parameters is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' As we are only interested in the asymptotic form of H(a), one can employ a perturbative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (98) yields �a0 a �B = φ 2+B 2 � (B + p + 2) C1φ p 2 + (B − p + 2) C2 φ p 2 � , (99) where B + p + 2 > 0 and B − p + 2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Therefore, in the large-a limit, the inversion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (99) leads to φ(a) = φ∞ + � i>1 φi �a0 a �i B ∼ φ∞ + φ1 �a0 a �B , (100) where φ∞ is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' By substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (100) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (96) and expanding for large a (properly accounting for the next-to-leading-order contributions), one finally obtains the asymptotic form of H(a), which reads H ∼ H∞ + H1 �a0 a �B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (101) Comparison to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (29) shows that m = 1 and n = B, and the corresponding hierarchy of δi’s is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (63) since δ1,∞ = lima→∞ ˙φ H∞φ∞ = 0 , (102) where ˙φ is given by (85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' One then obtains Φ = 3 − 2B, ns − 1 = 2B , (103) and, when 0 < B < 3/2, ns − 1 = 2B , (104) which are the same results as GR with a minimally coupled inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Indeed, in the a → ∞ limit, the homogeneous inflaton is frozen at a certain value and one essentially recovers the evolution of the minimally coupled case, where “Newton’s constant” is now reproduced by the (constant) asymptotic value of the inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Furthermore the a depen- dence of the solution is a consequence of the fact that the CR condition (84) is independent of the specific inflationary model, provided H∞ and φ∞ are found to be (finite) constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Jordan and Einstein frame mapping In the previous section, we found the same asymptotic behaviour for the spectra in the minimally coupled case and in the IG case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This result was obtained in spite of the fact that CR condition is not frame invariant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=', the CR condition in the Einstein Frame (EF) is not mapped, in general, into a CR condition in the Jordan Frame (JF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In this section, we briefly review this statement and discuss its consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' It is well known that, by a suitable conformal transformation and a redefinition of the scalar field (inflaton), one can map a minimally coupled theory (defined in the so-called EF) into a non-minimally coupled one (in the JF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In partcular, for IG, the mapping is given by the following transformation rules (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' [14]): a(t) = MP √ξσ ˜a(t), N(t) = MP √ξσ ˜ N(t) , (105) and φ = MP � 1 + 6ξ ξ ln σ σ0 , ˜V (φ(σ)) = MP 2 ξ2σ4 V (σ) , (106) where the tilde refers to the Einstein frame, φ is the scalar field in the EF, σ is that in the JF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The mapping induces the following transformations of the Hubble parameter: ˜H = d˜a/dt ˜N˜a = (1 + δ1) MP √ξσ H , (107) where H(t) = da(t)/dt a(t)N(t), ǫi+1 = dǫi/dt ǫiN(t)H(t), δi+1 = dδi/dt δiN(t)H(t) (108) are the Hubble and SR parameters in the JF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' From the relation (105), one also finds that d d ln ˜a = (1 + δ1)−1 d d ln a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (109) It is now straightforward to obtain the relations between SR parameters in the two frames: ˜ǫ1 ≡ −d ln ˜H d ln ˜a = − (1 + δ1)−1 d d ln a ln � (1 + δ1) MP √ξσ H � = δ1 + ǫ1 − δ1δ2 1+δ1 1 + δ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (110) Given the relation (56), one then finds ˜ǫ1 = (1 + 6ξ)δ2 1 2ξ(1 + δ1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (111) From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (56) and (111), given that CR for a minimally coupled inflaton has ˜ǫ1,∞ = 0, one concludes that, correspondingly, in the JF one has δ1,∞ = 0 and ǫ1,∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' If we differentiate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (111), we obtain the following relations among other SR parameters in the two frames ˜ǫ2 = 2δ2 (1 + δ1)2 , (112) ˜ǫ3 = δ3 − 2δ1δ2 + δ1δ3 (1 + δ1)2 , (113) ˜ǫ4 = δ1δ2 � 2δ2 − 2δ1δ2 + 3δ3 + 3δ1δ3 − (1 + δ1)2 δ3δ4 � (1 + δ1)2 (2δ1δ2 − δ1δ3 − δ3) , (114) and further ˜ǫi’s can be found by iterating the procedure but are useless for what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' 15 Similarly, one can directly calculate the relations of the ˜δi’s with the dynamical variables in the JF: ˜δ1 ≡ ˙φ ˜N ˜Hφ = � 1 + 6ξ ξ MP φ δ1 1 + δ1 , (115) and ˜δ2 ≡ d˜δ1/dt ˜N ˜H˜δ1 = −˜δ1 + δ2 (1 + δ1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (116) From the last relation and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (56), one has that the CR condition in the EF, ˜δ2 + ˜δ1 − ˜ǫ1 + B = 0 , (117) is mapped into the following condition in the JF δ2 + (B − 1) δ1 − ǫ1 + B = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' (118) Notice that only for B = 2 the CR condition is frame invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Nonetheless, both equations reduce to δ2,∞ = ˜δ2,∞ = −B at late times, and the evolution is indistinguishable, at least as far as the homogeneous degrees of freedom and the inflationary spectra are concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' We conclude that, whereas the scalar spectral index ns − 1 is frame invariant, Φ is generally not frame invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This can be checked directly by substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' However, assuming CR holds in the EF, one verifies that Φ and ns − 1 are both frame invariant in the asymptotic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Results summary Inflation Asymptotic Growing Blue-Tilted Model Svolution Solution Spectral Index GR dS n > 3 0 < n < 3 GR PL − − IG dS, δ1,∞ = 0 n > 3/2 0 < n < 3/2 IG dS, δ1,∞ ̸= 0 CR+GR dS B > 3/2 0 < B < 3/2 CR+IG dS, δ1,∞ = 0 B > 3/2 0 < B < 3/2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' CONCLUSIONS In this article, we have analyzed the effects of different transient phases, which may occur during inflation due to a particularity of the inflaton potential, on the primordial inflationary spectrum of scalar perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' These transients have been studied in the last few years as sources of amplification of the amplitude of the curvature spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' It is important to notice that if the amplitude of scalar perturbations grows large enough, it may induce gravitational collapse and consequently seed the formation of primordial black holes after inflation ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In the literature, several mechanisms for such an amplification during inflation have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' In particular, the presence of an ultra slow-roll or, more generally, a constant-roll phase has been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Whereas in the former case the amplification is due to the existence of a growing solution to the equation of motion of the curvature perturbations, in the latter case the amplification can also be generated by a blue-tilted spectrum in absence of the growing solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The purpose of this paper was precisely to examine general features of the aforementioned models starting from a rather generic ansatz for the Hubble parameter as a function of the scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This general description of the transient phase is model independent, and many results obtained can be readily applied to several modified gravity models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The matter-gravity dynamics is described in terms of the hierarchies of SR parameters, both at the homogeneous level and at the level of perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' These hierarchies, when the transient phase that describes the approach to some inflationary attractor is considered, have been shown to take a peculiar form wherein either odd or even terms of the hierarchy are null and the remaining ones are different for zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' This general feature is a peculiarity of the asymptotic form of the SR parameters close to the attractor, and it is then used as a simplifying assumption 16 throughout the entire article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' The resulting hierarchies, in the large-a limit and for the cases considered, were used to calculate the behavior of the primordial curvature spectrum as the parametrisation of H(a) was varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Then, when possible, the corresponding inflaton potential was fully reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' An overview of the spectra enhancement results was presented in Table (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' For simplicity, only the induced gravity case has been considered here as a generalisation of general relativity with a minimally coupled inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Induced gravity is particularly relevant since both Higgs and Starobinsky inflationary models (which are in good agreement with observations) take place in the ‘induced gravity phase’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' We note that while transient evolutions that have the de Sitter universe as a limit (such as USR) can lead to an amplification, the results differ when power-law inflation is considered as the limit of a transitory dynamics and, for the cases we were able to solve explicitly, no modification of the scalar spectrum was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Finally, the constant-roll case was discussed in more detail as an application of the preceding results, and the issue of the transition from the Einstein frame to the Jordan frame was also scrutinized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Chapline, Nature 253 (1975) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='5489, 251-252 doi:10.' metadata={'source': 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Starobinsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' De Vega and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=' Sanchez (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE3T4oBgHgl3EQfXQpr/content/2301.04477v1.pdf'} +page_content=') Current trends in field theory quantum gravity and strings, Lecture Notes in Physics 246 Verlag, Heidelberg, 1986), pp.' 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Tessore1, A. Soulain1, G. Pantolmos1, J. Bouvier1, C. Pinte1, 2, and K. Perraut1 +1 Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France +2 School of Physics and Astronomy, Monash University, VIC 3800, Australia +xx/xx/xx; yy/yy/yy +ABSTRACT +Aims. We aim to assess the complementarity between spectroscopic and interferometric observations in the characterisation of the +inner star-disc interaction region of young stars. +Methods. We use the code MCFOST to solve the non-LTE problem of line formation in non-axisymmetric accreting magnetospheres. +We compute the Brγ line profile originating from accretion columns for models with different magnetic obliquities. We also derive +monochromatic synthetic images of the Brγ line emitting region across the line profile. This spectral line is a prime diagnostics of +magnetospheric accretion in young stars and is accessible with the long baseline near-infrared interferometer GRAVITY installed at +the ESO Very Large Telescope Interferometer. +Results. We derive Brγ line profiles as a function of rotational phase and compute interferometric observables, visibilities and phases, +from synthetic images. The line profile shape is modulated along the rotational cycle, exhibiting inverse P Cygni profiles at the time +the accretion shock faces the observer. The size of the line’s emission region decreases as the magnetic obliquity increases, which is +reflected in a lower line flux. We apply interferometric models to the synthetic visibilities in order to derive the size of the line-emitting +region. We find the derived interferometric size to be more compact than the actual size of the magnetosphere, ranging from 50 to +90% of the truncation radius. Additionally, we show that the rotation of the non-axisymmetric magnetosphere is recovered from the +rotational modulation of the Brγ-to-continuum photo-centre shifts, as measured by the differential phase of interferometric visibilities. +Conclusions. Based on the radiative transfer modelling of non-axisymmetric accreting magnetospheres, we show that simultaneous +spectroscopic and interferometric measurements provide a unique diagnostics to determine the origin of the Brγ line emitted by young +stellar objects and are ideal tools to probe the structure and dynamics of the star-disc interaction region. +Key words. Radiative transfer – Line: profiles – Stars: variables: T Tauri, Herbig Ae/Be – Accretion, accretion disks +1. Introduction +The early evolution of low mass stars (M∗ < 2 M⊙) during the +classical T Tauri (CTT) phase depends on the interaction be- +tween the star and its accretion disc, on a distance of a few stel- +lar radii. At the truncation radius, matter from the disc surface is +channelled onto the stellar surface following the magnetic field +lines and forming an accretion funnel or column (Ghosh et al. +1977; Zanni & Ferreira 2009; Romanova & Owocki 2016; Pan- +tolmos et al. 2020). The star-disc interaction is responsible for +accretion and ejection phenomena that have a strong impact on +spectral lines formed in the close vicinity of the star’s surface. +Ghosh et al. (1977) developed an analytical model of mag- +netospheric accretion around a rotating neutron star with a dipo- +lar magnetic field. Hartmann et al. (1994) applied this magne- +tospheric accretion model to the formation of emission lines in +the spectrum of T Tauri stars. This fundamental paper sets the +general theoretical framework for the density and temperature +distributions in aligned axisymmetric magnetospheres. The cou- +pling between this representation of magnetospheric accretion in +T Tauri systems with radiative transfer calculations has provided +a crucial tool to interpret spectroscopic, photometric, and inter- +ferometric observations. The sensitivity of hydrogen lines to the +parameters of the magnetospheric models was studied in detail +by Muzerolle et al. (2001), improving the earlier calculations by +Hartmann et al. (1994). +Near-infrared observations of the Brackett γ (Brγ) line with +the Very Large Telescope Interferometer (VLTI) GRAVITY in- +strument (Gravity Collaboration et al. 2017) also probe the inner +part of the star-disc interaction region (Gravity Collaboration +et al. 2020; Bouvier et al. 2020a). However, it is still difficult +to associate the characteristic sizes derived from interferometry +with the actual size of the magnetospheric accretion region, a +key parameter in our understanding of the star-disc interaction. +In this paper, we aim at studying the formation of the Brγ +line and compute its spectroscopic and interferometric signa- +tures for non-axisymmetric models of the inner star-disc inter- +action region, akin to state-of-art MHD simulations (Romanova +& Owocki 2016). In particular, we want to clarify the meaning +of the sizes inferred through near-infrared interferometric obser- +vations and how they compare with the overall size of the mag- +netospheric accretion region. +In sections §2 and §3, we describe the model used to com- +pute the line formation in accreting magnetospheres. We discuss +spectroscopic and interferometric signatures in sections §4 and +§5, respectively. +2. Radiative transfer framework +We use the code MCFOST 1(Pinte et al. 2006, 2009; Tessore +et al. 2021) to compute emergent line fluxes from multidimen- +1 https://github.com/cpinte/mcfost +Article number, page 1 of 13 +arXiv:2301.11628v1 [astro-ph.SR] 27 Jan 2023 + +A&A proofs: manuscript no. bt_spidi2 +sional models of magnetospheres for a 20-level hydrogen atom. +The atomic model, with 19 bound levels and the ground state of +HII, consists of 171 bound-bound transitions (atomic lines) and +19 bound-free transitions (continua). We focus here on the Brγ +line at 2.1661 µm although, the Balmer lines Hα and Hβ and +the Paschen β line (Paβ) are modelled as well. These specific +hydrogen lines are commonly used to characterise accretion and +ejection phenomena in young systems (Folha & Emerson 2001; +Alencar et al. 2012; Bouvier et al. 2020a; Pouilly et al. 2020; +Sousa et al. 2021). The method to solve for the non-LTE pop- +ulations of hydrogen and the microphysics are the same as in +Tessore et al. (2021). The updated version of the code we use +now simultaneously solves the charge equation and the statis- +tical equilibrium equations, which has been proven to increase +the convergence in chromospheric conditions (Leenaarts et al. +2007). We tested our code for different magnetospheric models +taken as benchmarks in Muzerolle et al. (2001) and Kurosawa +et al. (2006). The results of this comparison are presented and +discussed in Appendix A. +3. Magnetospheric accretion model +Matter from the circumstellar disc is channelled onto the stel- +lar surface along the dipolar magnetic field lines. The stellar +magnetic field truncates the disc at a distance Rt from the star, +the truncation radius. In practice, the interaction between the +stellar magnetic field and the disc takes place over a small re- +gion between Rt and Rt + δr. Both Rt and δr are used to de- +fine the size of the disc region magnetically connected to the +star. As the gas approaches the stellar surface, it decelerates in +a shock and is heated at coronal temperatures. Theoretical mod- +els of accretion shocks by Calvet & Gullbring (1998) show that +the optically thin emission of the pre/post-shock dominates be- +low the Balmer jump and that the optically thick emission of +the heated photosphere contributes to the total continuum emis- +sion at larger wavelengths. In the following, we only consider +the contribution of the heated photosphere to the shock radia- +tion. The shock2 temperature is computed from the energy of the +gas infalling onto the stellar surface following the prescription +of Romanova et al. (2004) unless specified. This approach as- +sumes energy conservation and that the shock radiates as a black +body, meaning that its temperature is determined by the specific +kinetic energy and enthalpy of the gas deposited at the stellar +surface. The shock temperature hence derived is of the order of +4500 K - 6000 K. +3.1. The stellar surface +The stellar surface is considered as the inner boundary of the +model and emits as a blackbody whose temperature is deter- +mined by the stellar parameters. Throughout the paper, the stellar +parameters are T∗ = 4, 000 K, M∗ = 0.5 M⊙, and R∗ = 2 R⊙. We +set the distance to the star at 140 pc, which is typical of the near- +est star forming regions such as Upper Scorpius (≈ 146 pc Galli +et al. 2018a) or Taurus (≈ 130 pc Galli et al. 2018b). +3.2. Geometry of the accretion funnels +We consider 3D non-axisymmetric models of the magneto- +spheric accretion region. These models are parametrised by the +same set of parameters as the axisymmetric magnetospheric +2 We assume that the shock region is unresolved and is part of the +stellar surface. +Fig. 1: Density distribution of a non-axisymmetric model with an +obliquity of 10◦. The rotation axis of the star Ω is shown with a +white arrow and the dipole axis, µ, with a red arrow. The density +is computed from Eqs. (1) and (2). The colour map scales with +the density. +model of Hartmann et al. (1994) (see also Muzerolle et al. 1998, +2001; Kurosawa et al. 2006; Lima et al. 2010; Kurosawa et al. +2011; Dmitriev et al. 2019).The density and the velocity fields of +the accretion columns are fully described with a set of indepen- +dent parameters: the mass accretion rate ˙M, the rotation period +Prot, Rt, and δr. +For our study, the value of ˙M, Rt and δr, and of the temper- +ature of the magnetosphere are fixed. The impact of these pa- +rameters on the line formation has been discussed thoroughly in +Muzerolle et al. (1998, 2001, see also App. A). The line’s re- +sponse to the mass accretion rate and to the temperature is an es- +sential proxy for understanding the physics of the star-disc inter- +action region. We use a mass accretion rate ˙M = 10−8 M⊙ yr−1, +a truncation radius Rt = 4 R∗, and δr = 1 R∗. The value of the +rotation period is deduced from the maximum truncation ra- +dius (Rt + δr), imposing that stable accretion occurs at 90% of +the corotation radius, consistent with the work of Blinova et al. +(2016). The rotation period is therefore fixed at Prot = 6 days, +corresponding to slowly rotating T Tauri stars (see Herbst et al. +2007; Bouvier et al. 2014, for a review). The rotational velocity +for that period is thus of the order of 80 km s−1 at the outer edge +of the magnetosphere. +When the magnetic field axis (µ) is misaligned with respect +to the rotational axis (Ω) of the star, the geometry of the accre- +tion flow changes dramatically. The equations for the magnetic +field components of a non-axisymmetric dipole, i.e. with a non- +zero obliquity, are provided in Mahdavi & Kenyon (1998). The +parameter βma describes the angle between the dipole moment +and the star’s rotational axis, the magnetic obliquity. +We approximate the density, ρ, along the non-axisymmetric +magnetic field lines with, +ρ = α B +� = αB ρaxi +Baxi +, +(1) +where α is a constant along a given field line and B the ana- +lytic misaligned dipolar field. �, ρaxi, and Baxi denote the veloc- +ity field, density, and dipolar magnetic field, respectively, and +they are taken from the axisymmetric model of Hartmann et al. +(1994). In other words, the 3D density structure is computed +from Eq. (1) under the assumption that the infalling gas has a +velocity field on the poloidal plane. The value of α is computed +Article number, page 2 of 13 + +1.2e-08 +5e-9 +[s-'b] +2e-9 +1e-9 +5e-10 +2.3e-10 +RtB. Tessore & A. Soulain et al.: Spectroscopic and interferometric signatures of magnetospheric accretion +from the numerical integration 3 of the mass flux over the shock +area, +˙M = +� +ρv · dS, +(2) +where dS is the surface element and v the velocity field. In our +model, the value of ˙M is an input parameter and is held constant. +Therefore, α is obtained to ensure consistency between Eqs. (1) +and (2). We compute five models with an obliquity βma ranging +from five to forty degrees in step of ten degrees, representative +of what has been measured for T Tauri stars with spectroscopy +(McGinnis et al. 2020) and spectropolarimetry (Donati et al. +2008, 2010, 2013; Johnstone et al. 2014; Pouilly et al. 2020). For +these non-axisymmetric models, the shortest field lines – defin- +ing the main accretion columns4 – carry most of the gas density. +We remove the longest field lines – the secondary columns – +in our modelling as in Esau et al. (2014). This yields models +with one crescent-shaped accretion spot per stellar hemisphere +reminiscent of numerical simulations of misaligned dipoles (Ro- +manova et al. 2003). +Figure 1 shows the density of a non-axisymmetric magneto- +sphere with an obliquity of 10◦. +3.3. Temperature of the funnels +The temperature of the magnetospheric accretion region is not +well constrained. The determination of the temperature by Mar- +tin (1996) from first principles was not able to reproduce the +observations. A self-consistent calculation of the temperature of +the magnetosphere is beyond the scope of this paper. Instead, +we adopt here the temperature profile of Hartmann et al. (1994), +which has been extensively used in the past to model line fluxes +from accreting T Tauri stars. The temperature is computed using +a volumetric heating rate (∝ r−3) and balancing the energy input +with the radiative cooling rates of Hartmann et al. (1982). The +exact balance between the heating and cooling mechanisms is +unknown. Instead, the temperature profile is normalised to a free +parameter, Tmax, that sets the value of the maximum temperature +in the funnel flow. In the following, we have set the temperature +maximum to Tmax = 8, 000 K. +4. Spectroscopic signatures +Thanks to the Doppler shift of the funnel flow, it is possible +to reconstruct the origin of the emission line by looking at the +brightness maps in various velocity channels. Figure 2 shows +the contribution of the different parts of the magnetosphere to +the total integrated Brγ line flux at a given velocity for an in- +clination of 30◦, matching the model illustrated in Fig. 1. At +those density and temperature, the continuum emission comes +from the stellar surface (Isurf/Imag > 100). Locally, the contin- +uum emission from the shock is three times larger than the emis- +sion from the star. Overall, given the small covering area of the +accretion shock (around 1%), the total continuum emission at the +frequency of the Brγ line is dominated by the star’s radiation, +Fshock/F∗ = 3%. The low-velocity components (< 50 km s−1) of +the line form in the regions where the projected velocity along +the line-of-sight is close to zero and near the disc. The geometry +3 For this 3D magnetospheric accretion model, an explicit formula for +the shock area does not exist (see also Mahdavi & Kenyon 1998) +4 Geometrically, the shortest field lines obey the following criterion +cos φ′ × z > 0 where φ′ is the azimuth in the frame aligned with the +dipole axis and z the coordinate parallel to the rotation axis. +of the non-axisymmetric model, defined in §3, is responsible for +a rotational modulation of the integrated line flux. Many classical +T Tauri stars shows modulated photometric variability (e.g Cody +et al. 2014) and, more directly related to the magnetospheric re- +gion, many also show rotational modulation of the longitudinal +component of the stellar magnetic field (e.g Donati et al. 2020). +Indeed, the periodic variability of optical and emission line pro- +files has been reported in various systems (for instance Sousa +et al. 2016; Alencar et al. 2018; Bouvier et al. 2020a), which in- +dicates that the emission region is stable on a timescale of several +rotation periods. +Figure 3 shows the variability of the Brγ line at different +phases of rotation at an inclination of 60◦ for different obliq- +uities. The origin of the rotational phase is defined such that at +phase 0.5, the accretion shock is facing the observer. The red- +shifted absorption seen for the Brγ line at phases 0.250, 0.47 and +0.69, results from a lower source function of the gas above the +shock (see App. A). From observations, red-shifted absorption +in the Paβ and Brγ lines are seen in less than 34% and 20% of +the line profiles, respectively (Folha & Emerson 2001). The in- +verse P Cygni profile disappears when the shock, or a significant +fraction of it, is hidden on the opposite side of the star. The line, +with either a double-peaked profile or a moderate red-shifted ab- +sorption, is reminiscent of Reipurth et al. (1996) cases II and IV. +While the profiles with redshifted absorption agree with observa- +tions, those that display an M-shape are usually not observed in +young stellar objects. This suggests that magnetospheric accre- +tion is not the only contribution to the profile, which can also be +impacted by various types of outflows (e.g., stellar, interface, and +disk winds Lima et al. 2010; Kurosawa et al. 2011). The optically +thick accretion disc is not included in our models. The effect of +the disc emission and absorption on the spectroscopic and inter- +ferometric observables will be discussed in a subsequent paper. +We also observe a decrease of the line flux as the obliquity +increases. Figure 4 shows the radius encompassing 90% of the +total line flux, R90, at an inclination of 60◦ for non-axisymmetric +models with different obliquities for the Hα, Hβ, Paβ and Brγ +lines. +5 +10 +15 +20 +25 +30 +35 +40 +ma [ ] +2.8 +3.0 +3.2 +3.4 +3.6 +3.8 +R90 [R ] +H +H +Pa +Br +Fig. 4: Radius encompassing 90% of the total flux (R90) for each +line as a function of the obliquity, βma. Hydrogen lines are la- +belled with different colours. +As βma increases, the volume of the magnetospheric accre- +tion region decreases because the arc length of the accreting field +lines shortens. Therefore, the total flux, for all lines, decreases +accordingly, independently of the viewing angle of the system. +However, we also note a dependence of R90 with the line. The +Article number, page 3 of 13 + +A&A proofs: manuscript no. bt_spidi2 +II +III +IV +I +200 +0 +200 +v [km. s +1] +1.0 +1.2 +1.4 +F/Fc +I +II +III +IV +V +V +0% +5% +30% +50% +80% +Fig. 2: Origin of the emission seen across the Brackett γ line. The contribution of individual images to the total line flux is indicated +on the central image showing the line profile. The brightness maps are in units of the maximum emission. The emission of the stellar +surface is saturated. Orange to red colours indicates the regions of maximum emission. The system is seen at an inclination of 30◦ +and an rotational phase of ∼0.25, similar to Fig. 1. +value of R90 represents the size of the emitting region in a given +line, which is a function of density and temperature, and of the +viewing angle. +5. Interferometric signatures +In this section, we compute the size of the Brγ line-emitting re- +gion inferred from interferometric observations, and we compare +it to model flux radii (see §4). +5.1. Interferometric observables +The interferometric observables are derived from the radiative +transfer (RT) model using the ASPRO25 software developed by +the Jean-Marie Mariotti Center (JMMC). These observables rep- +resent what would be observed with GRAVITY in the near- +infrared. We consider the configuration obtained with the Very +Large Telescope (i.e. 4x8m telescopes), encompassing a range +of baselines from 35 to 135 m. With a typical night of 8 hours, +we compute one observing point per hour for the six baselines +5 Available at https://www.jmmc.fr +of the VLTI to increase the Fourier sampling, namely u-v cov- +erage, which is crucial for the fitting part of our approach. As +described in Bourgès & Duvert (2016), we derive the observ- +ables from the RT images (see Fig. 2) by computing the com- +plex visibility in each spectral channel around the Brγ line and +interpolating them to match GRAVITY’s spectral resolution (R += 4000). Specifically, we simulate a total of 37 spectral chan- +nels (from 2.161 to 2.171 µm with a step of 2.8 10−4 µm) for the +six projected baselines repeated eight times. Within this range, +31 spectral channels are used to measure the K-band continuum +and six channels sample the Brγ line emitting region. +Figure B.1 illustrates the resulting u-v plane projected on- +sky for a typical object observed at the VLTI with a declination +of -34◦ (e.g. TW Hydrae). +Figure 5 shows the interferometric observables along the ro- +tational cycle for a model with an inclination of 60◦ and an obliq- +uity of 10◦. The two main observables are: the modulus of the +complex visibility – the visibility amplitude – and the differen- +tial phase – its argument – dispersed in wavelength. The phase +is normalised to zero in the continuum. The visibility amplitude +can then be used to estimate the object’s size, while the phase +measures the photo-centre shifts between the line-emitting re- +Article number, page 4 of 13 + +B. Tessore & A. Soulain et al.: Spectroscopic and interferometric signatures of magnetospheric accretion +250 +125 +0 +125 +250 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +1.4 +phase = 0.03 +250 +125 +0 +125 +250 +phase = 0.25 +250 +125 +0 +125 +250 +phase = 0.47 +250 +125 +0 +125 +250 +phase = 0.69 +250 +125 +0 +125 +250 +phase = 0.92 +ma = 5 +ma = 10 +ma = 20 +ma = 30 +ma = 40 +v [km s +1] +F/Fc +Fig. 3: Brackett γ line variability along the rotational cycle. Each column corresponds to a specific rotational phase. At phase 0, +the shock area is unseen on the stellar surface, while phase of 0.5, the shock is fully seen on the visible hemisphere. The colours +correspond to different values of the obliquity. All fluxes are computed with an inclination of 60◦. +gion and the continuum. The phase can only be used as a rela- +tive measurement (e.g. between the line and the continuum), the +absolute phase being lost due to a combination of atmospheric +and instrumental effects. We repeat the simulated observations +and compute nine datasets over a rotational cycle sampled every +40 degrees ( 0.11 in phase). In this study, we are interested in the +line’s emitting region only. Therefore, we use pure line quanti- +ties, instead of total visibilities and phases, to remove the contri- +bution from the stellar surface (see appendix B for the derivation +of the pure line interferometric quantities). +5.2. Physical characteristic and sizes +Once the interferometric observables are computed, we apply +standard modelling methods to interpret the data (Berger 2003). +Firstly, we average the visibility amplitude of the six spectral +channels available within the Brγ line6. We use the average vis- +ibilities to recover the global size of the Brγ emitting region, +where the different velocities probe specific parts of the mov- +ing material within the magnetosphere. Then, we fit the aver- +aged visibility amplitude using elongated Gaussian or uniform +disc models. Such models are typically used in interferometry to +estimate the system’s characteristic size and on-sky orientation. +The source’s brightness distribution is defined by its half-flux ra- +dius in the case of a Gaussian disc or its radius for the uniform +disc model, and an elongation factor and a position angle. In the +following, we adopt the definition of "radius" for both models, +which corresponds to the half-flux semi-major axis for the Gaus- +sian model and the semi-major axis for the uniform disc model. +The recovered sizes and orientations are represented in the top +panel of Fig. 5. While neither model can fully account for the +size of the magnetosphere, the uniform disc probes a larger area +of the magnetosphere, while the Gaussian disc seems limited to +the most luminous parts. We note that the fit of the visibility is +6 Five and four spectral channels only were used at phase 0.25 and +0.47, respectively, due to a limited line-to-continuum ratio (see Ap- +pendix B for details). +equally good for both models and, thus, does not allow us to dis- +criminate between the models from the synthetic visibilities only +(middle-top, Fig. 5). +In order to quantify the physical meaning of the interfero- +metric measurements, we compare the interferometric sizes with +reference flux radii of the RT models. We set these radii to repre- +sent 50, 80, 90, and 99% of the total flux emitted by the magne- +tospheric accretion region. Figure 6 compares the sizes derived +with interferometry to the characteristic radii of the RT models. +We find that the size derived from the uniform disc model is +modulated around an average value of 3.5 R∗ corresponding to +90% of the Brγ emitting region. The size obtained by interferom- +etry appears to be modulated by the position of the funnel flows +close to the star, with a minimum located around phase 0.8. The +Gaussian model exhibits the same modulation but with a lower +amplitude (2.1 ± 0.4 R∗) and appears sensitive to the magneto- +sphere’s innermost region, close to the 50% flux radius. The size +derived from the uniform disc model emerges as being the most +appropriate to recover the reference model size, accounting for +at least 80% of the total flux emitted by the magnetosphere. +Article number, page 5 of 13 + +A&A proofs: manuscript no. bt_spidi2 +0.0 +0.2 +0.4 +0.6 +0.8 +Phase +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +Radius [R +] +Rt + +r +50% +80% +90% +99% +Uniform disc +Gaussian disc +Fig. 6: Interferometric radii as a function of the rotational phase. +Uniform and Gaussian disc models are shown with green and +yellow markers, respectively. Blue lines correspond to the radii +encompassing 50, 80, 90 and 99% of the total RT model’s flux. +The blue shaded areas represent the standard deviation of these +radii across the rotational phase. +The red shaded area indicates the inner (Rt) and outer radius (Rt+ +δr) of the RT model. +The derived orientations obtained from interferometry seem +to be particularly representative of the position of the accretion +funnel flow and the on-sky orientation of the Brγ emitting re- +gion (Fig. 5). The measured position angle agrees with the mag- +netosphere’s orientation, particularly when the shock faces the +observer (phase = 0.5). Nevertheless, it appears somewhat haz- +ardous to decipher the shape and orientation of the emitting re- +gion across the rotational cycle from this observable only, as +different magnetospheric configurations can be described by a +very similar interferometric model (e.g. phases 0.03 and 0.25). +A stronger constraint on the orientation of the funnel flows arises +from differential phase measurements. +5.3. Differential phases and photo-centre shifts +From the differential phases, we can derive the photo-centre shift +between the continuum and the Brγ line emitting region. In the +regime of marginally resolved sources, there is a direct relation- +ship between the projected photo-centre displacement vector (P) +and the phase along each baseline (Lachaume 2003): +φi = −2π Bi +λ P, +(3) +where φi is the differential phase measured for the ith baseline, +Bi is the length of the corresponding baseline, and λ is the effec- +tive wavelength of the spectral channel. A four telescope beam- +combiner like GRAVITY gives access to six projected baselines +that enable us to accurately retrieve the value and orientation +of the photo-centre shifts in each spectral channel (Le Bouquin +et al. 2009; Waisberg et al. 2017). Such a measurement results in +a position-velocity plot of the displacement of the photo-centre +across the Brγ line relative to the continuum. This is illustrated +in the bottom panels of Figure 5. +The photo-centre shifts trace the accretion funnel flow’s di- +rection and follow the stellar rotation. For instance, when the +northern accretion shock (N-shock) is located behind the star +(phase = 0), the accreting material falls onto the stellar surface in +the direction of the observer. Accordingly, the photo-centre mea- +sured in the blue-shifted part of the line profile (≃ -75 km s−1) +lies on the blue-shifted part of the velocity map, corresponding +to the approaching funnel flow. Equivalently, the photo-centre +measured in positive velocity channels of the line profile (≃ ++75 km s−1) is shifted towards the receding funnel flow. In con- +trast, when the shock faces the observer (phase = 0.5), the veloc- +ity map goes from blue to red in the east-west direction, and the +photo-centre shifts recover this trend as demonstrated at phase +0.47. +We can thus identify three privileged directions and shapes +of the photo-centre shifts: – linear north-south at phase ≃ 0 (N- +shock behind), – S-shape at phase 0.25 and 0.69 and – linear +east-west at phase ≃ 0.5 (N-shock in front). The differential +phase is, therefore, a key ingredient to recover the geometry and +orientation of the line-emitting region, tracing the moving mate- +rial along a rotational cycle. +5.4. Signal-to-noise considerations +As a proof-of-concept, the results presented above assume infi- +nite signal-to-noise ratio. The goal is to predict the spectroscopic +and interferometric signatures of the magnetospheric accretion +process. Thus, the models predict typical visibility amplitudes +ranging from 1 down to 0.97 (see Fig. 5). Such a modest inter- +ferometric signal requires a measurement accuracy of about 1% +to be securely detected. Similarly, the models predict a deviation +of the differential phases by 1 to 2 degrees (Fig. 5), which re- +quires an accuracy of order of a fraction of a degree to yield a +robust detection. Recent interferometric studies performed with +VLTI/GRAVITY in the K-band demonstrate that these levels of +accuracy can be routinely obtained indeed with reasonable ex- +posure times on young stellar objects (e.g. Bouvier et al. 2020b; +Gravity Collaboration et al. 2020, 2022), or active galactic nuclei +(Gravity Collaboration et al. 2018). +6. Summary and conclusion +We presented non-LTE radiative transfer modelling of the Brack- +ett γ line emission for non-axisymmetric models of accreting +magnetospheres. We used the equations of a misaligned dipo- +lar magnetic field to derive the geometry of the magnetospheric +accretion region for different obliquities of the magnetic dipole. +We used MCFOST to compute radiative signatures of the Brγ +line along a full stellar rotational cycle. Further, we derived near- +infrared interferometric observables for the line, comparable to +what the GRAVITY instrument has already measured for T Tauri +stars. +The main conclusions of this study are the following: +1) The total flux in the line, and the line-to-continuum ratio, +depends on the obliquity of the dipole. As the obliquity in- +creases, the size of the emitting region decreases, leading +to a lower integrated flux. Also, projection effects make the +emission region of lines forming close to the stellar surface +appearing narrower. +2) The Brγ line total flux varies with the rotational phase due to +the non-axisymmetry of the models induced by the magnetic +obliquity. The line profiles exhibit a red-shifted absorption, +that is an inverse P Cygni profile, when a significant fraction +of the accretion shock is aligned with the observer’s line of +sight. When the shock is hidden on the opposite side of the +star, the line profiles exhibit a double-peaked shape, reminis- +cent of the lines formed in rotating envelope. The latter is +due to the relatively large rotational velocity of the magneto- +spheric model (∼80 km s−1). +Article number, page 6 of 13 + +B. Tessore & A. Soulain et al.: Spectroscopic and interferometric signatures of magnetospheric accretion +3) Near-infrared interferometric observations in the Brγ line di- +rectly probe the size of the magnetospheric accretion region. +The Gaussian disc model is sensitive to the brightest parts of +the magnetosphere, up to 50% of the truncation radius, while +a uniform disc model grasps 90% of the magnetosphere. It is +of prime importance to consider this aspect when estimating +magnetospheric radius from interferometric measurements. +In both cases, the measured radius varies with the rotational +phase (due to the non-axisymmetry of the dipole). A robust +interferometric estimate of the magnetospheric radius there- +fore requires monitoring the system over a full rotational cy- +cle. +4) The combined knowledge of the differential phase and the +associated photo-centre shifts gives hints on the object ori- +entation and geometry. More specifically, the relative direc- +tion of the photo-centre shifts indicates the changing orien- +tation of the accreting material along the rotational cycle in +the non-axisymmetric case. +Near-infrared interferometry of the Brackett γ line is used +to characterise the inner star-disc interaction region, and offers a +good estima te of the size of the line’s forming region, at sub-au +precision. Comparing this size with reference model radii, such +as the truncation radius, allows us to distinguish between mul- +tiple origins of the Brγ line, within or beyond these radii (e.g. +magnetosphere, stellar and disc winds, jets). Further, simultane- +ous spectroscopic and interferometric observations along a rota- +tional cycle, have the potential to unveil the geometry and ori- +entation of the line’s emitting region. The variability of the line +associated with the photo-centre shifts, provides a unique and +unambiguous proxy of the physical processes occurring in the +magnetosphere of young accreting systems, within a few hun- +dredths of an astronomical unit around the central star. +Article number, page 7 of 13 + +A&A proofs: manuscript no. bt_spidi2 +Fig. 5: Synthetic interferometric measurements and modelling across the rotational phase of the system. Top: integrated images over the Brγ line. Green (uniform disc) and +yellow (Gaussian disc) ellipses are the characteristic sizes measured with a GRAVITY-like instrument. Middle-top: Pure line visibility amplitude observables associated with +the corresponding models. The visibility variation (for a given baseline) as the u-v plane rotates is the specific signature of an elongated object. Middle-bottom: Pure phase +visibility across the line profile for the six baselines of the VLTI. Colours encode the observing time. Bottom: Velocity map of the radiative transfer model. The coloured dots +represent the measurement of the photo-centre derived from the phase visibility in each available spectral channel (see appendix B). In each figure, the magnetosphere and the +stellar surface have been normalised independently, for display purpose. +Article number, page 8 of 13 + +phase = 0.03 +phase = 0.47 +phase = 0.92 +rud = 3.60 R* +phase = 0.25 +rud = 3.88 R* +phase = 0.69 +rud = 3.06 R* +rud = 3.30 R* +rgd = 2.12 R +rgd = 2.43 R, +rgd = 2.29 R* +rgd = 1.80 R* +1.95 R +0.8 +0.8 +0.8 +0.8 +0.8 +0.6 +0.6 +0.6 +0.6 +0.6 +0. +0.01 AU +0.01 AU +0.01 AU +0.01 AU +0.01 AU +1.00- +1.00- +1.00 +1.00 +1.00 +0.99 +m0.991 +≥0.99 - +visibil +visibil +visibil +visibil +visibil +.-GD fit +..GD fit +.. GD fit +.. GD fit +.-GD fit +UD fit +UD fit +UD fit +UD fit +UD fit +UT1-UT2 +In-TIn +UT1-UT2 +UT1-UT2 +UT1-UT2 +UT1-UT3 +UT1-UT4 +UT1-UT4 +UT1-UT4 +UT1-UT4 +UT1-UT4 +UT2-UT3 +UT2-UT3 +UT2-UT3 +UT2-UT3 +UT2-UT3 +-L60 +F260 +L60 +L60 +-L60 +. +UT2-UT4 +UT2-UT4 +UT2-UT4 +UT2-UT4 +UT3-UT4 +40 +120 +2 0 +50 +50 +50 +Diff. Φ [deg] +Diff. [deg] +Diff. [deg] +Diff. [deg] +Diff. [deg +UT2 +Φ [deg] +p[deg] +[deg] +[deg] +[deg] +Diff. +[deg] +[deg] +[deg] +[deg] +[deg] +Diff. +Diff. +Diff. +Diff. +Diff. +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +100 +.00 +100 +100 +elocity [km/s] +Velocity [km/s] +elocity [km/s] +F00m +F00m +F00m +-00m +-00m +75 +75 +75 +75 +200- +200- +Photocenter shift[μas] +-002 +200 +Photocenter shift [μas] +50 +[μas] +[μas] +[μas] +100- +100- +100- +100- +hotocenter shift [" +100- +hotocenter shift [ +hotocenter shift [ +25 +25 +25 +25 +-0 +-100 +-100- +-100- +-100- +-100- +-50 +-50 +-50 +-50 +-50 +-200- +-200- +-200- +200- +-75 +-75 +-75 +-75 +-75 +-300- +-300- +-300- +-300 -200 - +-300 -200 - +-300 -200 - +-300 -200 - +-300 -200 - +-100 +300 +-100 +300 +-100 +300 +-100 +300 +-100 +300 +Photocenter shift [μasB. Tessore & A. Soulain et al.: Spectroscopic and interferometric signatures of magnetospheric accretion +Appendix A: benchmark +We present here the comparison between line profiles obtained +with MCFOST and previous studies. The magnetospheric model +corresponds to the axisymmetric and compact configuration of +Muzerolle et al. (2001) with a fixed shock temperature at 8 000 +K, a rotation period of 5 days and the following canonical T Tauri +parameters: M∗ = 0.8 M⊙, R∗ = 2 R⊙ and T∗ = 4 000 K. The in- +clination of the system is 60 degrees. The continuum emission +of the stellar surface (shock and photosphere) is constant for all +models. Figures A.1, A.2, A.3, and A.4 show the Hα, Hβ, Paβ +and Brγ lines profiles for different values of Tmax and ˙M. An in- +verse P Cygni profile, with a red-shifted absorption, is seen for +all lines although it is dependent on the value of the mass ac- +cretion rate and of the maximum temperature. For a given mass +accretion rate, an increase of the maximum temperature results +in a higher line emission peak and a shallower red-shifted ab- +sorption. As the temperature increases, the line source function +increases, which is the cause of a higher emission above the +continuum emission. The appearance of the red-shifted absorp- +tion component is caused by absorption from the gas above the +stellar surface. It is controlled by the ratio between the source +function of the line in the accretion funnel and that of the un- +derlying continuum from the stellar surface, especially at low +mass accretion rates and temperatures. Eventually, for the high- +est mass accretion rate and temperature, the lines become so op- +tically thick that the red-shifted absorption is washed out by the +large wings of the line. The red-shifted absorption is more pro- +nounced for lines forming closer to the accretion shock like the +Hβ line. At a temperature larger than 8,000 K and a mass accre- +tion rate above 10−8 M⊙ yr−1, the continuum emission from the +magnetosphere becomes important and the line-to-continuum ra- +tio decreases. This effect is seen for instance in the Hα line (see +Fig. A.1). When the mass accretion rate increases for a given +temperature, the density of the magnetosphere increases. As a +consequence, the line source function increases. At high temper- +ature and high density, the background continuum emission of +the magnetosphere dominates for certain wavelengths, and ab- +sorption occurs. The latter effect is seen in the Paβ (Fig. A.3) and +Brγ (Fig. A.4) lines where the strong continuum contribution at +the disc surface leads to absorption at low velocities, where the +lines source function is small. These results are consistent with +the previous studies and demonstrate the robustness of our code +for modelling the close environment of T Tauri stars (Tessore +et al. 2021). +Appendix B: Derivation of the interferometric +pure-line phase and visibility +We focus on the magnetospheric emission probed by the Brγ line +and, therefore, aim to remove any additional contributions (stel- +lar photosphere, the accretion shocks, dusty disc, etc...). Follow- +ing Kraus et al. (2008); Bouvier et al. (2020b), we compute the +continuum-subtracted observables, the so-called pure line visi- +bility and phase, by using the emission line profiles computed in +Sect. 3. This is of prime importance in the case of Brγ line as +the magnetospheric emission is quite faint in the infrared (≈ 1.3 +excess flux compared to the continuum, Fig. 3). The deriva- +tion of the pure line quantities is only possible if the source is +marginally resolved (i.e, size < λ/2B). +In this case, the pure line visibility Vline and phase Φline are +given by: +VLine = FL/CVTot − VCont +FL/C − 1 +, +(B.1) +ΦLine = arcsin +� +FL/C +FL/C − 1 +VTot +VLine +sin ΦTot +� +. +(B.2) +Where FL/C denotes the line-to-continuum flux ratio as taken +from the normalised spectrum (Fig. 3), VCont is the visibility +computed in the continuum (star+shock only), and VTot, ΦTot +are the total complex quantities measured by GRAVITY. In +Eq. (B.1), we note that in the case when FL/C is close to one, +the derived VLine cannot exist (converges to infinity). Such non- +ideal profiles appear if the red absorption becomes too important. +Therefore, we assume to discard the affected spectral channels +for phases 0.25 and 0.47, where FL/C is too close to one: – one +point (v = 53 km s−1) at phases 0.25 and – two points (v = 15, +53 km s−1) at phase 0.47. +Acknowledgements. The authors thank Claudio Zanni, Lucas Labadie, Cather- +ine Dougados, and Alexander Wojtczak for fruitful discussions. This project +has received funding from the European Research Council (ERC) under the +European Union’s Horizon 2020 research and innovation programme (grant +agreement No 742095; SPIDI: Star-Planets-Inner Disk-Interactions, http:// +www.spidi-eu.org). B. Tessore thanks the french minister of Europe and +foreign affairs and the minister of superior education, research and innova- +tion (MEAE and MESRI) for research funding through FASIC partnership. +C. Pinte acknowledges funding from the Australian Research Council via +FT170100040 and DP180104235. The numerical simulations presented in this +paper were performed with the Dahu supercomputer of the GRICAD infrastruc- +ture (https://gricad.univ-grenoble-alpes.fr), which is supported by Grenoble re- +search communities. +Article number, page 9 of 13 + +A&A proofs: manuscript no. bt_spidi2 +0.8 +1.0 +1.2 +M = 10 +9.0 M /yr +1.0 +1.5 +M = 10 +8.0 M /yr +1 +2 +6000 K +M = 10 +7.0 M /yr +0.75 +1.00 +1.25 +1 +2 +2 +4 +7000 K +1.0 +1.5 +2.0 +2 +4 +6 +5 +10 +8000 K +1000 +500 +0 +500 +1000 +2 +4 +1000 +500 +0 +500 +1000 +5 +10 +15 +1000 +500 +0 +500 +1000 +1 +2 +10000 K +H +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +Fig. A.1: Dependence of Hα line flux with mass accretion rates ˙M and maximum temperatures Tmax. The inclination of the system +is 60◦. +0.9 +1.0 +M = 10 +9.0 M /yr +0.8 +1.0 +M = 10 +8.0 M /yr +0.75 +1.00 +1.25 +6000 K +M = 10 +7.0 M /yr +0.8 +1.0 +1.0 +1.5 +1 +2 +3 +7000 K +0.75 +1.00 +1.25 +1 +2 +3 +2 +4 +6 +8000 K +1000 +500 +0 +500 +1000 +1.0 +1.5 +2.0 +1000 +500 +0 +500 +1000 +2.5 +5.0 +7.5 +1000 +500 +0 +500 +1000 +1 +2 +3 +10000 K +H +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +Fig. A.2: Same as Fig. A.1 for Hβ +Article number, page 10 of 13 + +B. Tessore & A. Soulain et al.: Spectroscopic and interferometric signatures of magnetospheric accretion +1.00 +1.02 +M = 10 +9.0 M /yr +0.98 +1.00 +1.02 +M = 10 +8.0 M /yr +0.9 +1.0 +1.1 +6000 K +M = 10 +7.0 M /yr +0.98 +1.00 +1.02 +1.0 +1.2 +1.0 +1.5 +7000 K +1.0 +1.1 +1.0 +1.5 +1 +2 +8000 K +1000 +500 +0 +500 +1000 +1.0 +1.2 +1.4 +1000 +500 +0 +500 +1000 +1 +2 +1000 +500 +0 +500 +1000 +1.0 +1.1 +1.2 +10000 K +Pa +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +Fig. A.3: Same as Fig. A.1 for Paβ. +0.99 +1.00 +1.01 +M = 10 +9.0 M /yr +0.99 +1.00 +1.01 +M = 10 +8.0 M /yr +1.00 +1.05 +6000 K +M = 10 +7.0 M /yr +0.99 +1.00 +1.01 +0.95 +1.00 +1.05 +1.0 +1.2 +1.4 +7000 K +1.000 +1.025 +1.0 +1.2 +1.0 +1.5 +8000 K +1000 +500 +0 +500 +1000 +1.0 +1.1 +1.2 +1000 +500 +0 +500 +1000 +1.0 +1.5 +1000 +500 +0 +500 +1000 +1.00 +1.05 +1.10 +10000 K +Br +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +v [km/s] +F/Fc +Fig. A.4: Same as Fig. A.1 for Brγ. +Article number, page 11 of 13 + +A&A proofs: manuscript no. bt_spidi2 +50 +25 +0 +25 +50 +U [M ] +60 +40 +20 +0 +20 +40 +60 +V [M ] +UT1-UT2 +UT1-UT3 +UT1-UT4 +UT2-UT3 +UT2-UT4 +UT3-UT4 +150 +100 +50 +0 +50 +100 +150 +V [m] - East +150 +100 +50 +0 +50 +100 +150 +U [m] (2.17 µm) - North +Fig. B.1: Fourier sampling (u-v coverage) of the simulated data. +The different colours correspond to the six different baselines +of the VLTI. 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P., et al. 2021, A&A, 649, A68 +Tessore, B., Pinte, C., Bouvier, J., & Ménard, F. 2021, A&A, 647, A27 +Waisberg, I., Dexter, J., Pfuhl, O., et al. 2017, ApJ, 844, 72 +Zanni, C. & Ferreira, J. 2009, A&A, 508, 1117 +Article number, page 13 of 13 + diff --git a/JNFJT4oBgHgl3EQfwC0U/content/tmp_files/load_file.txt b/JNFJT4oBgHgl3EQfwC0U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1143b6325a02f90a539ae520a1d951e8d0c6e49 --- /dev/null +++ b/JNFJT4oBgHgl3EQfwC0U/content/tmp_files/load_file.txt @@ -0,0 +1,1066 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf,len=1065 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' bt_spidi2 ©ESO 2023 January 30, 2023 Spectroscopic and interferometric signatures of magnetospheric accretion in young stars B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Tessore1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Soulain1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Pantolmos1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Bouvier1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Pinte1, 2, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Perraut1 1 Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 2 School of Physics and Astronomy, Monash University, VIC 3800, Australia xx/xx/xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' yy/yy/yy ABSTRACT Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We aim to assess the complementarity between spectroscopic and interferometric observations in the characterisation of the inner star-disc interaction region of young stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We use the code MCFOST to solve the non-LTE problem of line formation in non-axisymmetric accreting magnetospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We compute the Brγ line profile originating from accretion columns for models with different magnetic obliquities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We also derive monochromatic synthetic images of the Brγ line emitting region across the line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' This spectral line is a prime diagnostics of magnetospheric accretion in young stars and is accessible with the long baseline near-infrared interferometer GRAVITY installed at the ESO Very Large Telescope Interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We derive Brγ line profiles as a function of rotational phase and compute interferometric observables, visibilities and phases, from synthetic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The line profile shape is modulated along the rotational cycle, exhibiting inverse P Cygni profiles at the time the accretion shock faces the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The size of the line’s emission region decreases as the magnetic obliquity increases, which is reflected in a lower line flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We apply interferometric models to the synthetic visibilities in order to derive the size of the line-emitting region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We find the derived interferometric size to be more compact than the actual size of the magnetosphere, ranging from 50 to 90% of the truncation radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Additionally, we show that the rotation of the non-axisymmetric magnetosphere is recovered from the rotational modulation of the Brγ-to-continuum photo-centre shifts, as measured by the differential phase of interferometric visibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Based on the radiative transfer modelling of non-axisymmetric accreting magnetospheres, we show that simultaneous spectroscopic and interferometric measurements provide a unique diagnostics to determine the origin of the Brγ line emitted by young stellar objects and are ideal tools to probe the structure and dynamics of the star-disc interaction region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Radiative transfer – Line: profiles – Stars: variables: T Tauri, Herbig Ae/Be – Accretion, accretion disks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Introduction The early evolution of low mass stars (M∗ < 2 M⊙) during the classical T Tauri (CTT) phase depends on the interaction be- tween the star and its accretion disc, on a distance of a few stel- lar radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' At the truncation radius, matter from the disc surface is channelled onto the stellar surface following the magnetic field lines and forming an accretion funnel or column (Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Zanni & Ferreira 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Romanova & Owocki 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Pan- tolmos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The star-disc interaction is responsible for accretion and ejection phenomena that have a strong impact on spectral lines formed in the close vicinity of the star’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (1977) developed an analytical model of mag- netospheric accretion around a rotating neutron star with a dipo- lar magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (1994) applied this magne- tospheric accretion model to the formation of emission lines in the spectrum of T Tauri stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' This fundamental paper sets the general theoretical framework for the density and temperature distributions in aligned axisymmetric magnetospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The cou- pling between this representation of magnetospheric accretion in T Tauri systems with radiative transfer calculations has provided a crucial tool to interpret spectroscopic, photometric, and inter- ferometric observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The sensitivity of hydrogen lines to the parameters of the magnetospheric models was studied in detail by Muzerolle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (2001), improving the earlier calculations by Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Near-infrared observations of the Brackett γ (Brγ) line with the Very Large Telescope Interferometer (VLTI) GRAVITY in- strument (Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2017) also probe the inner part of the star-disc interaction region (Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' However, it is still difficult to associate the characteristic sizes derived from interferometry with the actual size of the magnetospheric accretion region, a key parameter in our understanding of the star-disc interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In this paper, we aim at studying the formation of the Brγ line and compute its spectroscopic and interferometric signa- tures for non-axisymmetric models of the inner star-disc inter- action region, akin to state-of-art MHD simulations (Romanova & Owocki 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In particular, we want to clarify the meaning of the sizes inferred through near-infrared interferometric obser- vations and how they compare with the overall size of the mag- netospheric accretion region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In sections §2 and §3, we describe the model used to com- pute the line formation in accreting magnetospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We discuss spectroscopic and interferometric signatures in sections §4 and §5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Radiative transfer framework We use the code MCFOST 1(Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2006, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Tessore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2021) to compute emergent line fluxes from multidimen- 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='com/cpinte/mcfost Article number, page 1 of 13 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='11628v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='SR] 27 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' bt_spidi2 sional models of magnetospheres for a 20-level hydrogen atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The atomic model, with 19 bound levels and the ground state of HII, consists of 171 bound-bound transitions (atomic lines) and 19 bound-free transitions (continua).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We focus here on the Brγ line at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1661 µm although, the Balmer lines Hα and Hβ and the Paschen β line (Paβ) are modelled as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' These specific hydrogen lines are commonly used to characterise accretion and ejection phenomena in young systems (Folha & Emerson 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Alencar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Pouilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Sousa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The method to solve for the non-LTE pop- ulations of hydrogen and the microphysics are the same as in Tessore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The updated version of the code we use now simultaneously solves the charge equation and the statis- tical equilibrium equations, which has been proven to increase the convergence in chromospheric conditions (Leenaarts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We tested our code for different magnetospheric models taken as benchmarks in Muzerolle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (2001) and Kurosawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The results of this comparison are presented and discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Magnetospheric accretion model Matter from the circumstellar disc is channelled onto the stel- lar surface along the dipolar magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The stellar magnetic field truncates the disc at a distance Rt from the star, the truncation radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In practice, the interaction between the stellar magnetic field and the disc takes place over a small re- gion between Rt and Rt + δr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Both Rt and δr are used to de- fine the size of the disc region magnetically connected to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' As the gas approaches the stellar surface, it decelerates in a shock and is heated at coronal temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Theoretical mod- els of accretion shocks by Calvet & Gullbring (1998) show that the optically thin emission of the pre/post-shock dominates be- low the Balmer jump and that the optically thick emission of the heated photosphere contributes to the total continuum emis- sion at larger wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In the following, we only consider the contribution of the heated photosphere to the shock radia- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The shock2 temperature is computed from the energy of the gas infalling onto the stellar surface following the prescription of Romanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (2004) unless specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' This approach as- sumes energy conservation and that the shock radiates as a black body, meaning that its temperature is determined by the specific kinetic energy and enthalpy of the gas deposited at the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The shock temperature hence derived is of the order of 4500 K - 6000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The stellar surface The stellar surface is considered as the inner boundary of the model and emits as a blackbody whose temperature is deter- mined by the stellar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Throughout the paper, the stellar parameters are T∗ = 4, 000 K, M∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 M⊙, and R∗ = 2 R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We set the distance to the star at 140 pc, which is typical of the near- est star forming regions such as Upper Scorpius (≈ 146 pc Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2018a) or Taurus (≈ 130 pc Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Geometry of the accretion funnels We consider 3D non-axisymmetric models of the magneto- spheric accretion region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' These models are parametrised by the same set of parameters as the axisymmetric magnetospheric 2 We assume that the shock region is unresolved and is part of the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 1: Density distribution of a non-axisymmetric model with an obliquity of 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The rotation axis of the star Ω is shown with a white arrow and the dipole axis, µ, with a red arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The density is computed from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The colour map scales with the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' model of Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (1994) (see also Muzerolle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 1998, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Kurosawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Lima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Kurosawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Dmitriev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='The density and the velocity fields of the accretion columns are fully described with a set of indepen- dent parameters: the mass accretion rate ˙M, the rotation period Prot, Rt, and δr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' For our study, the value of ˙M, Rt and δr, and of the temper- ature of the magnetosphere are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The impact of these pa- rameters on the line formation has been discussed thoroughly in Muzerolle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (1998, 2001, see also App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The line’s re- sponse to the mass accretion rate and to the temperature is an es- sential proxy for understanding the physics of the star-disc inter- action region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We use a mass accretion rate ˙M = 10−8 M⊙ yr−1, a truncation radius Rt = 4 R∗, and δr = 1 R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The value of the rotation period is deduced from the maximum truncation ra- dius (Rt + δr), imposing that stable accretion occurs at 90% of the corotation radius, consistent with the work of Blinova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The rotation period is therefore fixed at Prot = 6 days, corresponding to slowly rotating T Tauri stars (see Herbst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2014, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The rotational velocity for that period is thus of the order of 80 km s−1 at the outer edge of the magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' When the magnetic field axis (µ) is misaligned with respect to the rotational axis (Ω) of the star, the geometry of the accre- tion flow changes dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The equations for the magnetic field components of a non-axisymmetric dipole, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' with a non- zero obliquity, are provided in Mahdavi & Kenyon (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The parameter βma describes the angle between the dipole moment and the star’s rotational axis, the magnetic obliquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We approximate the density, ρ, along the non-axisymmetric magnetic field lines with, ρ = α B � = αB ρaxi Baxi , (1) where α is a constant along a given field line and B the ana- lytic misaligned dipolar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' �, ρaxi, and Baxi denote the veloc- ity field, density, and dipolar magnetic field, respectively, and they are taken from the axisymmetric model of Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In other words, the 3D density structure is computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (1) under the assumption that the infalling gas has a velocity field on the poloidal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The value of α is computed Article number, page 2 of 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content="2e-08 5e-9 [s-'b] 2e-9 1e-9 5e-10 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='3e-10 RtB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Tessore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Soulain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' : Spectroscopic and interferometric signatures of magnetospheric accretion from the numerical integration 3 of the mass flux over the shock area, ˙M = � ρv · dS, (2) where dS is the surface element and v the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In our model, the value of ˙M is an input parameter and is held constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Therefore, α is obtained to ensure consistency between Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We compute five models with an obliquity βma ranging from five to forty degrees in step of ten degrees, representative of what has been measured for T Tauri stars with spectroscopy (McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2020) and spectropolarimetry (Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2008, 2010, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Johnstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Pouilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' For these non-axisymmetric models, the shortest field lines – defin- ing the main accretion columns4 – carry most of the gas density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We remove the longest field lines – the secondary columns – in our modelling as in Esau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' This yields models with one crescent-shaped accretion spot per stellar hemisphere reminiscent of numerical simulations of misaligned dipoles (Ro- manova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Figure 1 shows the density of a non-axisymmetric magneto- sphere with an obliquity of 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Temperature of the funnels The temperature of the magnetospheric accretion region is not well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The determination of the temperature by Mar- tin (1996) from first principles was not able to reproduce the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A self-consistent calculation of the temperature of the magnetosphere is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Instead, we adopt here the temperature profile of Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (1994), which has been extensively used in the past to model line fluxes from accreting T Tauri stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The temperature is computed using a volumetric heating rate (∝ r−3) and balancing the energy input with the radiative cooling rates of Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The exact balance between the heating and cooling mechanisms is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Instead, the temperature profile is normalised to a free parameter, Tmax, that sets the value of the maximum temperature in the funnel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In the following, we have set the temperature maximum to Tmax = 8, 000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Spectroscopic signatures Thanks to the Doppler shift of the funnel flow, it is possible to reconstruct the origin of the emission line by looking at the brightness maps in various velocity channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Figure 2 shows the contribution of the different parts of the magnetosphere to the total integrated Brγ line flux at a given velocity for an in- clination of 30◦, matching the model illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' At those density and temperature, the continuum emission comes from the stellar surface (Isurf/Imag > 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Locally, the contin- uum emission from the shock is three times larger than the emis- sion from the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Overall, given the small covering area of the accretion shock (around 1%), the total continuum emission at the frequency of the Brγ line is dominated by the star’s radiation, Fshock/F∗ = 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The low-velocity components (< 50 km s−1) of the line form in the regions where the projected velocity along the line-of-sight is close to zero and near the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The geometry 3 For this 3D magnetospheric accretion model, an explicit formula for the shock area does not exist (see also Mahdavi & Kenyon 1998) 4 Geometrically, the shortest field lines obey the following criterion cos φ′ × z > 0 where φ′ is the azimuth in the frame aligned with the dipole axis and z the coordinate parallel to the rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' of the non-axisymmetric model, defined in §3, is responsible for a rotational modulation of the integrated line flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Many classical T Tauri stars shows modulated photometric variability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='g Cody et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2014) and, more directly related to the magnetospheric re- gion, many also show rotational modulation of the longitudinal component of the stellar magnetic field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='g Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Indeed, the periodic variability of optical and emission line pro- files has been reported in various systems (for instance Sousa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Alencar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2020a), which in- dicates that the emission region is stable on a timescale of several rotation periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Figure 3 shows the variability of the Brγ line at different phases of rotation at an inclination of 60◦ for different obliq- uities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The origin of the rotational phase is defined such that at phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5, the accretion shock is facing the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The red- shifted absorption seen for the Brγ line at phases 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='250, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='47 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='69, results from a lower source function of the gas above the shock (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' From observations, red-shifted absorption in the Paβ and Brγ lines are seen in less than 34% and 20% of the line profiles, respectively (Folha & Emerson 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The in- verse P Cygni profile disappears when the shock, or a significant fraction of it, is hidden on the opposite side of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The line, with either a double-peaked profile or a moderate red-shifted ab- sorption, is reminiscent of Reipurth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (1996) cases II and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' While the profiles with redshifted absorption agree with observa- tions, those that display an M-shape are usually not observed in young stellar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' This suggests that magnetospheric accre- tion is not the only contribution to the profile, which can also be impacted by various types of outflows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=', stellar, interface, and disk winds Lima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Kurosawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The optically thick accretion disc is not included in our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The effect of the disc emission and absorption on the spectroscopic and inter- ferometric observables will be discussed in a subsequent paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We also observe a decrease of the line flux as the obliquity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Figure 4 shows the radius encompassing 90% of the total line flux, R90, at an inclination of 60◦ for non-axisymmetric models with different obliquities for the Hα, Hβ, Paβ and Brγ lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 5 10 15 20 25 30 35 40 ma [ ] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='8 R90 [R ] H H Pa Br Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 4: Radius encompassing 90% of the total flux (R90) for each line as a function of the obliquity, βma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Hydrogen lines are la- belled with different colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' As βma increases, the volume of the magnetospheric accre- tion region decreases because the arc length of the accreting field lines shortens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Therefore, the total flux, for all lines, decreases accordingly, independently of the viewing angle of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' However, we also note a dependence of R90 with the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The Article number, page 3 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' bt_spidi2 II III IV I 200 0 200 v [km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' s 1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='4 F/Fc I II III IV V V 0% 5% 30% 50% 80% Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2: Origin of the emission seen across the Brackett γ line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The contribution of individual images to the total line flux is indicated on the central image showing the line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The brightness maps are in units of the maximum emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The emission of the stellar surface is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Orange to red colours indicates the regions of maximum emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The system is seen at an inclination of 30◦ and an rotational phase of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='25, similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' value of R90 represents the size of the emitting region in a given line, which is a function of density and temperature, and of the viewing angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Interferometric signatures In this section, we compute the size of the Brγ line-emitting re- gion inferred from interferometric observations, and we compare it to model flux radii (see §4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Interferometric observables The interferometric observables are derived from the radiative transfer (RT) model using the ASPRO25 software developed by the Jean-Marie Mariotti Center (JMMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' These observables rep- resent what would be observed with GRAVITY in the near- infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We consider the configuration obtained with the Very Large Telescope (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 4x8m telescopes), encompassing a range of baselines from 35 to 135 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' With a typical night of 8 hours, we compute one observing point per hour for the six baselines 5 Available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='jmmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='fr of the VLTI to increase the Fourier sampling, namely u-v cov- erage, which is crucial for the fitting part of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' As described in Bourgès & Duvert (2016), we derive the observ- ables from the RT images (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2) by computing the com- plex visibility in each spectral channel around the Brγ line and interpolating them to match GRAVITY’s spectral resolution (R = 4000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Specifically, we simulate a total of 37 spectral chan- nels (from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='161 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='171 µm with a step of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='8 10−4 µm) for the six projected baselines repeated eight times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Within this range, 31 spectral channels are used to measure the K-band continuum and six channels sample the Brγ line emitting region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1 illustrates the resulting u-v plane projected on- sky for a typical object observed at the VLTI with a declination of -34◦ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' TW Hydrae).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Figure 5 shows the interferometric observables along the ro- tational cycle for a model with an inclination of 60◦ and an obliq- uity of 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The two main observables are: the modulus of the complex visibility – the visibility amplitude – and the differen- tial phase – its argument – dispersed in wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The phase is normalised to zero in the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The visibility amplitude can then be used to estimate the object’s size, while the phase measures the photo-centre shifts between the line-emitting re- Article number, page 4 of 13 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Tessore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Soulain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' : Spectroscopic and interferometric signatures of magnetospheric accretion 250 125 0 125 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='4 phase = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='03 250 125 0 125 250 phase = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='25 250 125 0 125 250 phase = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='47 250 125 0 125 250 phase = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='69 250 125 0 125 250 phase = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='92 ma = 5 ma = 10 ma = 20 ma = 30 ma = 40 v [km s 1] F/Fc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 3: Brackett γ line variability along the rotational cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Each column corresponds to a specific rotational phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' At phase 0, the shock area is unseen on the stellar surface, while phase of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5, the shock is fully seen on the visible hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The colours correspond to different values of the obliquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' All fluxes are computed with an inclination of 60◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' gion and the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The phase can only be used as a rela- tive measurement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' between the line and the continuum), the absolute phase being lost due to a combination of atmospheric and instrumental effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We repeat the simulated observations and compute nine datasets over a rotational cycle sampled every 40 degrees ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='11 in phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In this study, we are interested in the line’s emitting region only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Therefore, we use pure line quanti- ties, instead of total visibilities and phases, to remove the contri- bution from the stellar surface (see appendix B for the derivation of the pure line interferometric quantities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Physical characteristic and sizes Once the interferometric observables are computed, we apply standard modelling methods to interpret the data (Berger 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Firstly, we average the visibility amplitude of the six spectral channels available within the Brγ line6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We use the average vis- ibilities to recover the global size of the Brγ emitting region, where the different velocities probe specific parts of the mov- ing material within the magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Then, we fit the aver- aged visibility amplitude using elongated Gaussian or uniform disc models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Such models are typically used in interferometry to estimate the system’s characteristic size and on-sky orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The source’s brightness distribution is defined by its half-flux ra- dius in the case of a Gaussian disc or its radius for the uniform disc model, and an elongation factor and a position angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In the following, we adopt the definition of "radius" for both models, which corresponds to the half-flux semi-major axis for the Gaus- sian model and the semi-major axis for the uniform disc model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The recovered sizes and orientations are represented in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' While neither model can fully account for the size of the magnetosphere, the uniform disc probes a larger area of the magnetosphere, while the Gaussian disc seems limited to the most luminous parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We note that the fit of the visibility is 6 Five and four spectral channels only were used at phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='25 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='47, respectively, due to a limited line-to-continuum ratio (see Ap- pendix B for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' equally good for both models and, thus, does not allow us to dis- criminate between the models from the synthetic visibilities only (middle-top, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In order to quantify the physical meaning of the interfero- metric measurements, we compare the interferometric sizes with reference flux radii of the RT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We set these radii to repre- sent 50, 80, 90, and 99% of the total flux emitted by the magne- tospheric accretion region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Figure 6 compares the sizes derived with interferometry to the characteristic radii of the RT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We find that the size derived from the uniform disc model is modulated around an average value of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 R∗ corresponding to 90% of the Brγ emitting region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The size obtained by interferom- etry appears to be modulated by the position of the funnel flows close to the star, with a minimum located around phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The Gaussian model exhibits the same modulation but with a lower amplitude (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='4 R∗) and appears sensitive to the magneto- sphere’s innermost region, close to the 50% flux radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The size derived from the uniform disc model emerges as being the most appropriate to recover the reference model size, accounting for at least 80% of the total flux emitted by the magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Article number, page 5 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' bt_spidi2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='8 Phase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 Radius [R ] Rt + r 50% 80% 90% 99% Uniform disc Gaussian disc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 6: Interferometric radii as a function of the rotational phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Uniform and Gaussian disc models are shown with green and yellow markers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Blue lines correspond to the radii encompassing 50, 80, 90 and 99% of the total RT model’s flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The blue shaded areas represent the standard deviation of these radii across the rotational phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The red shaded area indicates the inner (Rt) and outer radius (Rt+ δr) of the RT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The derived orientations obtained from interferometry seem to be particularly representative of the position of the accretion funnel flow and the on-sky orientation of the Brγ emitting re- gion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The measured position angle agrees with the mag- netosphere’s orientation, particularly when the shock faces the observer (phase = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Nevertheless, it appears somewhat haz- ardous to decipher the shape and orientation of the emitting re- gion across the rotational cycle from this observable only, as different magnetospheric configurations can be described by a very similar interferometric model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' phases 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='03 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A stronger constraint on the orientation of the funnel flows arises from differential phase measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Differential phases and photo-centre shifts From the differential phases, we can derive the photo-centre shift between the continuum and the Brγ line emitting region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In the regime of marginally resolved sources, there is a direct relation- ship between the projected photo-centre displacement vector (P) and the phase along each baseline (Lachaume 2003): φi = −2π Bi λ P, (3) where φi is the differential phase measured for the ith baseline, Bi is the length of the corresponding baseline, and λ is the effec- tive wavelength of the spectral channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A four telescope beam- combiner like GRAVITY gives access to six projected baselines that enable us to accurately retrieve the value and orientation of the photo-centre shifts in each spectral channel (Le Bouquin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Waisberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Such a measurement results in a position-velocity plot of the displacement of the photo-centre across the Brγ line relative to the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' This is illustrated in the bottom panels of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The photo-centre shifts trace the accretion funnel flow’s di- rection and follow the stellar rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' For instance, when the northern accretion shock (N-shock) is located behind the star (phase = 0), the accreting material falls onto the stellar surface in the direction of the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Accordingly, the photo-centre mea- sured in the blue-shifted part of the line profile (≃ -75 km s−1) lies on the blue-shifted part of the velocity map, corresponding to the approaching funnel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Equivalently, the photo-centre measured in positive velocity channels of the line profile (≃ +75 km s−1) is shifted towards the receding funnel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In con- trast, when the shock faces the observer (phase = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5), the veloc- ity map goes from blue to red in the east-west direction, and the photo-centre shifts recover this trend as demonstrated at phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We can thus identify three privileged directions and shapes of the photo-centre shifts: – linear north-south at phase ≃ 0 (N- shock behind), – S-shape at phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='25 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='69 and – linear east-west at phase ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 (N-shock in front).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The differential phase is, therefore, a key ingredient to recover the geometry and orientation of the line-emitting region, tracing the moving mate- rial along a rotational cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Signal-to-noise considerations As a proof-of-concept, the results presented above assume infi- nite signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The goal is to predict the spectroscopic and interferometric signatures of the magnetospheric accretion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Thus, the models predict typical visibility amplitudes ranging from 1 down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='97 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Such a modest inter- ferometric signal requires a measurement accuracy of about 1% to be securely detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Similarly, the models predict a deviation of the differential phases by 1 to 2 degrees (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 5), which re- quires an accuracy of order of a fraction of a degree to yield a robust detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Recent interferometric studies performed with VLTI/GRAVITY in the K-band demonstrate that these levels of accuracy can be routinely obtained indeed with reasonable ex- posure times on young stellar objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2020, 2022), or active galactic nuclei (Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Summary and conclusion We presented non-LTE radiative transfer modelling of the Brack- ett γ line emission for non-axisymmetric models of accreting magnetospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We used the equations of a misaligned dipo- lar magnetic field to derive the geometry of the magnetospheric accretion region for different obliquities of the magnetic dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' We used MCFOST to compute radiative signatures of the Brγ line along a full stellar rotational cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Further, we derived near- infrared interferometric observables for the line, comparable to what the GRAVITY instrument has already measured for T Tauri stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The main conclusions of this study are the following: 1) The total flux in the line, and the line-to-continuum ratio, depends on the obliquity of the dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' As the obliquity in- creases, the size of the emitting region decreases, leading to a lower integrated flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Also, projection effects make the emission region of lines forming close to the stellar surface appearing narrower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2) The Brγ line total flux varies with the rotational phase due to the non-axisymmetry of the models induced by the magnetic obliquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The line profiles exhibit a red-shifted absorption, that is an inverse P Cygni profile, when a significant fraction of the accretion shock is aligned with the observer’s line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' When the shock is hidden on the opposite side of the star, the line profiles exhibit a double-peaked shape, reminis- cent of the lines formed in rotating envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The latter is due to the relatively large rotational velocity of the magneto- spheric model (∼80 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Article number, page 6 of 13 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Tessore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Soulain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' : Spectroscopic and interferometric signatures of magnetospheric accretion 3) Near-infrared interferometric observations in the Brγ line di- rectly probe the size of the magnetospheric accretion region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The Gaussian disc model is sensitive to the brightest parts of the magnetosphere, up to 50% of the truncation radius, while a uniform disc model grasps 90% of the magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' It is of prime importance to consider this aspect when estimating magnetospheric radius from interferometric measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In both cases, the measured radius varies with the rotational phase (due to the non-axisymmetry of the dipole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A robust interferometric estimate of the magnetospheric radius there- fore requires monitoring the system over a full rotational cy- cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 4) The combined knowledge of the differential phase and the associated photo-centre shifts gives hints on the object ori- entation and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' More specifically, the relative direc- tion of the photo-centre shifts indicates the changing orien- tation of the accreting material along the rotational cycle in the non-axisymmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Near-infrared interferometry of the Brackett γ line is used to characterise the inner star-disc interaction region, and offers a good estima te of the size of the line’s forming region, at sub-au precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Comparing this size with reference model radii, such as the truncation radius, allows us to distinguish between mul- tiple origins of the Brγ line, within or beyond these radii (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' magnetosphere, stellar and disc winds, jets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Further, simultane- ous spectroscopic and interferometric observations along a rota- tional cycle, have the potential to unveil the geometry and ori- entation of the line’s emitting region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The variability of the line associated with the photo-centre shifts, provides a unique and unambiguous proxy of the physical processes occurring in the magnetosphere of young accreting systems, within a few hun- dredths of an astronomical unit around the central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Article number, page 7 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' bt_spidi2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 5: Synthetic interferometric measurements and modelling across the rotational phase of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Top: integrated images over the Brγ line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Green (uniform disc) and yellow (Gaussian disc) ellipses are the characteristic sizes measured with a GRAVITY-like instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Middle-top: Pure line visibility amplitude observables associated with the corresponding models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The visibility variation (for a given baseline) as the u-v plane rotates is the specific signature of an elongated object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Middle-bottom: Pure phase visibility across the line profile for the six baselines of the VLTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Colours encode the observing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Bottom: Velocity map of the radiative transfer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The coloured dots represent the measurement of the photo-centre derived from the phase visibility in each available spectral channel (see appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In each figure, the magnetosphere and the stellar surface have been normalised independently, for display purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Article number, page 8 of 13 phase = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='03 phase = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='47 phase = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='92 rud = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='60 R* phase = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='25 rud = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='88 R* phase = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='69 rud = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='06 R* rud = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='30 R* rgd = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='12 R rgd = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='43 R, rgd = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='29 R* rgd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='80 R* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='95 R 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='99 m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='991 ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='99 - visibil visibil visibil visibil visibil .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='-GD fit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='.GD fit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='. GD fit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='. GD fit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='-GD fit UD fit UD fit UD fit UD fit UD fit UT1-UT2 In-TIn UT1-UT2 UT1-UT2 UT1-UT2 UT1-UT3 UT1-UT4 UT1-UT4 UT1-UT4 UT1-UT4 UT1-UT4 UT2-UT3 UT2-UT3 UT2-UT3 UT2-UT3 UT2-UT3 L60 F260 L60 L60 L60 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' UT2-UT4 UT2-UT4 UT2-UT4 UT2-UT4 UT3-UT4 40 120 2 0 50 50 50 Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Φ [deg] Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' [deg] Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' [deg] Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' [deg] Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' [deg UT2 Φ [deg] p[deg] [deg] [deg] [deg] Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' [deg] [deg] [deg] [deg] [deg] Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 100 100 100 100 100 100 100 100 100 100 100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='elocity [km/s] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='Velocity [km/s] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='elocity [km/s] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='F00m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='F00m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='F00m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='200- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='200- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='Photocenter shift[μas] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='Photocenter shift [μasB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Tessore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Soulain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' : Spectroscopic and interferometric signatures of magnetospheric accretion Appendix A: benchmark We present here the comparison between line profiles obtained with MCFOST and previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The magnetospheric model corresponds to the axisymmetric and compact configuration of Muzerolle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (2001) with a fixed shock temperature at 8 000 K, a rotation period of 5 days and the following canonical T Tauri parameters: M∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='8 M⊙, R∗ = 2 R⊙ and T∗ = 4 000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The in- clination of the system is 60 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The continuum emission of the stellar surface (shock and photosphere) is constant for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Figures A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='3, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='4 show the Hα, Hβ, Paβ and Brγ lines profiles for different values of Tmax and ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' An in- verse P Cygni profile, with a red-shifted absorption, is seen for all lines although it is dependent on the value of the mass ac- cretion rate and of the maximum temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' For a given mass accretion rate, an increase of the maximum temperature results in a higher line emission peak and a shallower red-shifted ab- sorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' As the temperature increases, the line source function increases, which is the cause of a higher emission above the continuum emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The appearance of the red-shifted absorp- tion component is caused by absorption from the gas above the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' It is controlled by the ratio between the source function of the line in the accretion funnel and that of the un- derlying continuum from the stellar surface, especially at low mass accretion rates and temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Eventually, for the high- est mass accretion rate and temperature, the lines become so op- tically thick that the red-shifted absorption is washed out by the large wings of the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The red-shifted absorption is more pro- nounced for lines forming closer to the accretion shock like the Hβ line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' At a temperature larger than 8,000 K and a mass accre- tion rate above 10−8 M⊙ yr−1, the continuum emission from the magnetosphere becomes important and the line-to-continuum ra- tio decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' This effect is seen for instance in the Hα line (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' When the mass accretion rate increases for a given temperature, the density of the magnetosphere increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' As a consequence, the line source function increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' At high temper- ature and high density, the background continuum emission of the magnetosphere dominates for certain wavelengths, and ab- sorption occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The latter effect is seen in the Paβ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='3) and Brγ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='4) lines where the strong continuum contribution at the disc surface leads to absorption at low velocities, where the lines source function is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' These results are consistent with the previous studies and demonstrate the robustness of our code for modelling the close environment of T Tauri stars (Tessore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Appendix B: Derivation of the interferometric pure-line phase and visibility We focus on the magnetospheric emission probed by the Brγ line and, therefore, aim to remove any additional contributions (stel- lar photosphere, the accretion shocks, dusty disc, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Follow- ing Kraus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (2020b), we compute the continuum-subtracted observables, the so-called pure line visi- bility and phase, by using the emission line profiles computed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' This is of prime importance in the case of Brγ line as the magnetospheric emission is quite faint in the infrared (≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='3 excess flux compared to the continuum, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The deriva- tion of the pure line quantities is only possible if the source is marginally resolved (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='e, size < λ/2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In this case, the pure line visibility Vline and phase Φline are given by: VLine = FL/CVTot − VCont FL/C − 1 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1) ΦLine = arcsin � FL/C FL/C − 1 VTot VLine sin ΦTot � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2) Where FL/C denotes the line-to-continuum flux ratio as taken from the normalised spectrum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 3), VCont is the visibility computed in the continuum (star+shock only), and VTot, ΦTot are the total complex quantities measured by GRAVITY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1), we note that in the case when FL/C is close to one, the derived VLine cannot exist (converges to infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Such non- ideal profiles appear if the red absorption becomes too important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Therefore, we assume to discard the affected spectral channels for phases 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='25 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='47, where FL/C is too close to one: – one point (v = 53 km s−1) at phases 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='25 and – two points (v = 15, 53 km s−1) at phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The authors thank Claudio Zanni, Lucas Labadie, Cather- ine Dougados, and Alexander Wojtczak for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 742095;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' SPIDI: Star-Planets-Inner Disk-Interactions, http:// www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='spidi-eu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Tessore thanks the french minister of Europe and foreign affairs and the minister of superior education, research and innova- tion (MEAE and MESRI) for research funding through FASIC partnership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Pinte acknowledges funding from the Australian Research Council via FT170100040 and DP180104235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The numerical simulations presented in this paper were performed with the Dahu supercomputer of the GRICAD infrastruc- ture (https://gricad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='univ-grenoble-alpes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='fr), which is supported by Grenoble re- search communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Article number, page 9 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' bt_spidi2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2 M = 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 M /yr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 M = 10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 M /yr 1 2 6000 K M = 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 M /yr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='25 1 2 2 4 7000 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 2 4 6 5 10 8000 K 1000 500 0 500 1000 2 4 1000 500 0 500 1000 5 10 15 1000 500 0 500 1000 1 2 10000 K H v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1: Dependence of Hα line flux with mass accretion rates ˙M and maximum temperatures Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The inclination of the system is 60◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 M = 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 M /yr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 M = 10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 M /yr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='25 6000 K M = 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 M /yr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 1 2 3 7000 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='25 1 2 3 2 4 6 8000 K 1000 500 0 500 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1000 500 0 500 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 1000 500 0 500 1000 1 2 3 10000 K H v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2: Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1 for Hβ Article number, page 10 of 13 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Tessore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Soulain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' : Spectroscopic and interferometric signatures of magnetospheric accretion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='02 M = 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 M /yr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='02 M = 10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 M /yr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1 6000 K M = 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 M /yr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 7000 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 1 2 8000 K 1000 500 0 500 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='4 1000 500 0 500 1000 1 2 1000 500 0 500 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2 10000 K Pa v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='3: Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1 for Paβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='01 M = 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 M /yr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='01 M = 10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 M /yr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='05 6000 K M = 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 M /yr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='4 7000 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 8000 K 1000 500 0 500 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='2 1000 500 0 500 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='5 1000 500 0 500 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='10 10000 K Br v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc v [km/s] F/Fc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='4: Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1 for Brγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Article number, page 11 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' bt_spidi2 50 25 0 25 50 U [M ] 60 40 20 0 20 40 60 V [M ] UT1-UT2 UT1-UT3 UT1-UT4 UT2-UT3 UT2-UT4 UT3-UT4 150 100 50 0 50 100 150 V [m] - East 150 100 50 0 50 100 150 U [m] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='17 µm) - North Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content='1: Fourier sampling (u-v coverage) of the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The different colours correspond to the six different baselines of the VLTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' The eight points per baseline represent a typical observational sequence with one data point per hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Article number, page 12 of 13 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Tessore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' Soulain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' : Spectroscopic and interferometric signatures of magnetospheric accretion References Alencar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' & Ferreira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} +page_content=' 2009, A&A, 508, 1117 Article number, page 13 of 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfwC0U/content/2301.11628v1.pdf'} diff --git a/JdE4T4oBgHgl3EQf7Q53/content/tmp_files/2301.05338v1.pdf.txt b/JdE4T4oBgHgl3EQf7Q53/content/tmp_files/2301.05338v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7ab2f5cdbc6496cbd0e970afebbcda2c387cb99e --- /dev/null +++ b/JdE4T4oBgHgl3EQf7Q53/content/tmp_files/2301.05338v1.pdf.txt @@ -0,0 +1,510 @@ +arXiv:2301.05338v1 [cs.DS] 13 Jan 2023 +Computing matching statistics on Wheeler DFAs +Alessio Conte1, Nicola Cotumaccio2,3, Travis Gagie3, Giovanni Manzini1, +Nicola Prezza4 and Marinella Sciortino5 +1 University of Pisa, Italy, alessio.conte@unipi.it, giovanni.manzini@unipi.it +2 GSSI, L’Aquila, Italy, nicola.cotumaccio@gssi.it +3 Dalhousie University, Halifax, Canada, nicola.cotumaccio@dal.ca, travis.gagie@dal.ca +4 Ca’ Foscari Unversity, Venice, Italy, nicola.prezza@unive.it +5 University of Palermo, Italy, marinella.sciortino@unipa.it +Abstract +Matching statistics were introduced to solve the approximate string matching problem, which is +a recurrent subroutine in bioinformatics applications. +In 2010, Ohlebusch et al. [SPIRE 2010] +proposed a time and space efficient algorithm for computing matching statistics which relies on +some components of a compressed suffix tree - notably, the longest common prefix (LCP) array. In +this paper, we show how their algorithm can be generalized from strings to Wheeler deterministic +finite automata. Most importantly, we introduce a notion of LCP array for Wheeler automata, thus +establishing a first clear step towards extending (compressed) suffix tree functionalities to labeled +graphs. +Introduction +Given a string T and a pattern π, the classical formulation of the pattern matching problem +requires to decide whether the pattern π occurs in the string T and, possibly, count the +number of such occurrences and report the positions where they occur. The invention of +the FM-index [1], which is based on the Burrows-Wheeler transform [2], opened a new +line of research in the pattern matching field. The indexing and compression techniques +behind the FM-index deeply rely on the idea of suffix sorting, and over the years have +been generalized from strings to trees [3], De Brujin graphs [4,5], Wheeler graphs [6,7] and +arbitrary graphs [8, 9]. In particular, the class of Wheeler graphs is probably the one that +captures the intuition behind the FM-index in the simplest way, and indeed the notion of +Wheeler order has relevant consequences in automata theory [7,10]. +However, in bioinformatics we are not only interested in exact pattern matching, but +also in a myriad of variations of the pattern matching problem [11]. In particular, matching +statistics were introduced to solve the approximate pattern matching problem [12]. A pow- +erful data structure that is able to address the variations of the pattern matching problem +at once is the suffix tree [13]. The main drawback of the suffix tree is its space consumption, +which is non-negligible both in theory and in practice. As a consequence, the suffix tree has +been replaced by the suffix array [14]. While suffix arrays do not have all the functionalities +of suffix trees, it has been shown that they can be augmented with some additional data +structures — notably, the longest common prefix (LCP) array — so that it is possible to +retrieve the full functionalities of a suffix trees [15]. All these components can be successfully +compressed, leading to the so-called compressed suffix trees [16]. + +The natural question is whether it is possible to provide suffix tree functionalities not +only to strings, but also to graphs, and in particular Wheeler graphs. In this paper, we +provide a first partial affirmative answer by considering the problem of computing matching +statistics. In 2010, Ohlebusch et al. [17] proposed a time and space efficient algorithm for +computing matching statistics which relies on some components of a compressed suffix tree. +In this paper, we show how their algorithm can be generalized from strings to Wheeler deter- +ministic finite automata. Most importantly, we introduce a notion of longest common prefix +(LCP) array for Wheeler automata, thus establishing an important step towards extending +(compressed) suffix tree functionalities to labeled graphs. +Notation and first definitions +Throughout the paper, we consider an alphabet Σ and a fixed total order ⪯ on Σ. We +denote by Σ∗ the set of all finite strings on Σ and by Σω the set of all (countably) infinite +strings on Σ. The empty word is ǫ. If α ∈ Σ∗, then αR is the reverse string of α. We extend +the total order ⪯ from Σ to Σ∗ ∪ Σω lexicographically. If i and j are integers, with i ≤ j, +define [i, j] = {i, i + 1, . . . , j − 1, j}. If T is a string, the i-th character of T is T[i], and +T[i..j] = T[i]..T[j]. +We will consider deterministic automata A = (Q, E, s0, F), where Q is the set of states, +E ⊆ Q × Q × Σ is the set of labeled edges, s0 ∈ Q is the initial state and F ⊆ Q is the set +of final states. The definition implies that for every u ∈ Q and for every a ∈ Σ there exists +at most one edge labeled a leaving u. Following [7,10], we assume that s0 has no incoming +edges, and every state is reachable from the initial state; moreover, all edges entering the +same state have the same label (input-consistency), so that for every u ∈ Q \ {s0} we can +let λ(u) be the label of all edges entering u. We define λ(s0) = #, where # ̸∈ Σ is a special +character for which we assume # ≺ a for every a ∈ Σ (the character # is an analogous of +the termination character $ used for suffix trees and suffix arrays). As a consequence, an +edge (u′, u, a) can be simply written as (u′, u), because it must be a = λ(u). +We assume familiarity with the notions of suffix array (SA), Burrows Wheeler transform +(BWT), FM-index and backward search [1]. +The matching statistics of a pattern π = π[1..m] with respect to a string T = T[1..n] are +defined as follows. Assume that T[n] = $ ̸∈ Σ, where $ ≺ a for every a ∈ Σ. Determining +the matching statistics of π with respect to T means determining, for 1 ≤ i ≤ m, (i) the +longest prefix π′ of π[i..m] which occurs in T, and (ii) the interval corresponding to the +set of all strings starting with π′ in the list of all lexicographically sorted suffixes. We can +describe (i) and (ii) by means of three values: the length ℓi of π′, and the endpoints li and +ri of the interval considered in (ii). For example, let T = mississippi$ (see Figure 1), and +π = stpissi. For i = 1, we have π′ = s, so ℓ1 = 1 and [l1, r1] = [9, 12] (suffixes starting with +s). For i = 2, we have π′ = ǫ, so ℓ2 = 0 and [l2, r2] = [1, n] = [1, 12] (all suffixes start with +the empty string). For i = 3, we have π′ = pi, so ℓ3 = 2, and [l3, r3] = [7, 7] (suffixes starting +with pi). For i = 4, we have π′ = issi, so ℓ4 = 4, and [l4, r4] = [4, 5] (suffixes starting with +issi). One can proceed analogously for i = 5, 6, 7. + +i +Sorted suffixes +LCP +SA +BWT +1 +$ +12 +i +2 +i$ +0 +11 +p +3 +ippi$ +1 +8 +s +4 +issippi$ +1 +5 +s +5 +ississippi$ +4 +2 +m +6 +mississippi$ +0 +1 +$ +7 +pi$ +0 +10 +p +8 +ppi$ +1 +9 +i +9 +sippi$ +0 +7 +s +10 +sissippi$ +2 +4 +s +11 +ssippi$ +1 +6 +i +12 +ssissippi$ +3 +3 +i +Figure 1: The sorted suffixes of “mississippi$” and the LCP, SA, and BWT arrays. +Computing matching statistics for strings +We will first describe the algorithm by Ohlebusch et al. [17], emphasizing the ideas that we +will generalize when switching to Wheeler DFAs. The algorithms computes the matching +statistics using a number of iterations linear in m by exploiting the backward search. We +start from the end of π, and we use the backward search (starting from the interval [1, n] +which corresponds to the set of suffixes prefixed by the empty string) to find the interval of +all occurrences of the last character of π in T (if any). Then, starting from the new interval, +we use the backward search to find all the occurrences of the suffix of length 2 of π in T (if +any), and so on. At some point, it may happen that for some i ≤ m+1 we have that π[i..m] +occurs in T, but the next application of the backward search returns the empty interval, so +that π[i − 1..m] does not occur in T (the case i = m + 1 corresponds to the initial setting +when π[i..m] is the empty string). We distinguish two cases: +• (Case 1) If li = 1 and ri = n, this means that all suffixes of T are prefixed by π[i..m]. +This may happen in particular if i = m + 1: this means that the first backward search +has been unsuccessful. We immediately conclude that character π[i−1] does not occur +in T, so ℓi−1 = 0 and [li−1, ri−1] = [1, n] (because all suffixes start with the empty +string). In this case, in the following iterations of the algorithm, we can simply discard +π[i − 1, m]: when for i′ ≤ i − 2 we will be searching for the longest prefix of π[i′, m] +occurring in T, it will suffice to search for the longest prefix of π[i′, i − 2] occurring in +T. +• (Case 2) If li > 1 or ri < n, this means that the number of suffixes of T starting with +π[i..m] is less than n. Now, every suffix starting with π[i..m] also starts with π[i..m−1]. +If the number of suffixes starting with π[i..m − 1] is equal to the number of suffixes +starting with π[i..m], then also π[i−1..m−1] does not occur in T. More in general, for +j ≤ m−1 we can have that π[i−1..j] occurs in T only if the number of suffixes starting +with π[i..j] is larger than the number of suffixes starting with π[i..m]. Since we are +interested in maximal matches, we want j to be as large as possible: we will show later + +how to compute the largest integer j such that the number of suffixes starting with +π[i..j] is larger than the number of suffixes starting with π[i..m]. Notice that j always +exists, because all n suffixes start with the empty string, but less than n suffixes start +with π[i..m]. After determining j we discard π[j + 1..m] (so in the following iterations +of the algorithm we will simply consider π[1..j]), and we recursively apply the backward +search starting from the interval associated with the occurrences of π[i..j] — we will +also see how to compute this interval. +Let us apply the above algorithm to T = mississippi$ and π = stpissi. We start with +the interval [1, n] = [1, 12], corresponding to the empty pattern, and character π[7] = i. A +backward step yields the interval [l7, r7] = [2, 5] (suffixes starting with i), so ℓ7 = 1. Now, we +apply a backward step from [2, 5] and π[6] = s, obtaining [l6, r6] = [9, 10] (suffixes starting +with si), so ℓ6 = 2. Again, we apply a backward step from [9, 10] and π[5] = s, obtaining +[l5, r5] = [11, 12] (suffixes starting with ssi), so ℓ5 = 3. Again, we apply a backward step +from [11, 12] and π[4] = i, obtaining [l4, r4] = [4, 5] (suffixes starting with issi), so ℓ4 = 4. +We now apply a backward step from [4, 5] and π[3] = p, and we obtain the empty interval. +This means that no suffix starts with pissi. Notice in Figure 1 that the number of suffixes +starting with issi is equal to the number of suffixes starting with iss or is, but the number +of suffixes starting with i is bigger. As a consequence, we consider the interval of all suffixes +starting with i — which is [2, 5] — and we apply a backward step with π[3] = p. This time +the backward step is successful, and we obtain [l3, r3] = [7, 7] (suffixes starting with pi), and +ℓ3 = 2. We now apply a backward step from [7, 7] and π[2] = t, obtaining the empty interval. +This means that no suffix starts with tpi. Notice in Figure 1 that the number of suffixes +starting with p is bigger than the number of suffixes starting with pi. The corresponding +interval is [7, 8], but a backward step with π[2] = t is still unsuccessful (so no suffix starts +with tp). +The number of suffixes starting with p is smaller than the number of suffixes +starting with the empty string (which is equal to n = 12), so we apply a backward step with +[1, 12] and π[2] = t. Since the backward step is still unsuccessful, we conclude that π[2] = t +does not occur in S, so [l2, r2] = [1, n] = [1, 12] and ℓ2 = 0. Finally, we start again from the +whole interval [1, 12], and a backward step with π[1] = s returns [l1, r1] = [9, 12] (suffixes +starting with s), so ℓ1 = 1. +It is easy to see that the number of iterations is linear in m. Indeed, every time we apply +a backward step, either we move to the left across π to compute a new matching statistic, +or we increase by at least 1 the length of the suffix of π which is forever discarded. This +implies that the number of iterations is bounded by 2|π| = 2m. +We are only left with showing (i) how to compute j and (ii) the interval of all suffixes +starting with π[i..j] in Case 2 of the algorithm. +To this end, we introduce the longest +common prefix (LCP) array LCP = LCP[2, n] of T. We define LCP[i] to be the length of the +longest common prefix of the (i − 1)-st lexicographically smallest suffix of T and the i-th +lexicographically smallest suffix of T. In Figure 1 we have LCP[5] = 4 because the fourth +lexicographically smallest suffix of T is issippi$, the fifth lexicographically smallest suffix of +T is ississippi$, and the longest common prefix of issippi$ and ississippi$ is issi, which has +length 4. Remember that in the example the backward search starting from [4, 5] (suffixes +starting with issi) and p was unsuccessful, so computing j means determining the longest +prefix of issi such that the the number of suffixes starting with such a prefix is bigger than 2. + +2 +5 +6 +7 +8 +9 +3 +4 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +1 +start +a +a +a +a +a +b +b +b +c +c +d +d +e +e +e +f +g +h +i +l +a +Figure 2: A Wheeler DFA. States are numbered according to their positions in the Wheeler +order. +This is easy to compute by using the LCP array: the longest such prefix is the one of length +max{LCP[4], LCP[6]} = max{1, 0} = 1, so that the desired prefix is i. As a consequence, +we are only left with showing how to compute the interval of all suffixes starting with the +prefix i — which is [2, 5]. Notice that in order to compute this interval, it is enough to +expand the interval [4, 6] in both directions as long as the LCP value does not go below 1. +Since LCP[4] = 1, LCP[3] = 1, and LCP[2] = 0, and we already know that LCP[6] = 0, we +conclude that the desired interval is [2, 5]. In other words, given a position t, we must be +able to compute the biggest integer k less than t such that LCP[k] < LCP[t], and the smallest +integer k bigger than t such that LCP[k] < LCP[t] (in our case, t = 4). These queries are +called PSV (“previous smaller value”) and NSV (“next smaller value”) queries. The LCP +array can be augmented in such a way that PSV and NSV queries can be solved efficiently: +different space-time trade-offs are possible, we refer the reader to [17] for details. +Matching statistics for Wheeler DFAs +Let us define Wheeler DFAs [7]. +Definition 1. Let A = (Q, E, s0, F) be a DFA. A Wheeler order on A is a total order ≤ on +Q such that s0 ≤ u for every u ∈ Q and: +(Axiom 1) If u, v ∈ Q and u < v, then λ(u) ⪯ λ(v). +(Axiom 2) If (u′, u), (v′, v) ∈ E, λ(u) = λ(v) and u < v, then u′ < v′. +A DFA A is Wheeler if it admits a Wheeler order. +It is immediate to check that this definition is equivalent to the one in [7], where it was +shown that if a DFA A admits a Wheeler order ≤, then ≤ is uniquely determined (that is, + +≤ is the Wheeler order on A). In the following, we fix a Wheeler DFA A = (Q, E, s0, F), +where we assume Q = {u1, . . . , un}, with u1 < u2 < · · · < un in the Wheeler order, and u1 +coincides with the initial state s0. See Figure 2 for an example. +We now show that a Wheeler order can be seen of as a permutation of the set of all states +playing the same role as the suffix array of a string. In the following, it will be expedient +to (conceptually) assume that s0 has a self-loop labeled # (this is consistent with Axiom 1, +because # ≺ a for every a ∈ Σ). This implies that every state has at least one incoming +edge, so for every state ui there exists at least one infinite string α ∈ Σω that can be read +starting from ui and following edges in a backward fashion (for example, in Figure 2 for u9 +such a string is cel### . . . ). We denote by Iui the set of all such strings. Formally: +Definition 2. Let i ∈ [1, n]. For every state ui ∈ Q define: +Iui = {α ∈ Σω | there exist integers f1, f2, . . . in [1, n] such that (i) f1 = i, +(ii) (ufk+1, ufk) ∈ E for every k ≥ 1 and (iii) α = λ(uf1)λ(uf2) . . . }. +For example, in Figure 2 we have Iu3 = {abdg### . . . , abeh### . . . , acei### . . . }. +The following lemma shows that the permutation of the states defined by the Wheeler +order is the one lexicographically sorting the strings entering each state, just like the permu- +tation defined by the suffix array lexicographically sorts the suffixes of the strings (a suffix +is seen as a string “leaving” a text position). +Lemma 3. Let i, j ∈ [1, n], with i < j. Let α ∈ Iui and β ∈ Iuj. Then, α ⪯ β. +Proof. Let f1, f2, . . . in [1, n] be such that (i) f1 = i, (ii) (ufk+1, ufk) ∈ E for every k ≥ 1 and +(iii) α = λ(uf1)λ(uf2) . . . . Analogously, let g1, g2, . . . in [1, n] be such that (i) g1 = j, (ii) +(ugk+1, ugk) ∈ E for every k ≥ 1 and (iii) β = λ(ug1)λ(ug2) . . . . Let α ̸= β. We must prove +that α ≺ β. Let p ≥ 1 be the smallest integer such that the p-th character of α is different +than the p-th character of β. In other words, we know that λ(uf1) = λ(ug1), λ(uf2) = λ(ug2), +. . . , λ(ufp−1) = λ(ugp−1), but λ(ufp) ̸= λ(ugp). We must prove that λ(ufp) ≺ λ(ugp). Since +λ(uf1) = λ(ug1) f1 = i < j = g1, and (uf2, uf1), (ug2, ug1) ∈ E, from Axiom 2 we obtain +f2 < g2. Since λ(uf2) = λ(ug2), f2 < g2, and (uf3, uf2), (ug3, ug2) ∈ E, from Axiom 2 we +obtain f3 < g3. By iterating this argument, we conclude fp < gp. By Axiom 1, we obtain +λ(ufp) ⪯ λ(ugp). Since λ(ufp) ̸= λ(ugp), we conclude λ(ufp) ≺ λ(ugp). +If we think of a string as a labeled path, then the suffix array sorts the strings that can +be read from each position by moving forward (that is, the suffixes of the string), while the +Wheeler order sorts the strings that can be read from each position by moving backward +towards the initial state. The underlying idea is the same: the forward vs backward difference +is only due to historical reasons [6]. To compute the matching statistics on Wheeler DFA +we reason as in the previous section replacing backward search with the forward search [6] +defined as follows: given an interval [i, j] in [1, n] and a ∈ Σ, find the (possibly empty) +interval [i′, j′] in [1, n] such that a state vk′ is reachable from some state vk, with i ≤ k ≤ j, +through an edge labeled a, if and only if i′ ≤ k′ ≤ j′ (this easily follows by using the axioms of +Definition 1). For a constant size alphabet, given [i, j] and a then [i′, j′] can be determined in +constant time. Given a string π ∈ Σ∗, if we start from the whole set of states and repeatedly +apply the forward search we reach the set of all states ui for which there exists α ∈ Iui + +prefixed by πR; this is an interval with respect to the Wheeler order: in the following we call +this interval T(π). +Because of the forward vs backward difference the problem of matching statistics will be +defined in a symmetrical way on Wheeler DFAs. Given a pattern π = π[1..m], for every +1 ≤ i ≤ m we want to determine (i) the longest suffix π′ of π[1..i] which occurs in the +Wheeler DFA A (that is, that can be read somewhere on A by concatenating edges), and +(ii) the endpoints of the interval T(π′). +Broadly speaking, we can apply the same idea of the algorithm for strings, but in a +symmetrical way. We start from the beginning of π (not from the end of π), and initially we +consider the whole set of states. We repeatedly apply the forward search (not the backward +search), until the forward search returns the empty interval for some i ≥ 0. This means that +π[1..i+1] does not occur in A. Then, if T(π[1..i]) is the whole set of states, we conclude that +the character π[i + 1] labels no edge in the graph. Otherwise, we must find the smallest j +such that T(π[1..i]) is strictly contained in T(π[j..i]) (that is, we must determine the longest +suffix π[j..i] of π[1..i] which reaches more states than π[1..i]). Then we must determine the +endpoints of the interval T(π[j..i]) so that we can go on with the forward search. +The challenge now is to find a way to solve the same subproblems that we identified in +Case 2 of the algorithm for strings. In other words, we must find a way to determine j and +find the endpoints of the interval T(π[j..i]). We will show that the solution is not as simple +as the one for the algorithm on strings. +The LCP array and matching statistics for Wheeler DFAs +We start observing that Iui may be an infinite set. For example, in Figure 2, we have +Iu2 = {aaaaa . . . , abdf### . . . , aabdf### . . . , aaabdf### . . . , . . . }. +In general, an infinite set of (lexicographically sorted) strings in Σω need not admit a +minimum or a maximum. For example, the set {baaaa . . . , abaaa . . . , aabaa . . . , aaaba . . . } +does not admit a minimum (but only the infimum string aaaaa . . . ). Nonetheless, Lemma 3 +implies that each Iui admits both a minimum and a maximum. For example, the minimum +is obtained as follows. Let f1 = i, and for every k ≥ 1, recursively let fk+1 be the smallest +integer in [1, n] such that (ufk+1, ufk) ∈ E. Then, the minimum of Iui is λ(uf1)λ(uf2) . . . , +and analogously one can determine the maximum. +In the following, we will denote the minimum and the maximum of Iui by mini and maxi, +respectively (for example, in Figure 2 we have min2 = aaaaa . . . , and max2 = abdf### . . . ). +Lemma 3 implies that: +min1 ⪯ max1 ⪯ min2 ⪯ max2 ⪯ · · · ⪯ maxn−1 ⪯ minn ⪯ maxn. +This suggests to generalize the LCP array as follows. Given α, β ∈ Σ∗ ∪ Σω, let lcp(α, β) be +the length of the longest common prefix of α and β (if α = β ∈ Σω, define lcp(α, β) = ∞). +Definition 4. The LCP-array of a Wheeler automaton A is the array LCPA = LCPA[2, 2n] +which contains the following 2n − 1 values in this order: lcp(min1, max1), lcp(max1, min2), +lcp(min2, max2), . . . , lcp(maxn−1, minn), lcp(minn, maxn). + +From the above characterization of mini and maxi, one can prove that for every entry +either LCPA[i] = ∞ or LCPA[i] < 3n (it follows from Fine and Wilf Theorem [18,19]), and +one can design a polynomial time algorithm to compute LCPA. +Unfortunately, the array LCPA alone is not sufficient for computing matching statistics. +Assume that T(π) = {ur, ur+1, . . . , us−1, us}, and that when we apply the forward search by +adding a character c, we obtain T(πc) = ∅. We must then determine the largest suffix π′ +of T(π) such that T(π) is strictly contained in T(π′). Suppose that every string in Iur is +prefixed by πR, and every string in Ius is prefixed by πR. In particular, both minr and maxs +are prefixed by πR. In this case, we can proceed like in the algorithm for strings: the desired +suffix π′ is the one having length max{lcp(maxr−1, minr), lcp(maxs, mins+1)}, which can be +determined using LCPA. However, in general, even if some string in Iur must be prefixed +by πR, the string minr need not be prefixed by πR, and similarly maxs need not be prefixed +by πR. The worst-case scenario occurs when r = s. Consider Figure 2, and assume that +π = heba. Then, we have r = s = 3 (note that abeh### . . . is a string in Iu3 prefixed by +πR). However, both min3 = abdg### . . . , and max3 = acei### . . . , are not prefixed by +πR. Notice that lcp(max2, min3) = 3 and lcp(max3, min4) = 3, but π′ is not the suffix of +length 3 of π. Indeed, since min3 is only prefixed by the prefix of πR of length 2, and max3 +is only prefixed by the prefix of πR of length 1, we conclude that it must be |π′| = 2. In +general, the desired suffix π′ is the one having length |π′| given by: +max +� +min{lcp(maxr−1, minr),lcp(minr, πR)}, min{lcp(πR, maxs),lcp(maxs, mins+1)} +� +. (1) +The above formula shows that, in order to compute π′, in addition to LCPA it suffices to +know the values lcp(minr, πR) and lcp(πR, maxs) (π′ is a suffix of π, so it is determined by +its length). We now show how our algorithm can efficiently maintain the current pattern π, +the set T(π) = {ur, ur+1, . . . , us−1, us} and the values lcp(minr, πR) and lcp(πR, maxs) during +the computation of the matching statistics. We assume that the input automaton is encoded +with the rank/select data structures supporting the execution of a step of forward search in +O(log |Σ|) time, see [6] for details. In addition, we will use the following result. +Lemma 5. Let A[1, n] be a sequence of values over an ordered alphabet Σ. Consider the +following queries: (i) given i, j ∈ [1..n], compute the minimum value in S[i..j], and (ii) +given t ∈ [1..n] and c ∈ Σ, determine the biggest k < t (or the smallest k > t) such that +A[k] < c. Then, A can be augumented with a data structure of 2n+o(n) bits such that query +(i) can be answered in constant time and query (ii) can be answered in O(log n) time. +Proof. There exists a data structure of 2n + o(n) bits that allows to solve range minimum +queries in constant time [20], so using A we can solve queries (i) in constant time. Now, let +us show how to solve queries (ii). Let f1 be the answer of query (i) on input i = ⌈t/2⌉ and +j = t − 1. If f1 < c, then we must keep searching in the interval [⌈t/2⌉, t − 1], otherwise, we +must keep searching in the interval [1, ⌈t/2⌉ − 1]. In other words, we can answer a query (ii) +by means of a binary search on [1, t − 1], which takes O(log t) (and so O(log n)) time. +Notice that query (ii) can be seen as a variant of PSV and NSV queries. In the following, +we assume that the array LCPA has been augmented with the data structure of Lemma 5. +At the beginning we have π = ǫ, so T(ǫ) = {1, 2, . . . , n} and trivially lcp(minr, πR) = +lcp(πR, maxs) = 0. At each iteration we perform a step of forward search computing T(πc) +given T(π); then we distinguish two cases according to whether T(πc) is empty or not. + +Case 1. T(πc) = {ur′, ur′+1, . . . , us′−1, us′} is not empty. In that case πc will become the +pattern at the next iteration. Since we already have T(πc) we are left with the task of com- +puting lcp(minr′, cπR) and lcp(cπR, maxs′). We only show how to compute lcp(minr′, cπR), +the latter computation being analogous. Let k be the smallest integer in [1, n] such that +(uk, ur′) ∈ E. Notice that we can easily compute k by means of standard rank/select opera- +tions on the compact data structure used to encode A. Since ur′ ∈ T(πc), it must be k ≤ s. +Moreover, the characterization of minr′ that we described above implies that minr′ = c mink, +hence lcp(minr′, cπR) = lcp(c mink, cπR) = 1 + lcp(mink, πR). To compute lcp(mink, πR) we +distinguish two subcases: +a) k > r, hence r < k ≤ s. Since ur, us ∈ T(π), there exist α ∈ Iur and β ∈ Ius both +prefixed by πR. But α ⪯ maxr ⪯ mink ⪯ mins ⪯ β, so mink is also prefixed by πR, +and we conclude lcp(mink, πR) = |π|. +b) k ≤ r. In this case, we have mink ⪯ maxk ⪯ mink+1 ≺ maxk+1 ⪯ · · · ⪯ minr ≺ πR, +and therefore lcp(mink, πR) is equal to +min{lcp(mink, maxk), lcp(maxk, mink+1), lcp(mink+1, maxk+1), . . . , lcp(minr, πR)}. +With the above formula we can compute lcp(mink, πR) using query (i) of Lemma 5 over +the range LCPA[2k, 2r − 1] and the value lcp(minr, πR). +Case 2. T(πc) is empty. In this case at the next iteration the pattern will be largest suffix +π′ of π such that T(π) is strictly contained in T(π′) = {ur′′, . . . , us′′}. We compute |π′| +using (1); if |π′| > lcp(minr, πR) we set r′′ = r, otherwise we apply query (ii) of Lemma 5 to +find the rightmost entry r′′ in LCPA[2, 2r − 1] smaller than |π′|. Computing s′′ is analogous. +Given T(π′) = {ur′′, ur′′+1, . . . , us′′−1, us′′}, where r′′ ≤ r, s ≤ s′′, and at least one inequal- +ity is strict, we want to compute lcp(minr′′, (π′)R) and lcp((π′)R, maxs′′). We only consider +lcp(minr′′, (π′)R), the latter computation being analogous. We distinguish two subcases: +a) r′′ = r. Then lcp(minr′′, (π′)R) = lcp(minr, (π′)R) = min{lcp(minr, πR), |π′|}. +b) r′′ < r. In particular, since ur′′ is the left endpoint of T(π′) and |T(π′)| ≥ 2, one can +prove like in Case 1a) that maxr′′ is prefixed by (π′)R. We immediately conclude that +lcp(minr′′, (π′)R) = min{lcp(minr′′, maxr′′), |π′|}, which can be immediately computed +since lcp(minr′′, maxr′′) is a value stored in LCPA. +We can summarize the above discussion as follows. +Theorem 6. Given a Wheeler DFA A, there exists a data structure occupying O(|A|) words +which can compute the pattern matching statistics of a pattern P in time O(|P| log |A|). +Funding +TG funded by National Institutes of Health (NIH) NIAID (grant no. HG011392), +the National Science Foundation NSF IIBR (grant no. 2029552) and a Natural Science and +Engineering Research Council (NSERC) Discovery Grant (grant no. RGPIN-07185-2020). +GM funded by the Italian Ministry of University and Research (PRIN 2017WR7SHH). MS +funded by the INdAM-GNCS Project (CUP E55F22000270001). NP funded by the European +Union (ERC, REGINDEX, 101039208). Views and opinions expressed are however those +of the author(s) only and do not necessarily reflect those of the European Union or the +European Research Council. Neither the European Union nor the granting authority can be +held responsible for them. + +References +[1] P. Ferragina and G. Manzini, “Opportunistic data structures with applications,” in Proc. 41st +Annual Symposium on Foundations of Computer Science (FOCS’00), 2000, pp. 390–398. +[2] M. Burrows and D. J. Wheeler, “A block-sorting lossless data compression algorithm,” Tech. +Rep., 1994. +[3] P. Ferragina, F. Luccio, G. Manzini, and S. Muthukrishnan, “Structuring labeled trees for +optimal succinctness, and beyond,” in proc. 46th Annual IEEE Symposium on Foundations of +Computer Science (FOCS’05), 2005, pp. 184–193. +[4] A. Bowe, T. Onodera, K. Sadakane, and T. Shibuya, “Succinct de Bruijn graphs,” in Algo- +rithms in Bioinformatics, Berlin, Heidelberg, 2012, pp. 225–235, Springer Berlin Heidelberg. +[5] V. M¨akinen, N. 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K¨ugell, “Computing matching statistics and maximal exact +matches on compressed full-text indexes,” in Proceedings of the 17th International Confer- +ence on String Processing and Information Retrieval (SPIRE’10), Berlin, Heidelberg, 2010, p. +347–358, Springer-Verlag. +[18] N. J. Fine and H. S. Wilf, “Uniqueness theorem for periodic functions,” Proc. Amer. Math. +Soc., , no. 16, pp. 109–114, 1965. +[19] S. Mantaci, A. Restivo, G. Rosone, and M. Sciortino, “An extension of the Burrows-Wheeler +transform,” Theor. Comput. Sci., vol. 387, no. 3, pp. 298–312, 2007. +[20] Johannes Fischer, +“Optimal succinctness for range minimum queries,” +in LATIN 2010: +Theoretical Informatics, Alejandro L´opez-Ortiz, Ed., Berlin, Heidelberg, 2010, pp. 158–169, +Springer Berlin Heidelberg. + diff --git a/JdE4T4oBgHgl3EQf7Q53/content/tmp_files/load_file.txt b/JdE4T4oBgHgl3EQf7Q53/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..01ebd8390feda22421c14a75aa9db5a479f542e9 --- /dev/null +++ b/JdE4T4oBgHgl3EQf7Q53/content/tmp_files/load_file.txt @@ -0,0 +1,538 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf,len=537 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='05338v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='DS] 13 Jan 2023 Computing matching statistics on Wheeler DFAs Alessio Conte1, Nicola Cotumaccio2,3, Travis Gagie3, Giovanni Manzini1, Nicola Prezza4 and Marinella Sciortino5 1 University of Pisa, Italy, alessio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='conte@unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='it, giovanni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='manzini@unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='it 2 GSSI, L’Aquila, Italy, nicola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='cotumaccio@gssi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='it 3 Dalhousie University, Halifax, Canada, nicola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='cotumaccio@dal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='ca, travis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='gagie@dal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='ca 4 Ca’ Foscari Unversity, Venice, Italy, nicola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='prezza@unive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='it 5 University of Palermo, Italy, marinella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='sciortino@unipa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='it Abstract Matching statistics were introduced to solve the approximate string matching problem, which is a recurrent subroutine in bioinformatics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In 2010, Ohlebusch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' [SPIRE 2010] proposed a time and space efficient algorithm for computing matching statistics which relies on some components of a compressed suffix tree - notably, the longest common prefix (LCP) array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In this paper, we show how their algorithm can be generalized from strings to Wheeler deterministic finite automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Most importantly, we introduce a notion of LCP array for Wheeler automata, thus establishing a first clear step towards extending (compressed) suffix tree functionalities to labeled graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Introduction Given a string T and a pattern π, the classical formulation of the pattern matching problem requires to decide whether the pattern π occurs in the string T and, possibly, count the number of such occurrences and report the positions where they occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The invention of the FM-index [1], which is based on the Burrows-Wheeler transform [2], opened a new line of research in the pattern matching field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The indexing and compression techniques behind the FM-index deeply rely on the idea of suffix sorting, and over the years have been generalized from strings to trees [3], De Brujin graphs [4,5], Wheeler graphs [6,7] and arbitrary graphs [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In particular, the class of Wheeler graphs is probably the one that captures the intuition behind the FM-index in the simplest way, and indeed the notion of Wheeler order has relevant consequences in automata theory [7,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' However, in bioinformatics we are not only interested in exact pattern matching, but also in a myriad of variations of the pattern matching problem [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In particular, matching statistics were introduced to solve the approximate pattern matching problem [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' A pow- erful data structure that is able to address the variations of the pattern matching problem at once is the suffix tree [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The main drawback of the suffix tree is its space consumption, which is non-negligible both in theory and in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' As a consequence, the suffix tree has been replaced by the suffix array [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' While suffix arrays do not have all the functionalities of suffix trees, it has been shown that they can be augmented with some additional data structures — notably, the longest common prefix (LCP) array — so that it is possible to retrieve the full functionalities of a suffix trees [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' All these components can be successfully compressed, leading to the so-called compressed suffix trees [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The natural question is whether it is possible to provide suffix tree functionalities not only to strings, but also to graphs, and in particular Wheeler graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In this paper, we provide a first partial affirmative answer by considering the problem of computing matching statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In 2010, Ohlebusch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' [17] proposed a time and space efficient algorithm for computing matching statistics which relies on some components of a compressed suffix tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In this paper, we show how their algorithm can be generalized from strings to Wheeler deter- ministic finite automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Most importantly, we introduce a notion of longest common prefix (LCP) array for Wheeler automata, thus establishing an important step towards extending (compressed) suffix tree functionalities to labeled graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Notation and first definitions Throughout the paper, we consider an alphabet Σ and a fixed total order ⪯ on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We denote by Σ∗ the set of all finite strings on Σ and by Σω the set of all (countably) infinite strings on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The empty word is ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' If α ∈ Σ∗, then αR is the reverse string of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We extend the total order ⪯ from Σ to Σ∗ ∪ Σω lexicographically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' If i and j are integers, with i ≤ j, define [i, j] = {i, i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , j − 1, j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' If T is a string, the i-th character of T is T[i], and T[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.j] = T[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.T[j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We will consider deterministic automata A = (Q, E, s0, F), where Q is the set of states, E ⊆ Q × Q × Σ is the set of labeled edges, s0 ∈ Q is the initial state and F ⊆ Q is the set of final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The definition implies that for every u ∈ Q and for every a ∈ Σ there exists at most one edge labeled a leaving u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Following [7,10], we assume that s0 has no incoming edges, and every state is reachable from the initial state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' moreover, all edges entering the same state have the same label (input-consistency), so that for every u ∈ Q \\ {s0} we can let λ(u) be the label of all edges entering u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We define λ(s0) = #, where # ̸∈ Σ is a special character for which we assume # ≺ a for every a ∈ Σ (the character # is an analogous of the termination character $ used for suffix trees and suffix arrays).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' As a consequence, an edge (u′, u, a) can be simply written as (u′, u), because it must be a = λ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We assume familiarity with the notions of suffix array (SA), Burrows Wheeler transform (BWT), FM-index and backward search [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The matching statistics of a pattern π = π[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m] with respect to a string T = T[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.n] are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Assume that T[n] = $ ̸∈ Σ, where $ ≺ a for every a ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Determining the matching statistics of π with respect to T means determining, for 1 ≤ i ≤ m, (i) the longest prefix π′ of π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m] which occurs in T, and (ii) the interval corresponding to the set of all strings starting with π′ in the list of all lexicographically sorted suffixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We can describe (i) and (ii) by means of three values: the length ℓi of π′, and the endpoints li and ri of the interval considered in (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' For example, let T = mississippi$ (see Figure 1), and π = stpissi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' For i = 1, we have π′ = s, so ℓ1 = 1 and [l1, r1] = [9, 12] (suffixes starting with s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' For i = 2, we have π′ = ǫ, so ℓ2 = 0 and [l2, r2] = [1, n] = [1, 12] (all suffixes start with the empty string).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' For i = 3, we have π′ = pi, so ℓ3 = 2, and [l3, r3] = [7, 7] (suffixes starting with pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' For i = 4, we have π′ = issi, so ℓ4 = 4, and [l4, r4] = [4, 5] (suffixes starting with issi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' One can proceed analogously for i = 5, 6, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' i Sorted suffixes LCP SA BWT 1 $ 12 i 2 i$ 0 11 p 3 ippi$ 1 8 s 4 issippi$ 1 5 s 5 ississippi$ 4 2 m 6 mississippi$ 0 1 $ 7 pi$ 0 10 p 8 ppi$ 1 9 i 9 sippi$ 0 7 s 10 sissippi$ 2 4 s 11 ssippi$ 1 6 i 12 ssissippi$ 3 3 i Figure 1: The sorted suffixes of “mississippi$” and the LCP, SA, and BWT arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Computing matching statistics for strings We will first describe the algorithm by Ohlebusch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' [17], emphasizing the ideas that we will generalize when switching to Wheeler DFAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The algorithms computes the matching statistics using a number of iterations linear in m by exploiting the backward search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We start from the end of π, and we use the backward search (starting from the interval [1, n] which corresponds to the set of suffixes prefixed by the empty string) to find the interval of all occurrences of the last character of π in T (if any).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Then, starting from the new interval, we use the backward search to find all the occurrences of the suffix of length 2 of π in T (if any), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' At some point, it may happen that for some i ≤ m+1 we have that π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m] occurs in T, but the next application of the backward search returns the empty interval, so that π[i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m] does not occur in T (the case i = m + 1 corresponds to the initial setting when π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m] is the empty string).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We distinguish two cases: (Case 1) If li = 1 and ri = n, this means that all suffixes of T are prefixed by π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' This may happen in particular if i = m + 1: this means that the first backward search has been unsuccessful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We immediately conclude that character π[i−1] does not occur in T, so ℓi−1 = 0 and [li−1, ri−1] = [1, n] (because all suffixes start with the empty string).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In this case, in the following iterations of the algorithm, we can simply discard π[i − 1, m]: when for i′ ≤ i − 2 we will be searching for the longest prefix of π[i′, m] occurring in T, it will suffice to search for the longest prefix of π[i′, i − 2] occurring in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' (Case 2) If li > 1 or ri < n, this means that the number of suffixes of T starting with π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m] is less than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Now, every suffix starting with π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m] also starts with π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' If the number of suffixes starting with π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m − 1] is equal to the number of suffixes starting with π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m], then also π[i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m−1] does not occur in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' More in general, for j ≤ m−1 we can have that π[i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.j] occurs in T only if the number of suffixes starting with π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.j] is larger than the number of suffixes starting with π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Since we are interested in maximal matches, we want j to be as large as possible: we will show later how to compute the largest integer j such that the number of suffixes starting with π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.j] is larger than the number of suffixes starting with π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Notice that j always exists, because all n suffixes start with the empty string, but less than n suffixes start with π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' After determining j we discard π[j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m] (so in the following iterations of the algorithm we will simply consider π[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.j]), and we recursively apply the backward search starting from the interval associated with the occurrences of π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.j] — we will also see how to compute this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Let us apply the above algorithm to T = mississippi$ and π = stpissi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We start with the interval [1, n] = [1, 12], corresponding to the empty pattern, and character π[7] = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' A backward step yields the interval [l7, r7] = [2, 5] (suffixes starting with i), so ℓ7 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Now, we apply a backward step from [2, 5] and π[6] = s, obtaining [l6, r6] = [9, 10] (suffixes starting with si), so ℓ6 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Again, we apply a backward step from [9, 10] and π[5] = s, obtaining [l5, r5] = [11, 12] (suffixes starting with ssi), so ℓ5 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Again, we apply a backward step from [11, 12] and π[4] = i, obtaining [l4, r4] = [4, 5] (suffixes starting with issi), so ℓ4 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We now apply a backward step from [4, 5] and π[3] = p, and we obtain the empty interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' This means that no suffix starts with pissi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Notice in Figure 1 that the number of suffixes starting with issi is equal to the number of suffixes starting with iss or is, but the number of suffixes starting with i is bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' As a consequence, we consider the interval of all suffixes starting with i — which is [2, 5] — and we apply a backward step with π[3] = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' This time the backward step is successful, and we obtain [l3, r3] = [7, 7] (suffixes starting with pi), and ℓ3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We now apply a backward step from [7, 7] and π[2] = t, obtaining the empty interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' This means that no suffix starts with tpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Notice in Figure 1 that the number of suffixes starting with p is bigger than the number of suffixes starting with pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The corresponding interval is [7, 8], but a backward step with π[2] = t is still unsuccessful (so no suffix starts with tp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The number of suffixes starting with p is smaller than the number of suffixes starting with the empty string (which is equal to n = 12), so we apply a backward step with [1, 12] and π[2] = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Since the backward step is still unsuccessful, we conclude that π[2] = t does not occur in S, so [l2, r2] = [1, n] = [1, 12] and ℓ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Finally, we start again from the whole interval [1, 12], and a backward step with π[1] = s returns [l1, r1] = [9, 12] (suffixes starting with s), so ℓ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' It is easy to see that the number of iterations is linear in m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Indeed, every time we apply a backward step, either we move to the left across π to compute a new matching statistic, or we increase by at least 1 the length of the suffix of π which is forever discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' This implies that the number of iterations is bounded by 2|π| = 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We are only left with showing (i) how to compute j and (ii) the interval of all suffixes starting with π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.j] in Case 2 of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' To this end, we introduce the longest common prefix (LCP) array LCP = LCP[2, n] of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We define LCP[i] to be the length of the longest common prefix of the (i − 1)-st lexicographically smallest suffix of T and the i-th lexicographically smallest suffix of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In Figure 1 we have LCP[5] = 4 because the fourth lexicographically smallest suffix of T is issippi$, the fifth lexicographically smallest suffix of T is ississippi$, and the longest common prefix of issippi$ and ississippi$ is issi, which has length 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Remember that in the example the backward search starting from [4, 5] (suffixes starting with issi) and p was unsuccessful, so computing j means determining the longest prefix of issi such that the the number of suffixes starting with such a prefix is bigger than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' 2 5 6 7 8 9 3 4 10 11 12 13 14 15 16 17 18 19 1 start a a a a a b b b c c d d e e e f g h i l a Figure 2: A Wheeler DFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' States are numbered according to their positions in the Wheeler order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' This is easy to compute by using the LCP array: the longest such prefix is the one of length max{LCP[4], LCP[6]} = max{1, 0} = 1, so that the desired prefix is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' As a consequence, we are only left with showing how to compute the interval of all suffixes starting with the prefix i — which is [2, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Notice that in order to compute this interval, it is enough to expand the interval [4, 6] in both directions as long as the LCP value does not go below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Since LCP[4] = 1, LCP[3] = 1, and LCP[2] = 0, and we already know that LCP[6] = 0, we conclude that the desired interval is [2, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In other words, given a position t, we must be able to compute the biggest integer k less than t such that LCP[k] < LCP[t], and the smallest integer k bigger than t such that LCP[k] < LCP[t] (in our case, t = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' These queries are called PSV (“previous smaller value”) and NSV (“next smaller value”) queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The LCP array can be augmented in such a way that PSV and NSV queries can be solved efficiently: different space-time trade-offs are possible, we refer the reader to [17] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Matching statistics for Wheeler DFAs Let us define Wheeler DFAs [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Let A = (Q, E, s0, F) be a DFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' A Wheeler order on A is a total order ≤ on Q such that s0 ≤ u for every u ∈ Q and: (Axiom 1) If u, v ∈ Q and u < v, then λ(u) ⪯ λ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' (Axiom 2) If (u′, u), (v′, v) ∈ E, λ(u) = λ(v) and u < v, then u′ < v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' A DFA A is Wheeler if it admits a Wheeler order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' It is immediate to check that this definition is equivalent to the one in [7], where it was shown that if a DFA A admits a Wheeler order ≤, then ≤ is uniquely determined (that is, ≤ is the Wheeler order on A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In the following, we fix a Wheeler DFA A = (Q, E, s0, F), where we assume Q = {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , un}, with u1 < u2 < · · · < un in the Wheeler order, and u1 coincides with the initial state s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' See Figure 2 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We now show that a Wheeler order can be seen of as a permutation of the set of all states playing the same role as the suffix array of a string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In the following, it will be expedient to (conceptually) assume that s0 has a self-loop labeled # (this is consistent with Axiom 1, because # ≺ a for every a ∈ Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' This implies that every state has at least one incoming edge, so for every state ui there exists at least one infinite string α ∈ Σω that can be read starting from ui and following edges in a backward fashion (for example, in Figure 2 for u9 such a string is cel### .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We denote by Iui the set of all such strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Formally: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Let i ∈ [1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' For every state ui ∈ Q define: Iui = {α ∈ Σω | there exist integers f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' in [1, n] such that (i) f1 = i, (ii) (ufk+1, ufk) ∈ E for every k ≥ 1 and (iii) α = λ(uf1)λ(uf2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' For example, in Figure 2 we have Iu3 = {abdg### .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , abeh### .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , acei### .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The following lemma shows that the permutation of the states defined by the Wheeler order is the one lexicographically sorting the strings entering each state, just like the permu- tation defined by the suffix array lexicographically sorts the suffixes of the strings (a suffix is seen as a string “leaving” a text position).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Let i, j ∈ [1, n], with i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Let α ∈ Iui and β ∈ Iuj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Then, α ⪯ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Let f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' in [1, n] be such that (i) f1 = i, (ii) (ufk+1, ufk) ∈ E for every k ≥ 1 and (iii) α = λ(uf1)λ(uf2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Analogously, let g1, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' in [1, n] be such that (i) g1 = j, (ii) (ugk+1, ugk) ∈ E for every k ≥ 1 and (iii) β = λ(ug1)λ(ug2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Let α ̸= β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We must prove that α ≺ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Let p ≥ 1 be the smallest integer such that the p-th character of α is different than the p-th character of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In other words, we know that λ(uf1) = λ(ug1), λ(uf2) = λ(ug2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , λ(ufp−1) = λ(ugp−1), but λ(ufp) ̸= λ(ugp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We must prove that λ(ufp) ≺ λ(ugp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Since λ(uf1) = λ(ug1) f1 = i < j = g1, and (uf2, uf1), (ug2, ug1) ∈ E, from Axiom 2 we obtain f2 < g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Since λ(uf2) = λ(ug2), f2 < g2, and (uf3, uf2), (ug3, ug2) ∈ E, from Axiom 2 we obtain f3 < g3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' By iterating this argument, we conclude fp < gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' By Axiom 1, we obtain λ(ufp) ⪯ λ(ugp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Since λ(ufp) ̸= λ(ugp), we conclude λ(ufp) ≺ λ(ugp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' If we think of a string as a labeled path, then the suffix array sorts the strings that can be read from each position by moving forward (that is, the suffixes of the string), while the Wheeler order sorts the strings that can be read from each position by moving backward towards the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The underlying idea is the same: the forward vs backward difference is only due to historical reasons [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' To compute the matching statistics on Wheeler DFA we reason as in the previous section replacing backward search with the forward search [6] defined as follows: given an interval [i, j] in [1, n] and a ∈ Σ, find the (possibly empty) interval [i′, j′] in [1, n] such that a state vk′ is reachable from some state vk, with i ≤ k ≤ j, through an edge labeled a, if and only if i′ ≤ k′ ≤ j′ (this easily follows by using the axioms of Definition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' For a constant size alphabet, given [i, j] and a then [i′, j′] can be determined in constant time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Given a string π ∈ Σ∗, if we start from the whole set of states and repeatedly apply the forward search we reach the set of all states ui for which there exists α ∈ Iui prefixed by πR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' this is an interval with respect to the Wheeler order: in the following we call this interval T(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Because of the forward vs backward difference the problem of matching statistics will be defined in a symmetrical way on Wheeler DFAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Given a pattern π = π[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.m], for every 1 ≤ i ≤ m we want to determine (i) the longest suffix π′ of π[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.i] which occurs in the Wheeler DFA A (that is, that can be read somewhere on A by concatenating edges), and (ii) the endpoints of the interval T(π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Broadly speaking, we can apply the same idea of the algorithm for strings, but in a symmetrical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We start from the beginning of π (not from the end of π), and initially we consider the whole set of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We repeatedly apply the forward search (not the backward search), until the forward search returns the empty interval for some i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' This means that π[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.i+1] does not occur in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Then, if T(π[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.i]) is the whole set of states, we conclude that the character π[i + 1] labels no edge in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Otherwise, we must find the smallest j such that T(π[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.i]) is strictly contained in T(π[j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.i]) (that is, we must determine the longest suffix π[j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.i] of π[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.i] which reaches more states than π[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.i]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Then we must determine the endpoints of the interval T(π[j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.i]) so that we can go on with the forward search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The challenge now is to find a way to solve the same subproblems that we identified in Case 2 of the algorithm for strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In other words, we must find a way to determine j and find the endpoints of the interval T(π[j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.i]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We will show that the solution is not as simple as the one for the algorithm on strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The LCP array and matching statistics for Wheeler DFAs We start observing that Iui may be an infinite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' For example, in Figure 2, we have Iu2 = {aaaaa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , abdf### .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , aabdf### .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , aaabdf### .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In general, an infinite set of (lexicographically sorted) strings in Σω need not admit a minimum or a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' For example, the set {baaaa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , abaaa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , aabaa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , aaaba .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' } does not admit a minimum (but only the infimum string aaaaa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Nonetheless, Lemma 3 implies that each Iui admits both a minimum and a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' For example, the minimum is obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Let f1 = i, and for every k ≥ 1, recursively let fk+1 be the smallest integer in [1, n] such that (ufk+1, ufk) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Then, the minimum of Iui is λ(uf1)λ(uf2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , and analogously one can determine the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In the following, we will denote the minimum and the maximum of Iui by mini and maxi, respectively (for example, in Figure 2 we have min2 = aaaaa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , and max2 = abdf### .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Lemma 3 implies that: min1 ⪯ max1 ⪯ min2 ⪯ max2 ⪯ · · · ⪯ maxn−1 ⪯ minn ⪯ maxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' This suggests to generalize the LCP array as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Given α, β ∈ Σ∗ ∪ Σω, let lcp(α, β) be the length of the longest common prefix of α and β (if α = β ∈ Σω, define lcp(α, β) = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The LCP-array of a Wheeler automaton A is the array LCPA = LCPA[2, 2n] which contains the following 2n − 1 values in this order: lcp(min1, max1), lcp(max1, min2), lcp(min2, max2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , lcp(maxn−1, minn), lcp(minn, maxn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' From the above characterization of mini and maxi, one can prove that for every entry either LCPA[i] = ∞ or LCPA[i] < 3n (it follows from Fine and Wilf Theorem [18,19]), and one can design a polynomial time algorithm to compute LCPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Unfortunately, the array LCPA alone is not sufficient for computing matching statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Assume that T(π) = {ur, ur+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , us−1, us}, and that when we apply the forward search by adding a character c, we obtain T(πc) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We must then determine the largest suffix π′ of T(π) such that T(π) is strictly contained in T(π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Suppose that every string in Iur is prefixed by πR, and every string in Ius is prefixed by πR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In particular, both minr and maxs are prefixed by πR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In this case, we can proceed like in the algorithm for strings: the desired suffix π′ is the one having length max{lcp(maxr−1, minr), lcp(maxs, mins+1)}, which can be determined using LCPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' However, in general, even if some string in Iur must be prefixed by πR, the string minr need not be prefixed by πR, and similarly maxs need not be prefixed by πR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' The worst-case scenario occurs when r = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Consider Figure 2, and assume that π = heba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Then, we have r = s = 3 (note that abeh### .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' is a string in Iu3 prefixed by πR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' However, both min3 = abdg### .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , and max3 = acei### .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , are not prefixed by πR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Notice that lcp(max2, min3) = 3 and lcp(max3, min4) = 3, but π′ is not the suffix of length 3 of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Indeed, since min3 is only prefixed by the prefix of πR of length 2, and max3 is only prefixed by the prefix of πR of length 1, we conclude that it must be |π′| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In general, the desired suffix π′ is the one having length |π′| given by: max � min{lcp(maxr−1, minr),lcp(minr, πR)}, min{lcp(πR, maxs),lcp(maxs, mins+1)} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' (1) The above formula shows that, in order to compute π′, in addition to LCPA it suffices to know the values lcp(minr, πR) and lcp(πR, maxs) (π′ is a suffix of π, so it is determined by its length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We now show how our algorithm can efficiently maintain the current pattern π, the set T(π) = {ur, ur+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , us−1, us} and the values lcp(minr, πR) and lcp(πR, maxs) during the computation of the matching statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We assume that the input automaton is encoded with the rank/select data structures supporting the execution of a step of forward search in O(log |Σ|) time, see [6] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In addition, we will use the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Let A[1, n] be a sequence of values over an ordered alphabet Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Consider the following queries: (i) given i, j ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.n], compute the minimum value in S[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.j], and (ii) given t ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content='.n] and c ∈ Σ, determine the biggest k < t (or the smallest k > t) such that A[k] < c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Then, A can be augumented with a data structure of 2n+o(n) bits such that query (i) can be answered in constant time and query (ii) can be answered in O(log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' There exists a data structure of 2n + o(n) bits that allows to solve range minimum queries in constant time [20], so using A we can solve queries (i) in constant time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Now, let us show how to solve queries (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Let f1 be the answer of query (i) on input i = ⌈t/2⌉ and j = t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' If f1 < c, then we must keep searching in the interval [⌈t/2⌉, t − 1], otherwise, we must keep searching in the interval [1, ⌈t/2⌉ − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In other words, we can answer a query (ii) by means of a binary search on [1, t − 1], which takes O(log t) (and so O(log n)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Notice that query (ii) can be seen as a variant of PSV and NSV queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In the following, we assume that the array LCPA has been augmented with the data structure of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' At the beginning we have π = ǫ, so T(ǫ) = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , n} and trivially lcp(minr, πR) = lcp(πR, maxs) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' At each iteration we perform a step of forward search computing T(πc) given T(π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' then we distinguish two cases according to whether T(πc) is empty or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' T(πc) = {ur′, ur′+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , us′−1, us′} is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In that case πc will become the pattern at the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Since we already have T(πc) we are left with the task of com- puting lcp(minr′, cπR) and lcp(cπR, maxs′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We only show how to compute lcp(minr′, cπR), the latter computation being analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Let k be the smallest integer in [1, n] such that (uk, ur′) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Notice that we can easily compute k by means of standard rank/select opera- tions on the compact data structure used to encode A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Since ur′ ∈ T(πc), it must be k ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Moreover, the characterization of minr′ that we described above implies that minr′ = c mink, hence lcp(minr′, cπR) = lcp(c mink, cπR) = 1 + lcp(mink, πR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' To compute lcp(mink, πR) we distinguish two subcases: a) k > r, hence r < k ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Since ur, us ∈ T(π), there exist α ∈ Iur and β ∈ Ius both prefixed by πR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' But α ⪯ maxr ⪯ mink ⪯ mins ⪯ β, so mink is also prefixed by πR, and we conclude lcp(mink, πR) = |π|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' b) k ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In this case, we have mink ⪯ maxk ⪯ mink+1 ≺ maxk+1 ⪯ · · · ⪯ minr ≺ πR, and therefore lcp(mink, πR) is equal to min{lcp(mink, maxk), lcp(maxk, mink+1), lcp(mink+1, maxk+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , lcp(minr, πR)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' With the above formula we can compute lcp(mink, πR) using query (i) of Lemma 5 over the range LCPA[2k, 2r − 1] and the value lcp(minr, πR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' T(πc) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In this case at the next iteration the pattern will be largest suffix π′ of π such that T(π) is strictly contained in T(π′) = {ur′′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , us′′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We compute |π′| using (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' if |π′| > lcp(minr, πR) we set r′′ = r, otherwise we apply query (ii) of Lemma 5 to find the rightmost entry r′′ in LCPA[2, 2r − 1] smaller than |π′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Computing s′′ is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Given T(π′) = {ur′′, ur′′+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' , us′′−1, us′′}, where r′′ ≤ r, s ≤ s′′, and at least one inequal- ity is strict, we want to compute lcp(minr′′, (π′)R) and lcp((π′)R, maxs′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We only consider lcp(minr′′, (π′)R), the latter computation being analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We distinguish two subcases: a) r′′ = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Then lcp(minr′′, (π′)R) = lcp(minr, (π′)R) = min{lcp(minr, πR), |π′|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' b) r′′ < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' In particular, since ur′′ is the left endpoint of T(π′) and |T(π′)| ≥ 2, one can prove like in Case 1a) that maxr′′ is prefixed by (π′)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We immediately conclude that lcp(minr′′, (π′)R) = min{lcp(minr′′, maxr′′), |π′|}, which can be immediately computed since lcp(minr′′, maxr′′) is a value stored in LCPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' We can summarize the above discussion as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Given a Wheeler DFA A, there exists a data structure occupying O(|A|) words which can compute the pattern matching statistics of a pattern P in time O(|P| log |A|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Funding TG funded by National Institutes of Health (NIH) NIAID (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' HG011392), the National Science Foundation NSF IIBR (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' 2029552) and a Natural Science and Engineering Research Council (NSERC) Discovery Grant (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' RGPIN-07185-2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' GM funded by the Italian Ministry of University and Research (PRIN 2017WR7SHH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' MS funded by the INdAM-GNCS Project (CUP E55F22000270001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' NP funded by the European Union (ERC, REGINDEX, 101039208).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQf7Q53/content/2301.05338v1.pdf'} +page_content=' Views and opinions expressed are however those 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diff --git a/JtE0T4oBgHgl3EQfSADl/vector_store/index.faiss b/JtE0T4oBgHgl3EQfSADl/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..c5c0d951037c446171351dce45d13d1dab394476 --- /dev/null +++ b/JtE0T4oBgHgl3EQfSADl/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f59368c19d801f0e2558e91d5333ee73c6a2f9a032a1cf6896a1d13bee534513 +size 3735597 diff --git a/JtE0T4oBgHgl3EQfiQGP/content/tmp_files/2301.02443v1.pdf.txt b/JtE0T4oBgHgl3EQfiQGP/content/tmp_files/2301.02443v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..313668cbedbf52f9725f595f86cfaf3595f8415c --- /dev/null +++ b/JtE0T4oBgHgl3EQfiQGP/content/tmp_files/2301.02443v1.pdf.txt @@ -0,0 +1,1720 @@ +1 + +Exploring Euroleague History using Basic Statistics + + + +Christos Katris1,2 + +1Adjunct Lecturer, Department of Mathematics, University of Patras +2Customs Officer (Statistician), Independent Authority for Public Revenue, Greece + +1chriskatris@upatras.gr, 2c.katris1@aade.gr + + + + + + + + + + + + + +2 + +Abstract + +In this paper are used historical statistical data to track the evolution of the game in the European-wide +top-tier level professional basketball club competition (until 2017-2018 season) and also are answered +questions by analyzing them. The term basic is referred because of the nature of the data (not available +detailed statistics) and of the level of aggregation (not disaggregation to individual level). We are +examining themes such as the dominance per geographic area, the level of the competition in the game, +the evolution of scoring pluralism and possessions in the finals, the effect of a top scorer in the +performance of a team and the existence of unexpected outcomes in final fours. For each theme under +consideration, available statistical data is specified and suitable statistical analysis is applied. The analysis +allows us to handle and answer the above themes and interesting conclusions are drawn. This paper can +be an example of statistical thinking in basketball problems by the means of using efficiently available +statistical data. + +Keywords: Statistical analysis, basketball statistics, Euroleague evolution. + +1. Introduction +The field of basketball is ideal for the application of statistical methods in order to extract useful +conclusions which can help in analyzing many aspects of the game. The origin of many ideas is from +persons outside academia. The book of Oliver (2004) was a worthy attempt to develop and apply +statistical concepts in the area of basketball. Much information is included on this book and can offer to a +reader a statistical way of thinking for the game of basketball. There are also many academic papers +which use advanced statistical methods for basketball analysis in themes such as performance evaluation +of players and teams, home advantage effect etc. The field of basketball analytics is not yet entirely +unified and new ideas which are based on quantitative analysis are appearing continuously from diverse +academic fields. In many cases, there are used advanced statistics for the analysis of many situations. The +majority of studies – not only with USA origin - are related to NBA and this is not just a coincidence. The + +3 + +tracking system of statistics is superior to other leagues in terms of quality (calculation of more advanced +statistics) and quantity (calculation of more statistical categories), and the discrepancy was larger +especially in the past. + + The paper of Kubatko et al (2007) presents the general accepted basics of the analysis of basketball. +Furthermore, most of the statistics are based on the concept of possessions. However, this is not the case +for other leagues, including Euroleague. Given the available statistical data is difficult or even impossible +(for older years) to calculate neither possessions nor advanced statistics. Only after 2001 in the modern +era of Euroleague, plenty of statistics are available. + +This paper is an attempt to utilize available statistical information through statistical analysis in order to +explore the evolution of the game in Euroleague. It is demonstrated that even simple available statistics +can offer insights about the game and can be extracted useful conclusions. Graphical analysis, statistical +hypothesis testing and correlation measures are our weapons in this chase of insights related to the +evolution of Euroleague. The next section is a brief description of Euroleague and are referred the sources +of statistical data. Section 3 is the main part of the paper and contains statistical analysis and methods to +deal with questions related to the historical evolution of the tournament. Finally, in Section 4 are +presented the conclusions of the analysis. + +2. A Brief History of Euroleague and Statistical Data +In this paper is examined the evolution of the European-wide top-tier level professional basketball club +competition. Briefly the history of the competition is following. The FIBA European Champions Cup +competition has established in 1958 and FIBA was organizing its operation until 2000. Then Euroleague +Basketball was created. The next year, the two competitions were unified again under the umbrella of +Euroleague Basketball (for more details: https://en.wikipedia.org/wiki/EuroLeague). Also the competition +has changed names across time. From 1958 to 1991 was the FIBA European Champions Cup, from 1991 +to 1996 the name of the competition was FIBA European League, from 1996 to 2000 the name was FIBA + +4 + +Euroleague. In season 2000-2001 there were 2 competitions: FIBA Suproleague which was organized by +FIBA and Euroleague which was organized by Euroleague Basketball. From the next year there was a +unique competition for the top-tier level under the name Euroleague which was organized by Euroleague +Basketball. In 2016 the name changed to EuroLeague. For the rest of the article the name Euroleague is +used for the whole competition. The concept of final four applied for 1965-1966 and 1966-1967 seasons +and was included permanently in the competition from the season of 1987-1988. In this paper, we +consider as final four teams before 1986-1987, the teams which have reached the semi-finals in order to +generate a consistent system for studying the evolution of the tournament. + +There is not a unique data source which contains all information from the beginning of the tournament in +1958. Statistical data sources which were used are: Wikipedia, http://pearlbasket.altervista.org, +http://www.linguasport.com and http://www.fibaeurope.com/ and http://www.euroleague.net/ for stats +after 2001. + +3. Statistical Analysis of Euroleague Historical Data +In this section is made an attempt to shed light to questions related to the historical evolution of the game +with the use of suitable statistical methods. The graphs are created in excel, whilst for the implementation +of the methods is used statistical software R. + + +3.1 Dominance on the Game per Geographic Area +Firstly, we can derive some quick conclusions about the dominance in the game in terms of geographic +location. Table 1 displays per country the winners, the runners-up and the number of teams which had +appeared to final fours. We consider only the teams which participated to final fours since 1958. From +this limited statistical information we will explore briefly the game over time. + + +5 + +Based on Table 1, we consider the following Geographic areas: Spain and Italy which are leading the +table in all categories are considered separately, Ex USSR and ex Yugoslavian countries form the next +area and every other country is assigned to a fourth group (other). +Fig.1 displays the titles per time period of teams from each geographic area and Fig.2 displays the +appearances in final fours of teams from each geographic area. + +Table 1. Titles and appearances per country +Country +Winner +Runner-Up +Final Four Appearances +Number of Teams +Spain +13 +16 +57 +6 +Italy +13 +13 +44 +9 +Greece +9 +7 +13 +5 +Russia1 +7 +6 +17 +2 +Israel +6 +9 +20 +1 +Croatia2 +5 +1 +3 +3 +Latvia1 +3 +1 +4 +1 +Turkey +1 +2 +6 +2 +Lithuania1 +1 +1 +3 +1 +Georgia1 +1 +1 +3 +1 +Bosnia2 +1 +0 +4 +1 +Serbia2 +1 +0 +10 +4 +France +1 +0 +9 +4 +Czech Republic3 +0 +3 +9 +2 +Bulgaria +0 +2 +2 +1 +Slovenia2 +0 +0 +3 +1 +Poland +0 +0 +2 +2 +Romania +0 +0 +1 +1 +Netherlands +0 +0 +1 +1 +Hungary +0 +0 +1 +1 +Belgium +0 +0 +1 +1 +1 Trophies won before 1991 were under the umbrella of Soviet Union +2 Trophies won before 1995 were under the umbrella of Yugoslavia +3 Trophies won before 1991 were under the umbrella of Czechoslovakia + + + + + + + + + + + + + + + + + + +6 + +Fig.1. Titles evolution per geographic area + + + +Fig.2. Appearances to Final Four per geographic area + + + + + + +1958-1970 +1971-1980 +1981-1990 +1991-2000 +2001-2010 +2011-2018 +Spain +4 +3 +0 +2 +2 +2 +Italy +2 +4 +5 +1 +1 +0 +Ex USSR and Yugoslavian +7 +2 +4 +3 +2 +1 +Other +0 +1 +1 +4 +6 +5 +0 +1 +2 +3 +4 +5 +6 +7 +8 +Titles +Titles Evolution per Geographic Area +1958-1970 +1971-1980 +1981-1990 +1991-2000 +2001-2010 +2011-2018 +Spain +10 +9 +7 +11 +10 +10 +Italy +6 +12 +9 +7 +9 +1 +Ex USSR and Yugoslavian +20 +9 +12 +6 +10 +9 +Other +16 +10 +12 +16 +15 +12 +0 +5 +10 +15 +20 +25 +Appearances +Appearances to Final Four per geographic area + +7 + +To test formally if there are significant differences to the appearances and to the titles per geographic +area, we perform Friedman tests with titles (or appearances) per geographic area as treatments and time +periods as blocks (a blocking factor is a source of variability which is not of primary interest). We want to +check for significant differences to the titles and appearances per geographic area. Note that we want to +overall check the titles and appearances and not the trend, and we consider the time periods as blocks in +order to reduce their effect to the variability of titles and appearances respectively. The non-parametric +Friedman test is used in order not to have distributional assumptions, because normality assumption (data +to follow normal distribution) does not seem very likely. Details about the test can be found in every book +of non-parametric statistics such as that of (Hollander and Wolfe, 1999). + +The null hypothesis (𝐻0) is that apart of the effect of time period (blocks) there is no difference in titles +(or appearances) are even between the considered regions. The level of significance is considered at 5% +(0.05). To reject the null hypothesis, the p-value should be less than 0.05. +Friedman Test + +Statistic +df +p-value +Appearances +6.5789 +3 +0.0866 +Titles +0.7627 +3 +0.8584 + +From the application of the test we do not have enough evidence to suppose significant differences +between the performance of geographic areas in appearances and titles. However there are trends which +have been described graphically and discussed previously. + + +3.2 Dominance of the Champion +Next, is examined the dominance of the champion to its opponents and is measured in terms of scoring +points. The considered data are the points per game for and against the champions after the quarterfinals +because the potential existence of weak teams in earlier rounds may lead to instability of point +performance. In Fig.3 there are displayed the Points per Game (PPG) of the champion team and of their +opponents, while Fig.4 displays the Point difference as % of the points of the opponents of the champion + +8 + +team. This is considered as a metric of the dominance of the champion team against its opponents in +terms of scoring. + + +Fig.3. PPG for and against the champion team + + + + + + +Fig.4. Point difference as % of the opponents of the champion team + + + + +30.00% +40.00% +50.00% +60.00% +70.00% +80.00% +90.00% +1958-1970 1971-1980 1981-1990 1991-2000 2001-2010 2011-2018 +CR4 +CR5 +CR6 +CR7 +CR8 +0 +20 +40 +60 +80 +100 +120 +1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2001 2005 2009 2013 2017 +Points per Game +Points per game for and against the Champions +champion +opponent + +9 + +The above graph displays the points scored by the champion minus the points scored by opponents on +average, as percentage of opponent points. This could show in a sense how dominant was a champion. +Only in six seasons the champions scored more than 20% of their opponent points with Real Madrid to be +the only team which scored more than 30% of their opponents’ points in 1978. Extreme cases like this +should be examined in more detail. For example this team scored only 75 points in the final. +To better track the change of the game over time, we calculate the average points of the champions and +opponents in every decade and the average points per team and we draw their evolution across time in +Table 2 and Fig.5. +Table 2. Euroleague For and Against Points for Champion after Quarterfinals on average +Time Period +Average Points per Time Period +champion +opponent +Points per Team +1958-1970 +82.61 +72.90 +77.75 +1971- 1980 +91.20 +77.88 +84.54 +1981-1990 +88.27 +81.92 +85.09 +1991-2000 +72.92 +65.61 +69.27 +2001-2010 +82.94 +74.38 +78.66 +2011-2018 +81.86 +75.18 +78.52 + + + +Fig.5. Evolution of Average Points per team + + + + + +60 +65 +70 +75 +80 +85 +90 +Points per Team +Points per Team + +10 + + +From the above graph we can see the changes of the mentality of the game across time. From 1958 to +1990 there was an upward trend in scoring, with a sudden drop in 90s, something which indicates a +significant change in game mentality, and a return to the levels of 1958-1970 period. +Finally, the Fig.6 displays the decade average of the difference of points scored by champions minus the +points scored by opponents as percentage of the opponent points. It’s an indication of the dominance of +the champions of every decade. On average, the champions were more dominant in 60s and 70s, but in +the 80s they scored only 8% more than their opponents, a clear sign that the competition was more intense +in this decade. We also notice that the competitiveness of the last years (2010-2018) tends to similar +levels. + +Fig.6. Evolution of Point difference as % of the opponents of the champion team + + + + + + +3.3 Analyze Scoring Pluralism in the Finals: Evolution of the style of the game +In this subsection we want to follow the evolution of the game as pictured in finals. We use raw data which +are the first scorers and the team points of the finals since 1958. It is commonly assumed that in the runner-up +team there is a more dominant scorer, in terms of first scorer points as % of team points. To test this +hypothesis we perform a Wilcoxon rank sum test for pairs of observations for data from all finals since +1958. The null hypothesis (𝐻0) of the test is that the differences between the pairs follows a symmetric +0.00% +2.00% +4.00% +6.00% +8.00% +10.00% +12.00% +14.00% +16.00% +18.00% +1958-1970 1971- 1980 1981-1990 1991-2000 2001-2010 2011-2018 +Dominance of the Champion +Point Difference pct. + +11 + +distribution around zero. This is a non parametric test and through its application we avoid the +distributional assumption of normality of the data. A detailed description of the test can be found in +(Hollander and Wolfe, 1999). The test suggests that there is no reason to assume that in a specific year is +more probable the runner-up team to have a more dominant player in the scoring in final. + +Wilcoxon Signed Rank Test for paired Samples +Statistic +p-value +824 +0.1494 + +Additionally, we explore whether first scorer in terms of % of team points appears randomly or is more +probable to appear in sequences either from the champion or from the runner-up team. This can be +achieved through the application of a runs test in the difference of first scorer points as % of team points +between the two teams and we generate from this variable a sequence of + signs (if the variable is larger +than a threshold) and – signs (if the variable is smaller than a threshold). A run of a sequence is defined as +a series consisting of adjacent equal elements. We are testing the null hypothesis (H0) that each element +in the sequence is independently drawn from the same distribution. The threshold in our case is set to +zero. A description of the test can be found in (Gibbons and Chakraborti, 2003) and its implementation +performed via the randtests package of R (Caeiro and Mateus, 2014). +Through the application of the test, we can decide if over and under zero values are random. There is no +sign of non-randomness for this variable, so we can assume that the first scorer appears randomly from +the champion or from the runner-up team. +Runs Test +Statistic +Observations>0 +Observations<0 +Runs +p-value +~0 +24 +40 +31 +~1 + +The above tests are for the whole time period and they don’t reveal anything about the evolution of the game. The +rest of this section examines the evolution of the game and the statistical tests are adjusted accordingly. + +At first, we present graphically the 10 year moving average of the points scored by the first scorer in final as % of +team points in Fig.7. It is displayed the 10 year moving average for decreasing the effect of extreme cases +and is easier to follow the trend of the game. + +12 + +Fig.7. Moving average (10-year) of the points scored by the first scorer in final as % of team points + + + + + + +We observe that the game has been transformed over time from finals with offences which are based on +top scorers to finals with more pluralism. From the beginning until the 80s the trend was the one player +star in scoring, but since then, there was a slow but continuing turn to games based on pluralism. + +At the next step, in Fig. 8 we present graphically the 10 year moving average of the difference between +the points of first scorer as % of team points for champion team minus the same metric of runner-up team. + +Fig.8. Moving average (10-year) of the difference of points of first scorers as % of their team points + + + + +0% +5% +10% +15% +20% +25% +30% +35% +40% +45% +1966-1967 +1981-1982 +1996-1997 +2008-2009 +Points scored by 1st scorer as % of team points +10 year Moving Avarage +-8.00% +-6.00% +-4.00% +-2.00% +0.00% +2.00% +4.00% +1966-1967 +1978-1979 +1990-1991 +2002-2003 +2013-2014 +10-year Moving Average of difference + 10-year Moving Average +of difference + +13 + +From 1998 there is a downward trend until 2009 and a new cycle begins after 2009 and evolves but in a +lower level than the past. After 2001 there is no single year where the champion team has a more +dominant scorer than the runner-up team in terms of 10 year moving averages. + +We make an assumption that there is a structural break in this variable and is very important to specify the +time when it happened because is a clue that the game has changed at this moment. To achieve this, is +performed a Zivot-Andrew test (Zivot and Andrews, 1992) to test for the existence of a structural break +(null hypothesis 𝐻0) against the hypothesis of nonstationarity. + + + +Andrew-Zivot Test* +Statistic +p-value +Potential Break +-4.3499 +>0.1 +1997-1998 final + *We assume both level and linear trend and 5 lags + + +The existence of a structural break is in favour compared to non-stationarity. Moreover, it is important to +specify when the structural break occurs. The potential structural break occurs in 1997-1998 final. The +history of Euroleague can be break into 2 periods: before and after 1998, let’s say after 1998 is the +modern period of Euroleague. For this reason, we perform again the Wilcoxon sign rank test and the Runs +test for the modern period of Euroleague. In the modern period of Euroleague, we can assume that there is +a more dominant scorer in the runner-up team, but we cannot predict this fact for a specific year. + +Wilcoxon Signed Rank Test for paired Samples +Statistic +p-value +42.5 +0.02056 + + + +Runs Test +Statistic +Observations>0 +Observations<0 +Runs +p-value +1.5607 +5 +12 +11 +0.1186 + + + + + + + +14 + +3.4 Pace in the Finals: The concept of possessions +In this subsection we include to our analysis the central concept of possessions (Kubatko et al, 2007). +Larger number of possessions displays a quicker pace of a game and the intension is to track the evolution +of the game. + +We assume that both teams have the same number of possessions, but there is no unique formula for the +calculation of exact possessions in a game. For this reason, there are considered two formulas and we +average them in order to approximate more accurately the actual possessions. The used formulas are the +possessions lost (1) and the possessions gained (2) respectively: + 𝑃𝑂𝑆𝑆𝑡 = 𝐹𝐺𝐴𝑡 + 𝜆 × 𝐹𝑇𝐴𝑡 − 𝑂𝑅𝐸𝐵𝑡 + 𝑇𝑂𝑡 (1) +𝑃𝑂𝑆𝑆𝑡 = 𝐹𝐺𝑀𝑡 + 𝜆 × 𝐹𝑇𝑀𝑡 + 𝐷𝑅𝐸𝐵𝑜 + 𝑇𝑂𝑡 (2) + +After the calculation of the positions, we perform a line graph for the 5 year moving average of the +possessions in order to track their evolution (Fig.9). There is a downward trend and stability in low game +pace in the 90s, but from the beginning of the millennium there is a growing trend in game pace and from +2002 only four times there were fewer than 70 positions. + +Fig.9. Evolution of possessions as 5-year moving average + + + +60 +62 +64 +66 +68 +70 +72 +74 +76 +1987-1988 +1997-1998 +2006-2007 +2016-2017 +Possessions +5 year moving average + +15 + +Considering a break at 1997-1998 we perform a Mann-Whitney test for the equality of possessions before +and after 1998. This test is a non parametric equivalent of a t-test for comparing the means of 2 groups +when the data do not follow Normal distribution. Details can be found on (Hollander and Wolfe, 1999). + +Possessions +Before 1998 +After 1998 +66.25 +71.33 +Mann-Whitney Test for possessions +Statistic +p-value +84 +0.01028 + + +We detect a significant difference in possessions before and after 1998 finals at the 5% significance level. +This result is in accordance with our assumption that the game has changed after 1998 final and supports +the assumption that the triumph of Zalgiris in 1999 was the start of the change. + + + +3.5 Correlation of First Scorer with Team Performance +Another interesting question is whether the existence of a first scorer of a tournament is correlated with +the performance of the team. The most popular opinion is that first scorers belong to weak teams which +do not have offensive many good offensive players. The other opinion is that a very gifted scorer can +affect the performance of the team positively and relies to the coach to keep the balance of the team. We +have the first scorers of the tournament after 1992 and we measure the strength of the link between their +scoring performances (PPG) with the success of their team in the season using the Pearson (r) and +Spearman (ρ) correlation coefficients (Hollander and Wolfe, 1999; Best and Roberts, 1975). The +Spearman coefficient is non parametric and correlates the ranks of the variables and assesses monotonic +relationships between them (instead of linear relationships which are assessed from Pearson correlation). +Except from the coefficient, we perform a statistical test for testing the null hypothesis that the coefficient +(either r or ρ) is zero, thus there is not significant correlation between the variables. We assign values for +the performance of the teams: 1 for regular season, 2 for Top 16, 3 for quarterfinals, 4 for final four, 4.5 if + +16 + +the team was runner-up and 5 if the team won the trophy. Table 3 displays the first scorer, the team +position and the assigned values of the position. + + +𝑯𝟎: 𝝆 = 𝟎 𝒐𝒓 𝒓 = 𝟎 + +Coefficient +Statistic +p-value +Pearson Correlation +-0.4002 +-2.2271 +0.03481 +Spearman Correlation +-0.3925 +5088.032 +0.03886 + +Both Pearson and Spearman correlation coefficients indicate that there is a significant negative +relationship between the first scorer and the performance of the team. This finding rather favors the first +opinion where the first scorers are rarely parts of top teams (exception of Nando De Colo in 2015-2016 +confirms the general rule). +Table 3. First scorer of the tournament, team position and assigned values of the position +Season +Player +PPG +Team +Performance +Assigned Score +1991-1992 +Nikos Galis +32.3 +Aris +Regular season +1 +1992-1993 +Zdravko Radulović +23.9 +Cibona +Regular season +1 +1993-1994 +Nikos Galis +23.8 +Panathinaikos +3rd place +4 +1994-1995 +Sašha Danilović +22.1 +Buckler Bologna +Quarterfinals +3 +1995-1996 +Joe Arlauckas +26.4 +Real Madrid +4th place +4 +1996-1997 +Carlton Myers +22.9 +Teamsystem Bologna +Quarterfinals +3 +1997-1998 +Peja Stojaković +20.9 +PAOK +Top 16 +2 +1998-1999 +İbrahim Kutluay +21.4 +Fenerbahçe +Top 16 +2 +1999-2000 +Miljan Goljović +20.2 +Pivovarna Laško +Regular season +1 +2000-2001 +(FIBA) +Miroslav Berić +23.3 +Partizan +Top 16 +2 +2000-2001 +(Euroleague) +Alphonso Ford +26 +Peristeri +Top 16 +2 +2001-2002 +Alphonso Ford +24.8 +Olympiacos +Top 16 +2 +2002-2003 +Miloš Vujanić +25.8 +Partizan +Regular season +1 +2003-2004 +Lynn Greer +25.1 +Śląsk Wrocław +Regular season +1 +2004-2005 +Charles Smith +20.7 +Scavolini Pesaro +Quarterfinals +3 +2005-2006 +Drew Nicholas +18.5 +Benetton Treviso +Top 16 +2 +2006-2007 +Igor Rakočević +16.2 +Tau Cerámica +4th place +4 +2007-2008 +Marc Salyers +21.8 +Roanne +Regular season +1 +2008-2009 +Igor Rakočević +18 +Tau Cerámica +Quarterfinals +3 +2009-2010 +Linas Kleiza +17.1 +Olympiacos +2nd place +4.5 +2010-2011 +Igor Rakočević +17.2 +Efes Pilsen +Top 16 +2 +2011-2012 +Bo McCalebb +16.9 +Montepaschi Siena +Quarterfinals +3 +2012-2013 +Bobby Brown +18.8 +Montepaschi Siena +Top 16 +2 + +17 + +2013-2014 +Keith Langford +17.6 +EA7 Milano +Quarterfinals +3 +2014-2015 +Taylor Rochestie +18.9 +Nizhny Novgorod +Top 16 +2 +2015-2016 +Nando de Colo +18.9 +CSKA Moscow +Winner +5 +2016-2017 +Keith Langford +21.8 +UNICS +Regular season +1 +2017-2018 +Alexey Shved +21.8 +Khimki +Quarterfinals +3 + +3.6 Unexpected Outcomes in the Final Fours +In this section it is examined the unexpected of the Final-Fours in terms of outcomes based on previous +attempts with the use of Binomial Distribution. Can we make the assumption that each final four is an +experiment with each team to have the same probabilities of winning the tournament (25%)? +To answer this question, we consider each final four as an experiment and teams are considered as +independent random variables. Each experiment can be described by the binomial distribution and the +whole situation with multinomial distribution (Forbes et al, 2011) which is a generalization of binomial +distribution and describes n trials. There is performed a multinomial goodness of fit test and to strengthen +the results a binomial test for each team, in order to decide if there is any significant difference from +binomial distribution. +Multinomial Testing +p-value + +0.54499±0.001575 +Binomial Testing* + + +p-value +Cibona +2 attempts - 2 trophies +0.0625 +Jugoplastica +4 attempts - 3 trophies +0.05078 +ASK Riga +4 attempts - 3 trophies +0.05078 +Panathinaikos +12 attempts - 6 trophies +0.08608 +*Only cases with p-value<0.1 + +Table on the appendix displays the final four teams, the expected titles according to Binomial distribution, +the observed values and their difference. Indeed there is no evidence that there are significant +discrepancies from the binomial distribution at the 5% level of significance. +In the modern period of Euroleague (1999-2018), again there is no evidence of significant discrepancy +from the multinomial distribution, but the case of Panathinaikos could be seen as an exception, with +significant larger success rate than the expected. + + +18 + +Multinomial Testing +p-value + +0.68173±0.001473 +Binomial Testing* + + +p-value +Panathinaikos +8 attempts - 5 trophies +0.02730 +*Only cases with p-value<0.1 + + +4 Summary and Conclusions +To sum up, in this paper is made an attempt to address questions related to historical evolution of +Euroleague using statistical analysis to draw conclusions. One main problem is the lack of plenty +available statistical data from the beginning of the competition. This paper demonstrates that by applying +suitable statistical designs we can draw interesting conclusions even with limited data. Firstly, is made a +brief exploration of the historical evolution of the Euroleague and the tracking statistics. +Then, some questions are answered and some conclusions are drawn which are briefly the following: +Although overall there is no difference in success between more traditional powers such as Italy, Spain +and ex USSR and Yugoslavian countries and other countries, there is a clear trend of other countries to +expand their presence (in terms of titles and final four appearances) in the tournament after the 90s. In +terms of scoring, there was an upward trend from the beginning of the competition, with a sudden drop in +90s, something which indicates a significant change in game mentality in terms of defence and/or game +pace. The champions were more dominant in 60s and 70s, but in the 80s they scored only 8% more than +their opponent, which indicates that the competition was more intense in this decade. The last years +(2010-2018), the competitiveness of the tournament tends to similar levels. +There is a popular belief that the first scorer in the majority of cases come from the runner-up team. +However, there is no reason to assume that in a specific year is more probable the runner-up team to have +a more dominant player in the scoring in the final. In the modern period of Euroleague (after 1998), we +can assume that there is a more dominant scorer in the runner-up team, but we cannot predict this fact for +a specific year. According to the game evolution in finals, we observe that the game has been transformed +from finals with offences which are based on top scorers to finals with more pluralism. From the +beginning until the 80s the trend was the one player star in scoring, but since then, there was a slow but + +19 + +continuing turn to games based on pluralism. Moreover, we detect a significant difference in possessions +before and after 1998 finals, which is in accordance with our assumption that the game has changed after +1998 final and supports the assumption that the triumph of Zalgiris in 1999 was the start of the change. +Furthermore, it is found a significant negative relationship between the first scorer and the performance of +his team. This finding favors the opinion that the first scorers are rarely parts of top teams. Finally, there +is no evidence to reject the hypothesis that in a final four there are equal chances of winning overall. In +modern era, again the hypothesis of the final four as a random experiment is not rejected, however in the +case of Panathinaikos we observe significantly higher success rate than the expected. + + +References +Best, D. J., & Roberts, D. E. 1975. “Algorithm AS 89: the upper tail probabilities of Spearman's rho.” Journal of the +Royal Statistical Society. Series C (Applied Statistics), 24(3), 377-379. +Caeiro, F., & Mateus, A. 2014. randtests: Testing randomness in R. R package version, 1. +Forbes, C., Evans, M., Hastings, N., & Peacock, B. 2011.Statistical distributions. John Wiley & Sons. +Gibbons, J. D., & Chakraborti, S. 2003. Nonparametric Statistical Inference. Marcel Dekker. Inc. New York. +Hollander, M., & Wolfe, D. A. 1999. Nonparametric statistical methods. 2nd Edition, John Wiley & Sons. +Kubatko, J., Oliver, D., Pelton, K., & Rosenbaum, D. T. 2007. “A starting point for analyzing basketball statistics.” +Journal of Quantitative Analysis in Sports, 3(3). +Oliver, D. (2004). Basketball on paper: rules and tools for performance analysis. Potomac Books, Inc. +Zivot, E., & Andrews, D. W. K. 2002. “Further evidence on the great crash, the oil-price shock, and the unit-root +hypothesis.” Journal of business & economic statistics, 20(1), 25-44. + + + + + +20 + +Appendix +Table. Euroleague Final Four Teams +Year +Winner +Runner-Up +3rd Place +4th Place +1958 +ASK Riga +Academic +Honved +Real Madrid +1958- 1959 +ASK Riga +Academic +Lech Poznan +OKK Beograd +1959-1960 +ASK Riga +Dinamo Tbilisi +Pologna Warzawa +Slovan Orbis Praha +1960-1961 +CSKA Moscow +ASK Riga +Real Madrid +Steaua Bucarest +1961-1962 +Dinamo Tbilisi +Real Madrid +ASK Olimpija +CSKA Moscow +1962-1963 +CSKA Moscow +Real Madrid +Dinamo Tbilisi +Spartak Brno +1963-1964 +Real Madrid +Spartak Brno +OKK Beograd +Simmenthal Milano +1964-1965 +Real Madrid +CSKA Moscow +Ignis Varese +OKK Beograd +1965-1966 +Simmenthal Milano +Slavia Praha +CSKA Moscow +AEK +1966-1967 +Real Madrid +Simmenthal Milano +ASK Olimpija +Slavia Praha +1967-1968 +Real Madrid +Spartak Brno +Simmenthal Milano +Zadar +1968-1969 +CSKA Moscow +Real Madrid +Spartak Brno +Standard Liege +1969-1970 +Ignis Varese +CSKA Moscow +Real Madrid +Slavia Praha +1970-1971 +CSKA Moscow +Ignis Varese +Real Madrid +Slavia Praha +1971-1972 +Ignis Varese +Jugoplastica1 +Real Madrid +Panathinaikos +1972-1973 +Ignis Varese +CSKA Moscow +Crvena Zvezda +Simmenthal Milano +1973-1974 +Real Madrid +Ignis Varese +Berck +Radniski Belgrade +1974-1975 +Ignis Varese +Real Madrid +Berck +Zadar +1975-1976 +Mobilgirgi Varese +Real Madrid +ASVEL +Forst Cantù +1976-1977 +Maccabi Tel Aviv +Mobilgirgi Varese +CSKA Moscow +Real Madrid +1977-1978 +Real Madrid +Mobilgirgi Varese +ASVEL +Maccabi Tel Aviv +1978-1979 +Bosna +Emerson Varese +Maccabi Tel Aviv +Real Madrid +1979-1980 +Real Madrid +Maccabi Tel Aviv +Bosna +Sinudyne Bologna3 +1980-1981 +Maccabi Tel Aviv +Sinudyne Bologna3 +Nashua EBBC +Bosna +1981-1982 +Squibb Cantù +Maccabi Tel Aviv +Partizan +FC Barcelona +1982-1983 +Ford Cantù +Billy Milano +Real Madrid +CSKA Moscow +1983-1984 +Virtus Roma +FC Barcelona +Jollycolombani Cantù +Bosna +1984-1985 +Cibona +Real Madrid +Maccabi Tel Aviv +CSKA Moscow +1985-1986 +Cibona +Zalgiris +Simac Milano +Real Madrid +1986-1987 +Tracer Milano +Maccabi Tel Aviv +Orthez +Zadar +1987-1988 +Tracer Milano +Maccabi Tel Aviv +Partizan +Aris +1988-1989 +Jugoplastica1 +Maccabi Tel Aviv +Aris +FC Barcelona +1989-1990 +Jugoplastica1 +FC Barcelona +Limoges +Aris +1990-1991 +POP 84 1 +FC Barcelona +Maccabi Tel Aviv +Scavolini Pezaro +1991-1992 +Partizan +Joventut +Phillips Milano +Estudiantes +1992-1993 +Limoges +Benneton Treviso +PAOK +Real Madrid +1993-1994 +Joventut +Olympiacos +Panathinaikos +FC Barcelona +1994-1995 +Real Madrid +Olympiacos +Panathinaikos +Limoges +1995-1996 +Panathinaikos +FC Barcelona +CSKA Moscow +Real Madrid +1996-1997 +Olympiacos +FC Barcelona +Smelt Olimpija +ASVEL +1997-1998 +Kinder Bologna3 +AEK +Benneton Treviso +Partizan +1998-1999 +Zalgiris +Kinder Bologna3 +Olympiacos +Teamsystem Bologna4 +1999-2000 +Panathinaikos +Maccabi Tel Aviv +Efes Pilsen +FC Barcelona +2000-2001 (FIBA) +Kinder Bologna3 +TAU Ceramica2 +AEK +Paf Wennington Bologna4 +2000-2001 (Euroleague) +Maccabi Tel Aviv +Panathinaikos +Efes Pilsen +CSKA Moscow +2001-2002 +Panathinaikos +Kinder Bologna3 +Benneton Treviso +Maccabi Tel Aviv +2002-2003 +FC Barcelona +Benneton Treviso +Montepaschi Siena +CSKA Moscow +2003-2004 +Maccabi Tel Aviv +Skipper Bologna4 +CSKA Moscow +Montepaschi Siena +2004-2005 +Maccabi Tel Aviv +TAU Ceramica2 +Panathinaikos +CSKA Moscow +2005-2006 +CSKA Moscow +Maccabi Tel Aviv +TAU Ceramica2 +FC Barcelona +2006-2007 +Panathinaikos +CSKA Moscow +Unicaja Malaga +TAU Ceramica2 +2007-2008 +CSKA Moscow +Maccabi Tel Aviv +Montepaschi Siena +Real Madrid +2008-2009 +Panathinaikos +CSKA Moscow +FC Barcelona +Olympiacos +2009-2010 +FC Barcelona +Olympiacos +CSKA Moscow +Partizan +2010-2011 +Panathinaikos +Maccabi Tel Aviv +Montepaschi Siena +Real Madrid +2011-2012 +Olympiacos +CSKA Moscow +FC Barcelona +Panathinaikos +2012-2013 +Olympiacos +Real Madrid +CSKA Moscow +FC Barcelona +2013-2014 +Maccabi Tel Aviv +Real Madrid +FC Barcelona +CSKA Moscow +2014-2015 +Real Madrid +Olympiacos +CSKA Moscow +Fenerbahce +2015-2016 +CSKA Moscow +Fenerbahce +Lokomotiv Kuban +Laboral Kutxa2 +2016-2017 +Fenerbahce +Olympiacos +CSKA Moscow +Real Madrid +2017-2018 +Real Madrid +Fenerbahce +Zalgiris +CSKA Moscow +1Croatia Split +2Club Deportivo Saski Baskonia, S.A.D. +3Virtus Pallacanestro Bologna +4 Fortitudo Pallacanestro Bologna 103 + +21 + +Table . Teams and appearances to final four +Team +Winner +Appearances +Expected Titles +Observed Titles +Difference +Lokomotiv Kuban +0 +1 +0.25 +0 +-0.25 +Unicaja Malaga +0 +1 +0.25 +0 +-0.25 +PAOK +0 +1 +0.25 +0 +-0.25 +Estudiantes +0 +1 +0.25 +0 +-0.25 +Scavolini Pezaro +0 +1 +0.25 +0 +-0.25 +Orthez +0 +1 +0.25 +0 +-0.25 +Nashua EBBC +0 +1 +0.25 +0 +-0.25 +Radniski Belgrade +0 +1 +0.25 +0 +-0.25 +Crvena Zvezda +0 +1 +0.25 +0 +-0.25 +Standard Liege +0 +1 +0.25 +0 +-0.25 +Steaua Bucarest +0 +1 +0.25 +0 +-0.25 +Lech Poznan +0 +1 +0.25 +0 +-0.25 +Honved +0 +1 +0.25 +0 +-0.25 +Pologna Warzawa +0 +1 +0.25 +0 +-0.25 +Efes Pilsen +0 +2 +0.5 +0 +-0.5 +Berck +0 +2 +0.5 +0 +-0.5 +Zadar +0 +3 +0.75 +0 +-0.75 +Olimpija Ljubliana +0 +3 +0.75 +0 +-0.75 +OKK Beograd +0 +3 +0.75 +0 +-0.75 +ASVEL +0 +3 +0.75 +0 +-0.75 +Aris +0 +3 +0.75 +0 +-0.75 +Montepaschi Siena +0 +4 +1 +0 +-1 +Fortitudo Bologna +0 +3 +0.75 +0 +-0.75 +AEK +0 +3 +0.75 +0 +-0.75 +Praha +0 +5 +1.25 +0 +-1.25 +Baskonia +0 +5 +1.25 +0 +-1.25 +Treviso +0 +4 +1 +0 +-1 +Brno +0 +4 +1 +0 +-1 +Academic +0 +2 +0.5 +0 +-0.5 +Limoges +1 +3 +0.75 +1 +0.25 +Partizan +1 +5 +1.25 +1 +-0.25 +Virtus Roma +1 +1 +0.25 +1 +0.75 +Bosna +1 +4 +1 +1 +0 +Zalgiris +1 +3 +0.75 +1 +0.25 +Joventut Badalona +1 +2 +0.5 +1 +0.5 +Dinamo Tbilisi +1 +3 +0.75 +1 +0.25 +Fenerbahce +1 +4 +1 +1 +0 +Cibona +2 +2 +0.5 +2 +1.5 +Cantu +2 +4 +1 +2 +1 +Virtus Bologna +2 +6 +1.5 +2 +0.5 +FC Barcelona +2 +16 +4 +2 +-2 +Split +3 +4 +1 +3 +2 +ASK Riga (Latvia) +3 +4 +1 +3 +2 +Olympia Milano +3 +10 +2.5 +3 +0.5 +Olympiacos +3 +10 +2.5 +3 +0.5 +Varese +5 +11 +2.75 +5 +2.25 +Panathinaikos +6 +12 +3 +6 +3 +Maccabi Tel Aviv +6 +20 +5 +6 +1 +CSKA Moscow +7 +29 +7.25 +7 +-0.25 +Real Madrid +10 +32 +8 +10 +2 + + + + + diff --git a/JtE0T4oBgHgl3EQfiQGP/content/tmp_files/load_file.txt b/JtE0T4oBgHgl3EQfiQGP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..445745e98fb9e542d0c426c690cf55ffdff9f2b1 --- /dev/null +++ b/JtE0T4oBgHgl3EQfiQGP/content/tmp_files/load_file.txt @@ -0,0 +1,1075 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf,len=1074 +page_content='1 Exploring Euroleague History using Basic Statistics Christos Katris1,2 1Adjunct Lecturer, Department of Mathematics, University of Patras 2Customs Officer (Statistician), Independent Authority for Public Revenue, Greece 1chriskatris@upatras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='gr, 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='katris1@aade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='gr 2 Abstract In this paper are used historical statistical data to track the evolution of the game in the European-wide top-tier level professional basketball club competition (until 2017-2018 season) and also are answered questions by analyzing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The term basic is referred because of the nature of the data (not available detailed statistics) and of the level of aggregation (not disaggregation to individual level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' We are examining themes such as the dominance per geographic area, the level of the competition in the game, the evolution of scoring pluralism and possessions in the finals, the effect of a top scorer in the performance of a team and the existence of unexpected outcomes in final fours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' For each theme under consideration, available statistical data is specified and suitable statistical analysis is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The analysis allows us to handle and answer the above themes and interesting conclusions are drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' This paper can be an example of statistical thinking in basketball problems by the means of using efficiently available statistical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Keywords: Statistical analysis, basketball statistics, Euroleague evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Introduction The field of basketball is ideal for the application of statistical methods in order to extract useful conclusions which can help in analyzing many aspects of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The origin of many ideas is from persons outside academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The book of Oliver (2004) was a worthy attempt to develop and apply statistical concepts in the area of basketball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Much information is included on this book and can offer to a reader a statistical way of thinking for the game of basketball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' There are also many academic papers which use advanced statistical methods for basketball analysis in themes such as performance evaluation of players and teams, home advantage effect etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The field of basketball analytics is not yet entirely unified and new ideas which are based on quantitative analysis are appearing continuously from diverse academic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' In many cases, there are used advanced statistics for the analysis of many situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The majority of studies – not only with USA origin - are related to NBA and this is not just a coincidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The 3 tracking system of statistics is superior to other leagues in terms of quality (calculation of more advanced statistics) and quantity (calculation of more statistical categories), and the discrepancy was larger especially in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The paper of Kubatko et al (2007) presents the general accepted basics of the analysis of basketball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Furthermore, most of the statistics are based on the concept of possessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' However, this is not the case for other leagues, including Euroleague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Given the available statistical data is difficult or even impossible (for older years) to calculate neither possessions nor advanced statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Only after 2001 in the modern era of Euroleague, plenty of statistics are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' This paper is an attempt to utilize available statistical information through statistical analysis in order to explore the evolution of the game in Euroleague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' It is demonstrated that even simple available statistics can offer insights about the game and can be extracted useful conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Graphical analysis, statistical hypothesis testing and correlation measures are our weapons in this chase of insights related to the evolution of Euroleague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The next section is a brief description of Euroleague and are referred the sources of statistical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Section 3 is the main part of the paper and contains statistical analysis and methods to deal with questions related to the historical evolution of the tournament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Finally, in Section 4 are presented the conclusions of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' A Brief History of Euroleague and Statistical Data In this paper is examined the evolution of the European-wide top-tier level professional basketball club competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Briefly the history of the competition is following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The FIBA European Champions Cup competition has established in 1958 and FIBA was organizing its operation until 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Then Euroleague Basketball was created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The next year, the two competitions were unified again under the umbrella of Euroleague Basketball (for more details: https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='org/wiki/EuroLeague).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Also the competition has changed names across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' From 1958 to 1991 was the FIBA European Champions Cup, from 1991 to 1996 the name of the competition was FIBA European League, from 1996 to 2000 the name was FIBA 4 Euroleague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' In season 2000-2001 there were 2 competitions: FIBA Suproleague which was organized by FIBA and Euroleague which was organized by Euroleague Basketball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' From the next year there was a unique competition for the top-tier level under the name Euroleague which was organized by Euroleague Basketball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' In 2016 the name changed to EuroLeague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' For the rest of the article the name Euroleague is used for the whole competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The concept of final four applied for 1965-1966 and 1966-1967 seasons and was included permanently in the competition from the season of 1987-1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' In this paper, we consider as final four teams before 1986-1987, the teams which have reached the semi-finals in order to generate a consistent system for studying the evolution of the tournament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' There is not a unique data source which contains all information from the beginning of the tournament in 1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Statistical data sources which were used are: Wikipedia, http://pearlbasket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='altervista.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='org, http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='linguasport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='com and http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='fibaeurope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='com/ and http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='euroleague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='net/ for stats after 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Statistical Analysis of Euroleague Historical Data In this section is made an attempt to shed light to questions related to the historical evolution of the game with the use of suitable statistical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The graphs are created in excel, whilst for the implementation of the methods is used statistical software R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='1 Dominance on the Game per Geographic Area Firstly, we can derive some quick conclusions about the dominance in the game in terms of geographic location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Table 1 displays per country the winners, the runners-up and the number of teams which had appeared to final fours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' We consider only the teams which participated to final fours since 1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' From this limited statistical information we will explore briefly the game over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' 5 Based on Table 1, we consider the following Geographic areas: Spain and Italy which are leading the table in all categories are considered separately, Ex USSR and ex Yugoslavian countries form the next area and every other country is assigned to a fourth group (other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='1 displays the titles per time period of teams from each geographic area and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='2 displays the appearances in final fours of teams from each geographic area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Titles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='appearances ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='per ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='country ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Country ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Winner ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Runner-Up ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Final ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Four ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Appearances ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Teams ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Spain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='57 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Italy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='13 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Titles evolution per geographic area Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Appearances to Final Four per geographic area ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='1958-1970 1971-1980 1981-1990 1991-2000 2001-2010 2011-2018 Spain 4 3 0 2 2 2 Italy 2 4 5 1 1 0 Ex USSR and Yugoslavian 7 2 4 3 2 1 Other 0 1 1 4 6 5 0 1 2 3 4 5 6 7 8 Titles Titles Evolution per Geographic Area 1958-1970 1971-1980 1981-1990 1991-2000 2001-2010 2011-2018 Spain 10 9 7 11 10 10 Italy 6 12 9 7 9 1 Ex USSR and Yugoslavian 20 9 12 6 10 9 Other 16 10 12 16 15 12 0 5 10 15 20 25 Appearances Appearances to Final Four per geographic area ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='To test formally if there are significant differences to the appearances and to the titles per geographic area,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' we perform Friedman tests with titles (or appearances) per geographic area as treatments and time periods as blocks (a blocking factor is a source of variability which is not of primary interest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' We want to check for significant differences to the titles and appearances per geographic area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Note that we want to overall check the titles and appearances and not the trend, and we consider the time periods as blocks in order to reduce their effect to the variability of titles and appearances respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The non-parametric Friedman test is used in order not to have distributional assumptions, because normality assumption (data to follow normal distribution) does not seem very likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Details about the test can be found in every book of non-parametric statistics such as that of (Hollander and Wolfe, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The null hypothesis (𝐻0) is that apart of the effect of time period (blocks) there is no difference in titles (or appearances) are even between the considered regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The level of significance is considered at 5% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' To reject the null hypothesis, the p-value should be less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Friedman Test Statistic df p value Appearances 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='5789 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='0866 Titles 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='7627 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='8584 From the application of the test we do not have enough evidence to suppose significant differences between the performance of geographic areas in appearances and titles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' However there are trends which have been described graphically and discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='2 Dominance of the Champion Next, is examined the dominance of the champion to its opponents and is measured in terms of scoring points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The considered data are the points per game for and against the champions after the quarterfinals because the potential existence of weak teams in earlier rounds may lead to instability of point performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='3 there are displayed the Points per Game (PPG) of the champion team and of their opponents, while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='4 displays the Point difference as % of the points of the opponents of the champion 8 team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' This is considered as a metric of the dominance of the champion team against its opponents in terms of scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' PPG for and against the champion team Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Point difference as % of the opponents of the champion team 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 1958-1970 1971-1980 1981-1990 1991-2000 2001-2010 2011-2018 CR4 CR5 CR6 CR7 CR8 0 20 40 60 80 100 120 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2001 2005 2009 2013 2017 Points per Game Points per game for and against the Champions champion opponent 9 The above graph displays the points scored by the champion minus the points scored by opponents on average, as percentage of opponent points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' This could show in a sense how dominant was a champion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Only in six seasons the champions scored more than 20% of their opponent points with Real Madrid to be the only team which scored more than 30% of their opponents’ points in 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Extreme cases like this should be examined in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' For example this team scored only 75 points in the final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' To better track the change of the game over time, we calculate the average points of the champions and opponents in every decade and the average points per team and we draw their evolution across time in Table 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Euroleague For and Against Points for Champion after Quarterfinals on average Time Period Average Points per Time Period champion opponent Points per Team 1958-1970 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='61 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='90 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='75 1971- 1980 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='20 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='88 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='54 1981-1990 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='27 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='92 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='09 1991-2000 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='92 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='61 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='27 2001-2010 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='94 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='38 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='66 2011-2018 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='86 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='18 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='52 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Evolution of Average Points per team 60 65 70 75 80 85 90 Points per Team Points per Team 10 From the above graph we can see the changes of the mentality of the game across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' From 1958 to 1990 there was an upward trend in scoring, with a sudden drop in 90s, something which indicates a significant change in game mentality, and a return to the levels of 1958-1970 period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Finally, the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='6 displays the decade average of the difference of points scored by champions minus the points scored by opponents as percentage of the opponent points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' It’s an indication of the dominance of the champions of every decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' On average, the champions were more dominant in 60s and 70s, but in the 80s they scored only 8% more than their opponents, a clear sign that the competition was more intense in this decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' We also notice that the competitiveness of the last years (2010-2018) tends to similar levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Evolution of Point difference as % of the opponents of the champion team 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='3 Analyze Scoring Pluralism in the Finals: Evolution of the style of the game In this subsection we want to follow the evolution of the game as pictured in finals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' We use raw data which are the first scorers and the team points of the finals since 1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' It is commonly assumed that in the runner-up team there is a more dominant scorer, in terms of first scorer points as % of team points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' To test this hypothesis we perform a Wilcoxon rank sum test for pairs of observations for data from all finals since 1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The null hypothesis (𝐻0) of the test is that the differences between the pairs follows a symmetric 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 1958-1970 1971- 1980 1981-1990 1991-2000 2001-2010 2011-2018 Dominance of the Champion Point Difference pct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' 11 distribution around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' This is a non parametric test and through its application we avoid the distributional assumption of normality of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' A detailed description of the test can be found in (Hollander and Wolfe, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The test suggests that there is no reason to assume that in a specific year is more probable the runner-up team to have a more dominant player in the scoring in final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Wilcoxon Signed Rank Test for paired Samples Statistic p-value 824 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='1494 Additionally, we explore whether first scorer in terms of % of team points appears randomly or is more probable to appear in sequences either from the champion or from the runner-up team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' This can be achieved through the application of a runs test in the difference of first scorer points as % of team points between the two teams and we generate from this variable a sequence of + signs (if the variable is larger than a threshold) and – signs (if the variable is smaller than a threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' A run of a sequence is defined as a series consisting of adjacent equal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' We are testing the null hypothesis (H0) that each element in the sequence is independently drawn from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The threshold in our case is set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' A description of the test can be found in (Gibbons and Chakraborti, 2003) and its implementation performed via the randtests package of R (Caeiro and Mateus, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Through the application of the test, we can decide if over and under zero values are random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' There is no sign of non-randomness for this variable, so we can assume that the first scorer appears randomly from the champion or from the runner-up team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Runs Test Statistic Observations>0 Observations<0 Runs p-value ~0 24 40 31 ~1 The above tests are for the whole time period and they don’t reveal anything about the evolution of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The rest of this section examines the evolution of the game and the statistical tests are adjusted accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' At first, we present graphically the 10 year moving average of the points scored by the first scorer in final as % of team points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' It is displayed the 10 year moving average for decreasing the effect of extreme cases and is easier to follow the trend of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Moving average (10-year) of the points scored by the first scorer in final as % of team points We observe that the game has been transformed over time from finals with offences which are based on top scorers to finals with more pluralism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' From the beginning until the 80s the trend was the one player star in scoring, but since then, there was a slow but continuing turn to games based on pluralism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' At the next step, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' 8 we present graphically the 10 year moving average of the difference between the points of first scorer as % of team points for champion team minus the same metric of runner-up team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Moving average (10-year) of the difference of points of first scorers as % of their team points 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 1966-1967 1981-1982 1996-1997 2008-2009 Points scored by 1st scorer as % of team points 10 year Moving Avarage -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='00% 1966-1967 1978-1979 1990-1991 2002-2003 2013-2014 10-year Moving Average of difference 10-year Moving Average of difference 13 From 1998 there is a downward trend until 2009 and a new cycle begins after 2009 and evolves but in a lower level than the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' After 2001 there is no single year where the champion team has a more dominant scorer than the runner-up team in terms of 10 year moving averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' We make an assumption that there is a structural break in this variable and is very important to specify the time when it happened because is a clue that the game has changed at this moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' To achieve this, is performed a Zivot-Andrew test (Zivot and Andrews, 1992) to test for the existence of a structural break (null hypothesis 𝐻0) against the hypothesis of nonstationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Andrew-Zivot Test* Statistic p-value Potential Break -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='3499 >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='1 1997-1998 final *We assume both level and linear trend and 5 lags The existence of a structural break is in favour compared to non-stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Moreover, it is important to specify when the structural break occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The potential structural break occurs in 1997-1998 final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The history of Euroleague can be break into 2 periods: before and after 1998, let’s say after 1998 is the modern period of Euroleague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' For this reason, we perform again the Wilcoxon sign rank test and the Runs test for the modern period of Euroleague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' In the modern period of Euroleague, we can assume that there is a more dominant scorer in the runner-up team, but we cannot predict this fact for a specific year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Wilcoxon Signed Rank Test for paired Samples Statistic p-value 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='02056 Runs Test Statistic Observations>0 Observations<0 Runs p value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='5607 5 12 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='1186 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='4 Pace in the Finals: The concept of possessions In this subsection we include to our analysis the central concept of possessions (Kubatko et al, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Larger number of possessions displays a quicker pace of a game and the intension is to track the evolution of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' We assume that both teams have the same number of possessions, but there is no unique formula for the calculation of exact possessions in a game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' For this reason, there are considered two formulas and we average them in order to approximate more accurately the actual possessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The used formulas are the possessions lost (1) and the possessions gained (2) respectively: 𝑃𝑂𝑆𝑆𝑡 = 𝐹𝐺𝐴𝑡 + 𝜆 × 𝐹𝑇𝐴𝑡 − 𝑂𝑅𝐸𝐵𝑡 + 𝑇𝑂𝑡 (1) 𝑃𝑂𝑆𝑆𝑡 = 𝐹𝐺𝑀𝑡 + 𝜆 × 𝐹𝑇𝑀𝑡 + 𝐷𝑅𝐸𝐵𝑜 + 𝑇𝑂𝑡 (2) After the calculation of the positions, we perform a line graph for the 5 year moving average of the possessions in order to track their evolution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' There is a downward trend and stability in low game pace in the 90s, but from the beginning of the millennium there is a growing trend in game pace and from 2002 only four times there were fewer than 70 positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Evolution of possessions as 5-year moving average 60 62 64 66 68 70 72 74 76 1987 1988 1997 1998 2006 2007 2016 2017 Possessions 5 year moving average 15 Considering a break at 1997-1998 we perform a Mann-Whitney test for the equality of possessions before and after 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' This test is a non parametric equivalent of a t-test for comparing the means of 2 groups when the data do not follow Normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Details can be found on (Hollander and Wolfe, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Possessions Before 1998 After 1998 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='25 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='33 Mann Whitney Test for possessions Statistic p value 84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='01028 We detect a significant difference in possessions before and after 1998 finals at the 5% significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' This result is in accordance with our assumption that the game has changed after 1998 final and supports the assumption that the triumph of Zalgiris in 1999 was the start of the change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='5 Correlation of First Scorer with Team Performance Another interesting question is whether the existence of a first scorer of a tournament is correlated with the performance of the team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The most popular opinion is that first scorers belong to weak teams which do not have offensive many good offensive players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The other opinion is that a very gifted scorer can affect the performance of the team positively and relies to the coach to keep the balance of the team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' We have the first scorers of the tournament after 1992 and we measure the strength of the link between their scoring performances (PPG) with the success of their team in the season using the Pearson (r) and Spearman (ρ) correlation coefficients (Hollander and Wolfe, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Best and Roberts, 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The Spearman coefficient is non parametric and correlates the ranks of the variables and assesses monotonic relationships between them (instead of linear relationships which are assessed from Pearson correlation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Except from the coefficient, we perform a statistical test for testing the null hypothesis that the coefficient (either r or ρ) is zero, thus there is not significant correlation between the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' We assign values for the performance of the teams: 1 for regular season, 2 for Top 16, 3 for quarterfinals, 4 for final four, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='5 if 16 the team was runner-up and 5 if the team won the trophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Table 3 displays the first scorer, the team position and the assigned values of the position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' 𝑯𝟎: 𝝆 = 𝟎 𝒐𝒓 𝒓 = 𝟎 Coefficient Statistic p value Pearson Correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='4002 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='2271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='03481 Spearman Correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='3925 5088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='03886 Both Pearson and Spearman correlation coefficients indicate that there is a significant negative relationship between the first scorer and the performance of the team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' This finding rather favors the first opinion where the first scorers are rarely parts of top teams (exception of Nando De Colo in 2015-2016 confirms the general rule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' First scorer of the tournament, team position and assigned values of the position Season Player PPG Team Performance Assigned Score 1991-1992 Nikos Galis 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='3 Aris Regular season 1 1992-1993 Zdravko Radulović 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='9 Cibona Regular season 1 1993-1994 Nikos Galis 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='8 Panathinaikos 3rd place 4 1994-1995 Sašha Danilović 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='1 Buckler Bologna Quarterfinals 3 1995-1996 Joe Arlauckas 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='4 Real Madrid 4th place 4 1996-1997 Carlton Myers 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='9 Teamsystem Bologna Quarterfinals 3 1997-1998 Peja Stojaković 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='9 PAOK Top 16 2 1998-1999 İbrahim Kutluay 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='4 Fenerbahçe Top 16 2 1999-2000 Miljan Goljović 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='2 Pivovarna Laško Regular season 1 2000-2001 (FIBA) Miroslav Berić 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='3 Partizan Top 16 2 2000-2001 (Euroleague) Alphonso Ford 26 Peristeri Top 16 2 2001-2002 Alphonso Ford 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='8 Olympiacos Top 16 2 2002-2003 Miloš Vujanić 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='8 Partizan Regular season 1 2003-2004 Lynn Greer 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='1 Śląsk Wrocław Regular season 1 2004-2005 Charles Smith 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='7 Scavolini Pesaro Quarterfinals 3 2005-2006 Drew Nicholas 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='5 Benetton Treviso Top 16 2 2006-2007 Igor Rakočević 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='2 Tau Cerámica 4th place 4 2007-2008 Marc Salyers 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='8 Roanne Regular season 1 2008-2009 Igor Rakočević 18 Tau Cerámica Quarterfinals 3 2009-2010 Linas Kleiza 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='1 Olympiacos 2nd place 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='5 2010-2011 Igor Rakočević 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='2 Efes Pilsen Top 16 2 2011-2012 Bo McCalebb 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='9 Montepaschi Siena Quarterfinals 3 2012-2013 Bobby Brown 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='8 Montepaschi Siena Top 16 2 17 2013 2014 Keith Langford 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='6 EA7 Milano Quarterfinals 3 2014 2015 Taylor Rochestie 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='9 Nizhny Novgorod Top 16 2 2015 2016 Nando de Colo 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='9 CSKA Moscow Winner 5 2016 2017 Keith Langford 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='8 UNICS Regular season 1 2017 2018 Alexey Shved 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='8 Khimki Quarterfinals 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='6 Unexpected Outcomes in the Final Fours In this section it is examined the unexpected of the Final-Fours in terms of outcomes based on previous attempts with the use of Binomial Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Can we make the assumption that each final four is an experiment with each team to have the same probabilities of winning the tournament (25%)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' To answer this question, we consider each final four as an experiment and teams are considered as independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Each experiment can be described by the binomial distribution and the whole situation with multinomial distribution (Forbes et al, 2011) which is a generalization of binomial distribution and describes n trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' There is performed a multinomial goodness of fit test and to strengthen the results a binomial test for each team, in order to decide if there is any significant difference from binomial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Multinomial Testing p-value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='54499±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='001575 Binomial Testing* p value Cibona 2 attempts 2 trophies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='0625 Jugoplastica 4 attempts 3 trophies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='05078 ASK Riga 4 attempts 3 trophies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='05078 Panathinaikos 12 attempts 6 trophies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='08608 Only cases with p value<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='1 Table on the appendix displays the final four teams, the expected titles according to Binomial distribution, the observed values and their difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Indeed there is no evidence that there are significant discrepancies from the binomial distribution at the 5% level of significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' In the modern period of Euroleague (1999-2018), again there is no evidence of significant discrepancy from the multinomial distribution, but the case of Panathinaikos could be seen as an exception, with significant larger success rate than the expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' 18 Multinomial Testing p value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='68173±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='001473 Binomial Testing* p value Panathinaikos 8 attempts 5 trophies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='02730 Only cases with p value<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='1 4 Summary and Conclusions To sum up, in this paper is made an attempt to address questions related to historical evolution of Euroleague using statistical analysis to draw conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' One main problem is the lack of plenty available statistical data from the beginning of the competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' This paper demonstrates that by applying suitable statistical designs we can draw interesting conclusions even with limited data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Firstly, is made a brief exploration of the historical evolution of the Euroleague and the tracking statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Then, some questions are answered and some conclusions are drawn which are briefly the following: Although overall there is no difference in success between more traditional powers such as Italy, Spain and ex USSR and Yugoslavian countries and other countries, there is a clear trend of other countries to expand their presence (in terms of titles and final four appearances) in the tournament after the 90s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' In terms of scoring, there was an upward trend from the beginning of the competition, with a sudden drop in 90s, something which indicates a significant change in game mentality in terms of defence and/or game pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The champions were more dominant in 60s and 70s, but in the 80s they scored only 8% more than their opponent, which indicates that the competition was more intense in this decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' The last years (2010-2018), the competitiveness of the tournament tends to similar levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' There is a popular belief that the first scorer in the majority of cases come from the runner-up team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' However, there is no reason to assume that in a specific year is more probable the runner-up team to have a more dominant player in the scoring in the final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' In the modern period of Euroleague (after 1998), we can assume that there is a more dominant scorer in the runner-up team, but we cannot predict this fact for a specific year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' According to the game evolution in finals, we observe that the game has been transformed from finals with offences which are based on top scorers to finals with more pluralism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' From the beginning until the 80s the trend was the one player star in scoring, but since then, there was a slow but 19 continuing turn to games based on pluralism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Moreover, we detect a significant difference in possessions before and after 1998 finals, which is in accordance with our assumption that the game has changed after 1998 final and supports the assumption that the triumph of Zalgiris in 1999 was the start of the change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Furthermore, it is found a significant negative relationship between the first scorer and the performance of his team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' This finding favors the opinion that the first scorers are rarely parts of top teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' Finally, there is no evidence to reject the hypothesis that in a final four there are equal chances of winning overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' In modern era, again the hypothesis of the final four as a random experiment is not rejected, however in the case of Panathinaikos we observe significantly higher success rate than the expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content=' References Best, D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Euroleague ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Final ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Four ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Teams ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Winner ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Runner-Up ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='3rd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Place ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='4th ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Place ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='1958 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='ASK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE0T4oBgHgl3EQfiQGP/content/2301.02443v1.pdf'} +page_content='Riga ' metadata={'source': 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+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +QIN WANG AND JIAWEN ZHANG +Abstract. In this paper, we investigate the ideal structure of uniform Roe algebras +for general metric spaces beyond the scope of Yu’s property A. Inspired by the +ideal of ghost operators coming from expander graphs and in contrast to the +notion of geometric ideal, we introduce a notion of ghostly ideal in a uniform Roe +algebra, whose elements are locally invisible in certain directions at infinity. We +show that the geometric ideal and the ghostly ideal are respectively the smallest +and the largest element in the lattice of ideals with a common invariant open +subset of the unit space of the coarse groupoid by Skandalis-Tu-Yu, and hence the +study of ideal structure can be reduced to classifying ideals between the geometric +and the ghostly ones. As an application, we provide a concrete description for +the maximal ideals in a uniform Roe algebra in terms of the minimal points in the +Stone- ˇCech boundary of the space. We also provide a criterion to ensure that the +geometric and the ghostly ideals have the same K-theory, which helps to recover +counterexamples to the Baum-Connes type conjectures. Moreover, we introduce +a notion of partial Property A for a metric space to characterise the situation in +which the geometric ideal coincides with the ghostly ideal. +Mathematics Subject Classification (2020): 47L20, 46L80, 51F30. +Keywords: Uniform Roe algebras, Coarse groupoids, Geometric and ghostly ideals, Max- +imal ideals, Partial Property A +1. Introduction +Roe algebras are C∗-algebras associated to metric spaces, which encode the +coarse geometry of the underlying spaces. They were introduced by Roe in his pi- +oneering work on higher index theory [31], where he discovered that the K-theory +of Roe algebras serves as a receptacle for higher indices of elliptic differential oper- +ators on open manifolds. Hence the computation for the K-theory of Roe algebras +becomes crucial in the study of higher index theory, and a pragmatic and prac- +tical approach is to consult the Baum-Connes type conjectures [3, 4, 22]. There +is also a uniform version of the Roe algebra, which equally plays a key role in +higher index theory (see [39, 41]). Over the last four decades, there have been a +number of excellent works around this topic (e.g., [17, 21, 24, 46, 47]), which lead +to significant progresses in topology, geometry and analysis (see, e.g., [32, 33]). +On the other hand, the analytic structure of (uniform) Roe algebras reflects the +coarse geometry of the underlying spaces, and the rigidity problem asks whether +the coarse geometry of a metric space can be fully determined by the associated +(uniform) Roe algebra. This problem was initially studied by ˇSpakula and Willett +in [42], followed by a series of works in the last decade [5, 6, 7, 8, 25]. Recently +this problem is completely solved in the uniform case by the profound work +Date: January 13, 2023. +QW is partially supported by NSFC (No. 11831006, 12171156), and the Science and Technology +Commission of Shanghai Municipality (No. 22DZ2229014). JZ is supported by NSFC11871342. +1 + +2 +QIN WANG AND JIAWEN ZHANG +[2], which again highlights the importance of uniform Roe algebras in coarse +geometry. Meanwhile, uniform Roe algebras have also attained rapidly-growing +interest from researchers in mathematical physics, especially in the theory of +topological materials and topological insulators (see, e.g., [18] and the references +therein). +Due to their importance, Chen and the first-named author initiated the study +of the ideal structure for (uniform) Roe algebras [11, 12, 13, 14, 15, 44]. They +succeeded in obtaining a full description for the ideal structure of the uniform Roe +algebra when the underlying space has Yu’s Property A (see [12, 14]). However, +the general picture is far from clear beyond the scope of Property A. +In the present paper, we aim to provide a systematic study on the ideal structure +of uniform Roe algebras for general discrete metric spaces. To outline our main +results, let us first explain some notions. +Let (X, d) be a discrete metric space of bounded geometry (see Section 2.2 for +precise definitions). Thinking of operators on ℓ2(X) as X-by-X matrices, we say +that such an operator has finite propagation if the non-zero entries appear only in an +entourage of finite width (measured by the metric on X) around the main diagonal +(see Section 2.3 for full details). The set of all finite propagation operators forms +a ∗-subalgebra of B(ℓ2(X)), and its norm closure is called the uniform Roe algebra of +X and denoted by C∗ +u(X). +There is another viewpoint on the uniform Roe algebra based on groupoids. +Recall from [39] that Skandalis, Tu and Yu introduced a notion of coarse groupoid +G(X) associated to a discrete metric space X, and they succeeded in relating +coarse geometry to the theory of groupoids. The coarse groupoid G(X) is a lo- +cally compact, Hausdorff, ´etale and principal groupoid (see Section 2.5 for precise +definitions), and the unit space of G(X) coincides with the Stone- ˇCech compacti- +fication βX of X. Moreover, the uniform Roe algebra C∗ +u(X) can be interpreted as +the reduced groupoid C∗-algebra of G(X) (see also [34, Chapter 10]). +In [12], Chen and the first-named author concentrated on a class of ideals in the +uniform Roe algebra in which finite propagation operators therein are dense, and +they showed that these ideals can be described geometrically using the coarse +groupoid. More precisely, recall that a subset U ⊆ βX is invariant if any element +γ in G(X) with source in U also has its range in U (see Section 2.4). As shown +in [12] (see also Section 4), for any ideal I in C∗ +u(X) one can associate an invariant +open subset U(I) of βX, and conversely for any invariant open subset U ⊆ βX one +can associate an ideal I(U) in C∗ +u(X). Furthermore, these two procedures provide +a one-to-one correspondence between invariant open subsets of βX and ideals in +C∗ +u(X) in which finite propagation operators therein are dense. +Based on [12], the first-named author introduced the following notion in [44, +Definition 1.4]: +Definition A (Definition 4.4). Let (X, d) be a discrete metric space of bounded +geometry. An ideal I in the uniform Roe algebra C∗ +u(X) is called geometric if the set +of all finite propagation operators in I is dense in I. +As explained above, [12, Theorem 6.3] (see also Proposition 4.9) indicates that +the geometric ideals in C∗ +u(X) can be fully determined by invariant open subsets +of βX, which explains the terminology. Consequently, the geometric ideals in + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +3 +C∗ +u(X) are easy to handle and they must have the form of I(U) for some invariant +open subset U ⊆ βX, called the geometric ideal associated to U (see Definition 4.6). +Moreover, it follows from [14, Theorem 4.4] that all ideals in C∗ +u(X) are geometric +when X has Yu’s Property A. +However, things get complicated beyond the context of Property A. As noticed +in [12, Remark 6.5], when X comes from a sequence of expander graphs then the +ideal IG consisting of all ghost operators are not geometric (see also [21]). Recall +that an operator T ∈ B(ℓ2(X)) is a ghost if T ∈ C0(X × X) when regarding T as a +function on X × X. Ghost operators are introduced by Yu, and they are crucial to +provide counterexamples to the coarse Baum-Connes conjecture ([21]). +Direct calculations show that the associated invariant open subsets for IG and +for the ideal of compact operators in B(ℓ2(X)) are the same, both of which equal X +(see also Example 4.8 and 5.5). Hence for a general metric space X and an invariant +open subset U ⊆ βX, there might be more than one ideal I in the uniform Roe +algebra C∗ +u(X) satisfying U(I) = U. Therefore, the study of the ideal structure for +C∗ +u(X) can be reduced to analyse the lattice (where the order is given by inclusion) +(1.1) +IU := {I is an ideal in C∗ +u(X) : U(I) = U} +for each invariant open subset U ⊆ βX. +One of the main contributions of the present paper is to find the smallest and +the largest elements in the lattice IU. Following the discussions in [12], it is easy to +see that I(U(I)) ⊆ I for any ideal I in C∗ +u(X), which implies that the geometric ideal +I(U) is the smallest element in IU (see Proposition 4.10). To explore the largest +element, we have to include every ideal I in C∗ +u(X) with U(I) = U. Inspired by the +definition of U(I) (see Equality (4.1)), we introduce the following key notion: +Definition B (Definition 5.1). Let (X, d) be a discrete metric space of bounded +geometry and U be an invariant open subset of βX. The ghostly ideal associated to +U is defined to be +˜I(U) := {T ∈ C∗ +u(X) : r(suppε(T)) ⊆ U for any ε > 0}, +where suppε(T) := {(x, y) ∈ X ×X : |T(x, y)| ≥ ε} and r : X ×X → X is the projection +onto the first coordinate. +We show that ˜I(U) is indeed an ideal in the uniform Roe algebra C∗ +u(X) (see +Lemma 5.2) and moreover, we obtain the following desired result: +Theorem C (Theorem 5.4). Let (X, d) be a discrete metric space of bounded geometry +and U be an invariant open subset of βX. Then any ideal I in C∗ +u(X) with U(I) = U sits +between I(U) and ˜I(U). More precisely, the geometric ideal I(U) is the smallest element +while the ghostly ideal ˜I(U) is the largest element in the lattice IU in (1.1). +Theorem C draws the border of the lattice IU in (1.1), as an important step to +study the ideal structure of uniform Roe algebras for general metric spaces. More +precisely, once we can bust every ideal between I(U) and ˜I(U) for each invariant +open subset U ⊆ βX, then we will obtain a full description for the ideal structure +of the uniform Roe algebra C∗ +u(X). We pose it as an open question in Section 9 and +hope this will be done in some future work. +Concerning the ghostly ideal ˜I(U), we also provide an alternative picture in +terms of limit operators developed in [43], showing that ˜I(U) consists of operators + +4 +QIN WANG AND JIAWEN ZHANG +which vanish in the (βX \ U)-direction (see Proposition 5.6). +Note that ghost +operators vanish in all directions (see Corollary 5.7), and hence operators in ˜I(U) +can be regarded as “partial” ghosts, which clarifies its terminology. Thanks to +this viewpoint, we discover the deep reason behind the counterexample to the +conjecture in [12], constructed by the first-named author in [44, Section 3] (see +Example 5.10). +As an application, we manage to describe maximal ideals in the uniform Roe +algebra. More precisely, it follows directly from Theorem C that maximal ideals +correspond to minimal invariant closed subsets of the Stone- ˇCech boundary +∂βX := βX \ X. Moreover using the theory of limit spaces1 developed in [43], +we prove the following: +Proposition D (Proposition 6.2, Corollary 6.3 and Lemma 6.5). Let (X, d) be a +strongly discrete metric space of bounded geometry and I be a maximal ideal in the +uniform Roe algebra C∗ +u(X). Then there exists a point ω ∈ ∂βX such that I coincides with +the ghostly ideal ˜I(βX \ X(ω)), where X(ω) is the limit space of ω. +A point ω ∈ ∂βX satisfying the condition in Proposition D is called a minimal +point (see Definition 6.4). We show that there exist a number of non-minimal +points in the boundary even for the simple case of X = Z: +Theorem E (Theorem 6.8). For the integer group Z with the usual metric, there exist +non-minimal points in the boundary ∂βZ. More precisely, for any sequence {hn}n∈N in Z +tending to infinity such that |hn − hm| → +∞ when n + m → ∞ and n � m, and any +ω ∈ ∂βZ with ω({hn}n∈N) = 1, then ω is not a minimal point. +We provide two approaches to prove Theorem E. One is topological, which +makes use of several constructions and properties of ultrafilters (recalled in Ap- +pendix A). The other is C∗-algebraic, which replies on a description of maximal +ideals in terms of limit operators (see Lemma 6.13) together with a recent result +by Roch [30]. +Returning to the lattice IU defined in (1.1), we already notice that generally IU +consists of more than one element. Hence it will be interesting and important to +explore when IU has only a single element, or equivalently (thanks to Theorem +C), when the geometric ideal I(U) coincides with the ghostly ideal ˜I(U). +To study this problem, we start with an extra picture for geometric and ghostly +ideals using the associated groupoid C∗-algebras (see Lemma 4.7 and Proposition +5.9). Based on these descriptions, we show that the amenability of the restriction +G(X)∂βX\U of the coarse groupoid ensures that I(U) = ˜I(U) (see Proposition 7.2). +Meanwhile, we also discuss the K-theory of the geometric and ghostly ideals and +provide a criterion to ensure that K∗(I(U)) = K∗(˜I(U)) for ∗ = 0, 1: +Proposition F (Proposition 7.8). Let X be a discrete metric space of bounded geometry +which can be coarsely embedded into some Hilbert space. Then for any countably generated +invariant open subset U ⊆ βX, we have an isomorphism +(ιU)∗ : K∗(I(U)) −→ K∗(˜I(U)) +1Note that the theory of limit spaces and limit operators developed in [43] only concerns strongly +discrete metric spaces of bounded geometry (see Section 2.2 for precise definitions). Although as +noticed in [43] this will not lose any generality, we put this assumption to simplify proofs. + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +5 +for ∗ = 0, 1, where ιU is the inclusion map. +Note that there is a technical condition of countable generatedness (see Def- +inition 7.4) used in Proposition F, which holds for a number of examples (see +Example 7.5) including X itself. However as shown in Example 7.11, there does +exist an invariant open subset which is not countably generated. Applying Propo- +sition F to the case of U = X, we partially recover [19, Proposition 35]. This is +crucial in the constructions of counterexamples to the Baum-Connes type conjec- +tures (see [39] for the coarse version and [20, Section 5] for the boundary version, +which is based on the example considered in [44, Section 3]). Hence presumably +Proposition F will find further applications in higher index theory. +Conversely, it is natural to ask whether I(U) = ˜I(U) implies that the restriction +groupoid G(X)∂βX\U is amenable. Note that when U = X, [36, Theorem 1.3] implies +that I(X) = ˜I(X) is equivalent to that X has Property A (see also Example 4.8 and +5.5), which is further equivalent to that the coarse groupoid G(X) is amenable +thanks to [39, Theorem 5.3]. Inspired by these works, we introduce the following +partial version of Property A: +Definition G (Definition 8.1). Let (X, d) be a discrete metric space of bounded +geometry and U ⊆ βX be an invariant open subset. We say that X has partial +Property A towards ∂βX \ U if G(X)∂βX\U is amenable. +Finally we reach the following, which recovers [36, Theorem 1.3] when U = X: +Theorem H (Theorem 8.16). Let (X, d) be a strongly discrete metric space of bounded +geometry and U ⊆ βX be a countably generated invariant open subset. Then the following +are equivalent: +(1) X has partial Property A towards βX \ U; +(2) ˜I(U) = I(U); +(3) the ideal IG of all ghost operators is contained in I(U). +The proof of Theorem H follows the outline of the case that U = X (cf. [36, +Theorem 1.3]), and is divided into several steps. Firstly, we unpack the groupoid +language of Definition G and provide a concrete geometric description similar to +the definition of Property A (see Proposition 8.2). Then we introduce a notion +of partial operator norm localisation property (Definition 8.9), which is a partial +version of the operator norm localisation property (ONL) introduced in [10]. +Parallel to Sako’s result that Property A is equivalent to ONL ([37]), we show that +partial Property A is equivalent to partial ONL (see Proposition 8.12). Finally +thanks to the assumption of countable generatedness, we conclude Theorem H. +We also remark that in the proof of Theorem H, we make use of the notion of +ideals in spaces introduced by Chen and the first-named author in [12] (see also +Definition 4.12) instead of using invariant open subsets of βX directly. This has the +advantage of playing within the given space rather than going to the mysterious +Stone- ˇCech boundary, which allows us to step over several technical gaps (see, +e.g., Remark 8.18). +The paper is organised as follows. In Section 2, we recall necessary background +knowledge in coarse geometry and groupoid theory. In Section 3, we recall the +theory of limit spaces and limit operators developed in [43], which will be an +important tool used throughout the paper. Section 4 is devoted to the notion + +6 +QIN WANG AND JIAWEN ZHANG +of geometric ideals (Definition A) studied in [12, 44], and we also discuss their +minimality in the lattice of ideals IU from (1.1). We introduce the key notion of +ghostly ideals (Definition B) in Section 5, prove Theorem C and provide several +characterisations for later use. Then we discuss maximal ideals in uniform Roe +algebras in Section 6, and prove Proposition D and Theorem E. In Section 7, we +study the problem when the geometric ideal coincides with the ghostly ideal, +discuss their K-theories and prove Proposition F. Then in Section 8, we introduce +the notion of partial Property A (Definition G) and prove Theorem H. Finally, +we list some open questions in Section 9, and provide Appendix A to record the +notion of ultrafilters and their properties used throughout the paper. +Acknowledgement. We would like to thank Baojie Jiang and J´an ˇSpakula for +some helpful discussions. +2. Preliminaries +2.1. Standard notation. Here we collect the notation used throughout the paper. +For a set X, denote by |X| the cardinality of X. For a subset A ⊆ X, denote by χA +the characteristic function of A, and set δx := χ{x} for x ∈ X. +When X is a locally compact Hausdorff space, we denote by C(X) the set of +complex-valued continuous functions on X, and by Cb(X) the subset of bounded +continuous functions on X. Recall that the support of a function f ∈ C(X) is the +closure of {x ∈ X : f(x) � 0}, written as supp(f), and denote by Cc(X) the set +of continuous functions with compact support. We also denote by C0(X) the set +of continuous functions vanishing at infinity, which is the closure of Cc(X) with +respect to the supremum norm ∥ f∥∞ := sup{| f(x)| : x ∈ X}. +When X is discrete, denote ℓ∞(X) := Cb(X) and ℓ2(X) the Hilbert space of +complex-valued square-summable functions on X. Denote by B(ℓ2(X)) the C∗- +algebra of all bounded linear operators on ℓ2(X), and by K(ℓ2(X)) the C∗-subalgebra +of all compact operators on ℓ2(X). +For a discrete space X, denote by βX its Stone- ˇCech compactification and ∂βX := +βX \ X the Stone- ˇCech boundary. +2.2. Notions from coarse geometry. Here we collect necessary notions from +coarse geometry, and guide readers to [27, 34] for more details. +For a discrete metric space (X, d), denote the closed ball by BX(x, r) := {y ∈ X : +d(x, y) ≤ r} for x ∈ X and r ≥ 0. +For a subset A ⊆ X and r > 0, denote the +r-neighbourhood of A in X by Nr(A) := {x ∈ X : dX(x, A) ≤ r}. For R > 0, denote the +R-entourage by ER := {(x, y) ∈ X × X : d(x, y) ≤ R}. +We saythat(X, d)hasbounded geometry ifforanyr > 0, the numbersupx∈X |BX(x, r)| +is finite. Also say that (X, d) is strongly discrete if the set {d(x, y) : x, y ∈ X} is a dis- +crete subset of R. +Convention. We say that “X is a space” as shorthand for “X is a strongly discrete +metric space of bounded geometry” (as in [43]) throughout the rest of this paper. +We remark that although our results hold without the assumption of strong +discreteness, we choose to add it so as to simplify the proofs. As discussed in [43, +Section 2], this will not lose any generality since one can always modify a discrete +metric space (using a coarse equivalence) to satisfy this assumption. + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +7 +Now we recall the notion of Property A introduced by Yu (see, e.g., [45, Propo- +sition 1.2.4] for the equivalence to Yu’s original definition): +Definition 2.1 ([47]). A space (X, d) is said to have Property A if for any ε, R > 0 +there exist an S > 0 and a function f : X × X → [0, +∞) satisfying: +(1) supp(f) ⊆ ES; +(2) for any x ∈ X, we have � +z∈X f(z, x) = 1; +(3) for any x, y ∈ X with d(x, y) ≤ R, then � +z∈X | f(z, x) − f(z, y)| ≤ ε. +Using a standard normalisation argument, we have the following: +Lemma 2.2. A space (X, d) has Property A if and only if for any ε, R > 0 there exist an +S > 0 and a function f : X × X → [0, +∞) satisfying: +(1) supp(f) ⊆ ES; +(2) for any x ∈ X, we have | � +z∈X f(z, x) − 1| ≤ ε; +(3) for any x, y ∈ X with d(x, y) ≤ R, then � +z∈X | f(z, x) − f(z, y)| ≤ ε. +We also need a characterisation for Property A using kernels. Recall that a kernel +on X is a function k: X × X → R. We say that k is of positive type if for any n ∈ N, +x1, . . . , xn ∈ X and λ1, . . . , λn ∈ R, we have: +n +� +i,j=1 +λiλjk(xi, xj) ≥ 0. +The following is well-known (see, e.g., [45, Proposition 1.2.4]): +Lemma 2.3. A space (X, d) has Property A if and only if for any R > 0 and ε > 0, there +exist S > 0 and a kernel k : X × X → R of positive type satisfying the following: +(1) for x, y ∈ X, we have k(x, y) = k(y, x) and k(x, x) = 1; +(2) for x, y ∈ X with d(x, y) ≥ S, we have k(x, y) = 0; +(3) for x, y ∈ X with d(x, y) ≤ R, we have |1 − k(x, y)| ≤ ε. +We also recall the notion of coarse embedding: +Definition 2.4. Let (X, dX) and (Y, dY) be metric spaces and f : X → Y be a map. +We say that f is a coarse embedding if there exist functions ρ± : [0, ∞) → [0, ∞) with +limt→+∞ ρ±(t) = +∞ such that for any x, y ∈ X we have +ρ−(dX(x, y)) ≤ dY(f(x), f(y)) ≤ ρ+(dX(x, y)). +If additionally there exists C > 0 such that Y = NC(f(X)), then we say that f is a +coarse equivalence and (X, dX), (Y, dY) are coarsely equivalent. +2.3. Uniform Roe algebras. Let (X, d) be a discrete metric space. Each operator +T ∈ B(ℓ2(X)) can be written in the matrix form T = (T(x, y))x,y∈X, where T(x, y) = +⟨Tδy, δx⟩ ∈ C. We also regard T ∈ B(ℓ2(X)) as a bounded function on X × X, i.e., +an element in ℓ∞(X × X). Denote by ∥T∥ the operator norm of T in B(ℓ2(X)), and +∥T∥∞ the supremum norm when regarding T as a function in ℓ∞(X × X). It is clear +that ∥T∥∞ ≤ ∥T∥ for any T ∈ B(ℓ2(X)). +Given an operator T ∈ B(ℓ2(X)), we define the support of T to be +supp(T) := {(x, y) ∈ X × X : T(x, y) � 0}, + +8 +QIN WANG AND JIAWEN ZHANG +and the propagation of T to be +prop(T) := sup{d(x, y) : (x, y) ∈ supp(T)}. +Definition 2.5. Let (X, d) be a space. +(1) The set of all finite propagation operators in B(ℓ2(X)) forms a ∗-algebra, +called the algebraic uniform Roe algebra of X and denoted by Cu[X]. For each +R ≥ 0, denote the subset +CR +u[X] := {T ∈ B(ℓ2(X)) : prop(T) ≤ R}. +It is clear that Cu[X] = � +R≥0 CR +u[X]. +(2) The uniform Roe algebra of X is defined to be the operator norm closure of +Cu[X] in B(ℓ2(X)), which forms a C∗-algebra and is denoted by C∗ +u(X). +The following notion was originally introduced by Yu: +Definition 2.6. An operator T ∈ C∗ +u(X) is called a ghost if T ∈ C0(X × X) when +regarding T as a function in ℓ∞(X × X). In other words, for any ε there exists a +finite subset F ⊆ X such that for any (x, y) � F × F, we have |T(x, y)| < ε. +It is easy to see that all the ghost operators in C∗ +u(X) form an ideal in C∗ +u(X), +denoted by IG. Intuitively speaking, a ghost operator is locally invisible at infinity +in all directions. This will be made more precise in the sequel. +2.4. Groupoids and C∗-algebras. We collect here some basic notions and termi- +nology on groupoids. Details can be found in [28], or [38] in the ´etale case. +Recall that a groupoid is a small category, in which every morphism is invertible. +More precisely, a groupoid consists of a set G, a subset G(0) called the unit space, two +maps s, r : G → G(0) called the source and range maps respectively, a composition +law: +G(2) := {(γ1, γ2) ∈ G × G : s(γ1) = r(γ2)} ∋ (γ1, γ2) �→ γ1γ2 ∈ G, +and an inverse map γ �→ γ−1. These operationssatisfya couple ofaxioms, including +associativity law and the fact that elements in G(0) act as units. +For x ∈ G(0), denote Gx := r−1(x) and Gx := s−1(x). For Y ⊆ G(0), denote GY +Y := +r−1(Y) ∩ s−1(Y). Note that GY +Y is a subgroupoid of G (in the sense that it is stable +under multiplication and inverse), called the reduction of G by Y. A subset Y is +said to be invariant if r−1(Y) = s−1(Y), and we write GY instead of GY +Y in this case. +A locally compact Hausdorff groupoid is a groupoid equipped with a locally +compact and Hausdorff topology such that the structure maps (composition and +inverse) are continuous with respect to the induced topologies. Such a groupoid +is called ´etale (also called r-discrete) if the range (hence the source) map is a local +homeomorphism. Clearly in this case, each fibre Gx (and Gx) is discrete with the +induced topology, and G(0) is clopen in G. The notion of ´etaleness for a groupoid +can be regarded as an analogue of discreteness in the group case. +Example 2.7. Let X be a set. The pair groupoid of X is X×X as a set, whose unit space +is {(x, x) ∈ X × X : x ∈ X} and identified with X for simplicity. The source map is +the projection onto the second coordinate and the range map is the projection onto +the first coordinate. The composition is given by (x, y)· (y, z) = (x, z) for x, y, z ∈ X. +When X is a discrete Hausdorff space, then X × X is a locally compact Hausdorff +´etale groupoid. + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +9 +Now we introduce the algebras associated to groupoids. Here we only focus +on the case of ´etaleness, and guide readers to [28] for the general case. +Let G be a locally compact, Hausdorff and ´etale groupoid with unit space G(0). +Note that the space Cc(G) is a ∗-involutive algebra with respect to the following +operations: for f, g ∈ Cc(G), +(f ∗ g)(γ) += +� +α∈Gs(γ) +f(γα−1)g(α), +f ∗(γ) += +f(γ−1). +Consider the following algebraic norm on Cc(G) defined by: +∥ f∥I := max +sup +x∈G(0) +� +γ∈Gx +| f(γ)|, sup +x∈G(0) +� +γ∈Gx +| f ∗(γ)| + . +The completion of Cc(G) with respect to the norm ∥ · ∥I is denoted by L1(G). +The maximal (full) groupoid C∗-algebra C∗ +max(G) is defined to be the completion of +Cc(G) with respect to the norm: +∥ f∥max := sup ∥π(f)∥, +where the supremum is taken over all ∗-representations π of L1(G). +In order to define the reduced counterpart, we recall that for each x ∈ G(0) the +regular representation at x, denoted by λx : Cc(G) → B(ℓ2(Gx)), is defined as follows: +(2.1) +� +λx(f)ξ +� +(γ) := +� +α∈Gx +f(γα−1)ξ(α), +where f ∈ Cc(G) and ξ ∈ ℓ2(Gx). +It is routine work to check that λx is a well-defined ∗-homomorphism. The reduced +norm on Cc(G) is +∥ f∥r := sup +x∈G(0) +∥λx(f)∥, +and the reduced groupoid C∗-algebra C∗ +r(G) is defined to be the completion of the +∗-algebra Cc(G) with respect to this norm. Clearly, each regular representation λx +can be extended to a homomorphism λx : C∗ +r(G) → B(ℓ2(Gx)) automatically. It +is also routine to check that there is a canonical surjective homomorphism from +C∗ +max(G) to C∗ +r(G). +2.5. Coarse groupoids. Let (X, d) be a space as in Section 2.2. The coarse groupoid +G(X) on X was introduced by Skandalis, Tu and Yu in [39] (see also [34, Chapter +10]) to relate coarse geometry to the theory of groupoids. As a topological space, +G(X) := +� +r>0 +Er +β(X×X) ⊆ β(X × X). +Recall from Example 2.7 that X × X is the pair groupoid with source and range +maps s(x, y) = y and r(x, y) = x. These maps extend to maps G(X) → βX, still +denoted by r and s. +Now consider (r, s) : G(X) → βX × βX. It was shown in [39, Lemma 2.7] that the +map (r, s) is injective, and hence G(X) can be endowed with a groupoid structure + +10 +QIN WANG AND JIAWEN ZHANG +induced by the pair groupoid βX × βX, called the coarse groupoid of X. Therefore, +G(X) can also be equivalently defined by +G(X) := +� +r>0 +Er +βX×βX ⊆ βX × βX, +with the weak topology. It was also shown in [39, Proposition 3.2] that the coarse +groupoid G(X) is locally compact, Hausdorff, ´etale and principal. Clearly, the unit +space of G(X) can be identified with βX. +Given f ∈ Cc(G(X)), then f is a continuous function supported on Er for some +r > 0; equivalently, we can interpret f as a bounded function on Er. Hence we +define an operator θ(f) on ℓ2(X) by setting its matrix coefficients to be θ(f)(x, y) := +f(x, y) for x, y ∈ X. We have the following: +Proposition 2.8 ([34, Proposition 10.29]). The map θ provides a ∗-isomorphism from +Cc(G(X)) to Cu[X], and extends to a C∗-isomorphism Θ : C∗ +r(G(X)) → C∗ +u(X). Note that +Θ maps the C∗-subalgebra C∗ +r(X × X) onto the compact operators K(ℓ2(X)). +Recall from Section 2.3 that an operator T ∈ B(ℓ2(X)) can be regarded as an +element in ℓ∞(X × X). +Following the notation from [12], we denote by T the +continuous extension of T on β(X × X) when regarding T ∈ ℓ∞(X × X). Then +supp(T) = supp(T). +Note that G(X) is open in β(X × X), hence C0(G(X)) is a subalgebra in C(β(X × X)). +Restricting to G(X), we also regard T as a function on G(X) and hence we can talk +of the value T(α, γ) for (α, γ) ∈ G(X). Moreover, we have the following: +Lemma 2.9. For T ∈ Cu[X], we have T ∈ Cc(G(X)) and θ(T) = T. For T ∈ C∗ +u(X), we +have T ∈ C0(G(X)) and Θ(T) = T. +Proof. The first statement is a direct corollary of Proposition 2.8. Since ∥T∥∞ ≤ ∥T∥ +for any T ∈ B(ℓ2(X)), the second follows from the first. +□ +2.6. Amenability and a-T-menability for groupoids. Amenable groupoids com- +prise a large class of groupoids with relatively nice properties, which are literally +the analogue of amenable groups in the world of groupoids. Here we only focus +on the case of ´etaleness, in which the notion of amenability behaves quite well. +A standard reference is [1] and another reference for just ´etale groupoids is [9, +Chapter 5.6]. +Definition 2.10 ([1]). A locally compact, Hausdorff and ´etale groupoid G is said +to be (topologically) amenable if for any ε > 0 and compact K ⊆ G, there exists +f ∈ Cc(G) with range in [0, 1] such that for any γ ∈ K we have +� +α∈Gr(γ) +f(α) = 1 +and +� +α∈Gr(γ) +| f(α) − f(αγ)| < ε. +Similar to the case of Property A, we have the following by a standard normal- +isation argument: + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +11 +Lemma 2.11. A locally compact, Hausdorff and ´etale groupoid G is amenable if and +only if for any ε > 0 and compact K ⊆ G, there exists f ∈ Cc(G) with range in [0, +∞) +such that for any γ ∈ K we have +��� +� +α∈Gr(γ) +f(α) − 1 +��� < ε +and +� +α∈Gr(γ) +| f(α) − f(αγ)| < ε. +Amenability for ´etale groupoids enjoy similar permanence properties as in the +case of groups. For example, open or closed subgroupoids of amenable ´etale +groupoids are amenable, and amenability is preserved under taking groupoid +extensions. See [1, Section 5] for details. Also recall that we have the following: +Proposition 2.12 ([9, Corollary 5.6.17]). Let G be a locally compact, Hausdorff, ´etale +and amenable groupoid. Then the natural quotient C∗ +max(G) → C∗ +r(G) is an isomorphism. +Now we recall the notion of a-T-menability for groupoids introduced by Tu +[40]. Let G be a locally compact, Hausdorff and ´etale groupoid. A continuous +function f : G → R is said to be of negative type if +(1) f|G(0) = 0; +(2) for any γ ∈ G, f(γ) = f(γ−1); +(3) Given γ1, · · · , γn ∈ G with the same range and λ1, · · · , λn ∈ R with �n +i=1 λi = +0, we have � +i,j λiλj f(γ−1 +i γj) ≤ 0. +A continuous function f : G → R is called locally proper if for any compact subset +K ⊆ G(0), the restriction of f on GK +K is proper. +Definition 2.13 ([40, Section 3.3]). A locallycompact, Hausdorff and ´etale groupoid +G is said to be a-T-menable if there exists a continuous locally proper function +f : G → R of negative type on G. +Analogous to the case of groups, Tu [40] proved that a locally compact, σ- +compact, Hausdorff and ´etale groupoid G is a-T-menable if and only if there exists +a continuous field of Hilbert spaces over G(0) with a proper affine action of G. We +also need the following significant result by Tu: +Proposition 2.14 ([40, Th´eor`eme 0.1]). Let G be a locally compact, σ-compact, Haus- +dorff, ´etale and a-T-menable groupoid. Then G is K-amenable, i.e., the quotient map +induces an isomorphism K∗(C∗ +max(G)) → K∗(C∗ +r(G)) for ∗ = 0, 1. +Finally, we record the following result for coarse groupoids: +Proposition 2.15 ([39, Theorem 5.3 and 5.4]). Let (X, d) be a space and G(X) be the +associated coarse groupoid. Then: +(1) X has Property A if and only if G(X) is amenable; +(2) X can be coarsely embedded into Hilbert space if and only if G(X) is a-T-menable. +3. Limit spaces and limit operators +In this section, we recall the theory of limit spaces and limit operators for metric +spaces developed by ˇSpakula and Willett in [43], which becomes an important +tool for later use. + +12 +QIN WANG AND JIAWEN ZHANG +Throughout the section, we always assume that (X, d) is a space (see “Conven- +tion” in Section 2.2) and G(X) is its coarse groupoid. We will freely use the notion +of ultrafilters on X, and related materials are recalled in Appendix A. +3.1. Limit spaces. First recall that a function t : D → R with D, R ⊆ X is called a +partial translation if t is a bijection from D to R, and supx∈X d(x, t(x)) is finite. The +graph of t is {(t(x), x) : x ∈ D}, denoted by gr(t). It is well-known that each entourage +E on X can be decomposed into finitely many graphs of partial translations (see, +e.g., [34, Lemma 4.10]) thanks to the bounded geometry of X. +Definition 3.1 ([43, Definition 3.2 and 3.6]). Fix an ultrafilter ω ∈ βX. A partial +translation t : D → R on X is compatible with ω if ω(D) = 1. In this case, regarding +t as a function from D to βX, we define the following thanks to Lemma A.2: +t(ω) := lim +ω t ∈ βX. +In other words, consider the extension t : D → R then t(ω) = t(ω). +An ultrafilter α ∈ βX is compatible with ω if there exists a partial translation t +compatible with ω and t(ω) = α. Denote by X(ω) the collection of all ultrafilters on +X compatible with ω. A compatible family for ω is a collection of partial translations +{tα}α∈X(ω) such that each tα is compatible with ω and tα(ω) = α. +Fix an ultrafilter ω on X, and a compatible family {tα}α∈X(ω). Define a function +dω : X(ω) × X(ω) → [0, ∞) by +dω(α, β) := lim +x→ω d(tα(x), tβ(x)). +It is shown in [43, Proposition 3.7] that dω is a uniformly discrete metric of bounded +geometry on X(ω) which does not depend on the choice of {tα}. +This leads to the following: +Definition 3.2 ([43, Definition 3.8]). For each non-principal ultrafilter ω on X, the +metric space (X(ω), dω) is called the limit space of X at ω, which is a space in the +sense of “Convention” in Section 2.2. +It is shown in [43, Proposition 3.9] that for any α ∈ X(ω), we have X(α) = X(ω) +as metric spaces. Also note that when ω is principal, i.e., ω ∈ X, then it is clear +that (X(ω), dω) = (X, d). +We recall the following result, which reveals that the local geometry of X can +be recaptured by those of the limit spaces. +Proposition 3.3 ([43, Proposition 3.10]). Let ω be a non-principal ultrafilter on X, and +{tα : Dα → Rα} a compatible family for ω. Then for each finite F ⊆ X(ω), there exists a +subset Y ⊆ X with ω(Y) = 1 such that for each y ∈ Y, there is a finite subset G(y) ⊆ X +such that the map +fy : F → G(y), α �→ tα(y) +is a surjective isometry. Such a collection {fy}y∈Y is called a local coordinate systerm +for F, and the maps fy are called local coordinates. +Furthermore, if F is a metric ball B(ω, r), then there exist Y ⊆ X with ω(Y) = 1 and a +local coordinate system {fy : F → G(y)}y∈Y such that each G(y) is the ball B(y, r). +As shown in [43, Appendix C], limit spaces can be described in terms of the +coarse groupoid G(X): + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +13 +Lemma 3.4 ([43, Lemma C.3]). Given a non-principal ultrafilter ω ∈ βX, the map +F : X(ω) → G(X)ω, +α �→ (α, ω) +is a bijection. Hence X(ω) is the smallest invariant subset of βX containing ω. Here we +consider G(X) as a subset of βX × βX, as explained in Section 2.5. +Consequently, we obtain the following: +Corollary 3.5. As a set, we have +G(X) = +� +X × X +� +⊔ +� +ω∈∂βX +� +X(ω) × X(ω) +� +. +Now we would like to provide a quantitative version for Corollary 3.5. First +we record the following observation, whose proof is straightforward and almost +identical to that of Lemma 3.4 (originally from [34, discussion in 10.18-10.24]): +Lemma 3.6. Let t : D → R be a partial translation on X. Then we have: +gr(t) +βX×βX = gr(t) ⊔ +� +ω∈∂βX +� +(α, ω) : ω(D) = 1 and α = t(ω) +� +. +In general, we have the following: +Lemma 3.7. For any S ≥ 0, we have: +ES +βX×βX = ES ⊔ +� +ω∈∂βX +� +(α, γ) ∈ X(ω) × X(ω) : dω(α, γ) ≤ S +� +. +Proof. As explained at the beginning of this subsection, we can decompose +ES = gr(t1) ⊔ · · · ⊔ gr(tN) +where each ti : Di → Ri is a partial translation. Hence we have +ES = gr(t1) ∪ · · · ∪ gr(tN). +Applying Lemma 3.6, we obtain that ES is contained in the right hand side in the +statement. +On the other hand, given ω ∈ ∂βX and α, γ ∈ X(ω) with dω(α, γ) ≤ S, then we +have (X(ω), dω) = (X(γ), dγ). Take a partial translation t : D → R such that γ(D) = 1 +and α = t(γ). Note that +dγ(α, γ) = lim +x→γ d(t(x), x) ≤ S. +Hence for D′ := {x ∈ D : d(t(x), x) ≤ S}, we have γ(D′) = 1. Consider the restriction +of t on D′, denoted by t′. Then t′ is also a partial translation and α = t′(γ). By +Lemma 3.6, we obtain that (α, γ) ∈ gr(t′), which is contained in ES as desired. +□ +Now we compute concrete examples of limit spaces. First we recall the case +of groups from [43, Appendix B]. Let Γ be a countable discrete group, equipped +with a left-invariant bounded geometry and strongly discrete metric d. For each +g ∈ Γ, denote +ρg : Γ → Γ, h �→ hg +the right translation map. Each ρg is a partial translation with full domain, and +hence is compatible with every ω ∈ βΓ. Moreover, we have the following: + +14 +QIN WANG AND JIAWEN ZHANG +Lemma 3.8 ([43, Lemma B.1]). For each non-principal ultrafilter ω ∈ βΓ, the map +bω : Γ −→ Γ(ω), g �→ ρg(ω) +is an isometric bijection. +Inspired by Lemma 3.8, we provide the following general method: +Lemma 3.9. Let {tλ : Dλ → Rλ}λ∈Λ be a family of partial translations on X satisfying +the following: for each S > 0, there exists a finite subset ΛS ⊆ Λ such that gr(tλ) ∩ gr(tµ) +is finite for λ � µ in ΛS and ES \ +� � +λ∈ΛS gr(tλ) +� +is finite. Then for any non-principal +ultrafilter ω on X and α ∈ X(ω), there exists λ ∈ Λ such that ω(Dλ) = 1 and α = tλ(ω). +Proof. By definition, we assume that α = t(ω) for some partial translation t : D → R +with ω(D) = 1. For λ ∈ Λ, set �Dλ := {x ∈ D ∩ Dλ : t(x) = tλ(x)}. Choose S > 0 such +that gr(t) ⊆ ES, and hence there exists a finite subset F ⊆ ES such that +gr(t) ⊆ +� � +λ∈ΛS +gr(tλ) +� +⊔ F. +This implies that D ⊆ +� � +λ∈ΛS �Dλ +� +⊔ F′ for some finite F′ ⊆ X. Since ω is non- +principal, ΛS is finite and �Dλ ∩ �Dµ is finite for any λ, µ ∈ ΛS, there exists a unique +λ ∈ ΛS such that ω(�Dλ) = 1. This implies that ω(Dλ) = 1 and α = tλ(ω), which +concludes the proof. +□ +Back to the case of the group Γ, the set {ρg : Γ → Γ}g∈Γ satisfies the condition in +Lemma 3.9, and hence the map bω in Lemma 3.8 is surjective. It is straightforward +to check that bω is isometric, which recovers the proof for Lemma 3.8. +Example 3.10. Consider X = N with the usual metric. For each k ∈ Z with k ≥ 0, +define a partial translation +ρk : N −→ N, +n �→ n + k. +For k ∈ Z with k < 0, define a partial translation +ρk : [−k, ∞) ∩ N −→ N, +n �→ n + k. +Then it is clear that the set {ρk}k∈Z satisfies the condition in Lemma 3.9. Now for a +non-principal ultrafilter ω on N, consider the map +bω : Z −→ N(ω), +k �→ ρk(ω). +Note that for k, l ∈ Z, we have +dω(ρk(ω), ρl(ω)) = lim +n→ω |(k + n) − (l + n)| = |k − l|, +which implies that bω is isometric. Moreover, Lemma 3.9 shows that bω is surjec- +tive. Therefore, every limit space of N is isometric to Z. This provides a detailed +proof for [43, Example 3.14(2)]. +Similar to the analysis in Example 3.10, we can also apply Lemma 3.9 to obtain +proofs for [43, Example 3.14(3)-(5)]. Details are left to readers. + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +15 +3.2. Limit operators. Now we recall the notion of limit operators for metric spaces +introduced by ˇSpakula and Willett: +Definition 3.11 ([43, Definition 4.4]). For a non-principal ultrafilter ω on X, fix a +compatible family {tα}α∈X(ω) for ω and let T ∈ C∗ +u(X). The limit operator of T at ω, +denoted by Φω(T), is an X(ω)-by-X(ω) indexed matrix defined by +Φω(T)αγ := lim +x→ω Ttα(x)tγ(x) +for +α, γ ∈ X(ω). +It was studied in [43, Chapter 4] that the above definition does not depend on +the choice of the compatible family {tα}α∈X(ω) for ω. Furthermore, the limit operator +Φω(T) is indeed a bounded operator on ℓ2(X(ω)), and belongs to the uniform Roe +algebra C∗ +u(X(ω)). +Recall from Lemma 2.9 that for an operator T ∈ C∗ +u(X), the continuous extension +T ∈ C0(G(X)). We have the following, which was implicitly mentioned in the +proof of [43, Lemma C.3]. +Lemma 3.12. For a non-principal ultrafilter ω on X and T ∈ C∗ +u(X), we have +Φω(T)αγ = T(α, γ) +for +α, γ ∈ X(ω). +Proof. Choose partial translations tα, tγ compatible with α, γ such that tα(ω) = α, +tγ(ω) = γ. By definition, we have +Φω(T)αγ = lim +x→ω Ttα(x)tγ(x) = lim +x→ω T(tα(x), tγ(x)) = T(α, γ) +where the last equality comes from the discussion before Lemma 2.9. +□ +Since the limit operator Φω(T) contains the information of the asymptotic be- +haviour of T “in the ω-direction”, we introduce the following: +Definition 3.13. For an ω ∈ ∂βX, we say that an operator T ∈ C∗ +u(X) is locally +invisible (or vanishes) in the ω-direction if Φω(T) = 0. For a subset V ⊆ ∂βX, we say +that T is locally invisible (or vanishes) in the V-direction if Φω(T) = 0 for any ω ∈ V. +Finally, we recall from Proposition 2.8 that there is a C∗-isomorphism Θ : +C∗ +r(G(X)) → C∗ +u(X). This allows us to relate limit operators to left regular rep- +resentations of C∗ +r(G(X)): +Lemma 3.14 ([43, Lemma C.3]). For a non-principal ultrafilter ω on X, let Wω : +ℓ2(G(X)ω) → ℓ2(X(ω)) be the unitary representation induced by F in Lemma 3.4. Then +we have the following commutative diagram: +C∗ +r(G(X)) +λω +� +Θ � +� +B(ℓ2(G(X)ω)) +AdWω +� +� +C∗ +u(X) +Φω +� B(ℓ2(X(ω))), +where λω is the left regular representation from (2.1). + +16 +QIN WANG AND JIAWEN ZHANG +4. Geometric ideals +In this section, we recall the notion of geometric ideals, which was originally +introduced by the first-named author in [44] (see also [12]). +Throughout the +section, let X be a space in the sense of “Convention” in Section 2.2. +Definition 4.1 ([12, Definition 3.1, 3.3]). For an operator T ∈ C∗ +u(X) and ε > 0, the +ε-support of T is defined to be +suppε(T) := {(x, y) ∈ X × X : |T(x, y)| ≥ ε}. +Also define the ε-truncation of T to be +Tε(x, y) := +� T(x, y), +if |T(x, y)| ≥ ε; +0, +otherwise. +It is clear that supp(Tε) = suppε(T). We also record the following elementary +result for later use. The proof is straightforward, hence omitted. +Lemma 4.2. Given T ∈ C∗ +u(X) and ε > 0, we have +supp(Tε) ⊆ { ˜ω ∈ β(X × X) : |T( ˜ω)| ≥ ε} ⊆ supp(Tε/2). +The following is a key result in [12]: +Proposition 4.3 ([12, Theorem 3.5]). Let I be an ideal in the uniform Roe algebra C∗ +u(X). +For each T ∈ I and ε > 0, we have Tε ∈ I ∩ Cu[X]. Moreover, we have +I ∩ Cu[X] = {Tε : T ∈ I, ε > 0} +where the closures are taken with respect to the operator norm. +Now we recall the notion of geometric ideals from [44]: +Definition 4.4. An ideal I in the uniform Roe algebra C∗ +u(X) is called geometric if +I ∩ Cu[X] is dense in I. +In [12], Chen and the first-named author provide a full description for geometric +ideals in C∗ +u(X) in terms of invariant open subsets of G(X)(0) = βX. To outline their +work, let us start with the following elementary observation: +Lemma 4.5. Let U be a non-empty invariant open subset of βX, then X ⊆ U. +Proof. Since X is dense in βX and U is open and non-empty, we obtain that +U ∩ X � ∅. Take an x ∈ U ∩ X, then the pair (x, y) ∈ G(X) for any y ∈ X. Thanks to +the invariance of U, we obtain that y ∈ U. This implies that X ⊆ U. +□ +Given an invariant open subset U ⊆ βX, denote G(X)U := G(X) ∩ s−1(U). Fol- +lowing [12], we define +Ic(U) : = {f ∈ Cc(G(X)) : f( ˜ω) = 0 for any ˜ω � G(X)U} += {T ∈ Cu[X] : T( ˜ω) = 0 for any ˜ω � G(X)U}. +Obviously, Ic(U) is a two-sided ideal in Cc(G(X)). Denote its closure in C∗ +r(G(X)) +by I(U), which is a geometric ideal in C∗ +r(G(X)) � C∗ +u(X) from the definition (see +also [12, Lemma 5.1]). This leads to the following: + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +17 +Definition 4.6. For an invariant open subset U ⊆ βX, the ideal I(U) is called the +geometric ideal associated to U. +For later use, we record the following alternative description for the geometric +ideal I(U): +Lemma 4.7. Let U be an invariant open subset of βX. Then the ideal I(U) is isomorphic +to the reduced groupoid C∗-algebra C∗ +r(G(X)U). +Proof. This was implicitly contained in the proof of [12, Proposition 5.5]. +For +convenience to the readers, we include a proof here. By definition, C∗ +r(G(X)U) is +isomorphic to the norm closure of Cc(G(X)U) in C∗ +r(G(X)). Note that +Cc(G(X)U) = {T ∈ Cu[X] : supp(T) ⊆ G(X)U} ⊆ Ic(U). +On the other hand, for T ∈ Ic(U) we have T = limε→0 Tε since T has finite propaga- +tion. Note that supp(Tε) ⊆ G(X)U, which implies Tε ∈ Cc(G(X)U). Hence we obtain +that Cc(G(X)U) and Ic(U) have the same closure in C∗ +r(G(X)), which concludes the +proof. +□ +Example 4.8. For U = X, it follows directly from definition that G(X)X = X × X. +Hence combining Proposition 2.8 and Lemma 4.7, we obtain that the geometric +ideal associated to X is I(X) = K(ℓ2(X)). On the other hand, for U = βX it is clear +that I(βX) = C∗ +u(X). +Conversely, following [12, Section 4 and 5] we can associate an invariant open +subset of βX to any ideal in the uniform Roe algebra. More precisely, let I be an +ideal in the uniform Roe algebra C∗ +u(X). Define: +(4.1) +U(I) := +� +T∈I,ε>0 +r(suppε(T)) = +� +T∈I∩Cu[X],ε>0 +r(suppε(T)), +where the second equality follows directly from Proposition 4.3. Also [12, Lemma +5.2] implies that U(I) is an invariant open subset of βX. Furthermore, as a special +case of [12, Theorem 6.3], we have the following: +Proposition 4.9. For a space (X, d), the map I �→ U(I) provides an isomorphism between +the lattice of all geometric ideals in C∗ +u(X) and the lattice of all invariant open subsets of +βX, with the inverse map given by U �→ I(U). +Proposition 4.9 shows that geometric ideals in C∗ +u(X) can be fully determined +by invariant open subsets of βX. In contrast, general ideals in C∗ +u(X) cannot be +characterised merely by the associated subsets of βX. For example, direct calcu- +lations show that the associated invariant open subsets for the ideal IG defined in +Section 2.3 and for the ideal of compact operators in B(ℓ2(X)) are the same, both +of which equal X (see also Example 4.8 and 5.5 below). +Hence as pointed out in Section 1, the study of the ideal structure for the +uniform Roe algebra can be reduced to analyse the lattice (where the order is +given by inclusion) +IU = {I is an ideal in C∗ +u(X) : U(I) = U} +in (1.1) for each invariant open subset U ⊆ βX. The following result busts the +smallest element in IU: + +18 +QIN WANG AND JIAWEN ZHANG +Proposition 4.10. Let U be an invariant open subset of βX. Then the geometric ideal +I(U) is the smallest element in the lattice IU in (1.1). +The proof of Proposition 4.10 follows directly from the following lemma: +Lemma 4.11. Let (X, d) be a space and I an ideal in C∗ +u(X). Then we have +I(U(I)) = I ∩ Cu[X], +where the closure is taken in C∗ +u(X). Hence we have I(U(I)) ⊆ I. +Proof. Denoting ˚I := I ∩ Cu[X], it is clear that ˚I ∩ Cu[X] = I ∩ Cu[X]. This implies +that ˚I is a geometric ideal in C∗ +u(X), and hence ˚I = I(U(˚I)) by Proposition 4.9. By +definition, we have +U(˚I) = +� +T∈˚I∩Cu[X],ε>0 +r(suppε(T)) = +� +T∈I∩Cu[X],ε>0 +r(suppε(T)) = U(I). +Therefore, we obtain that I(U(I)) = I(U(˚I)) = ˚I = I ∩ Cu[X] as required. +□ +We would like to recall another description for geometric ideals based on the +notion of ideals in spaces introduced in [12]. It has the advantage of playing +within the given metric space, rather than going to the mysterious Stone- ˇCech +boundary, and hence will help us to step over several technical gaps in Section 8. +Definition 4.12 ([12, Definition 6.1]). An ideal in a space (X, d) is a collection L of +subsets of X satisfying the following: +(1) if Y ∈ L and Z ⊆ Y, then Z ∈ L; +(2) if R ≥ 0 and Y ∈ L, then NR(Y) ∈ L; +(3) if Y, Z ∈ L, then Y ∪ Z ∈ L. +For an ideal L in X, we define +U(L) := +� +Y∈L +Y +βX. +Conversely, given an invariant open subset U of βX, we define +L(U) := {Y ⊆ X : Y +βX ⊆ U}. +As a special case of [12, Theorem 6.3], we have the following: +Proposition 4.13. For a space (X, d), the map L �→ U(L) provides an isomorphism +between the lattice of all ideals in X and the lattice of all invariant open subsets of βX, +with the inverse map given by U �→ L(U). +Combining Proposition 4.9 and 4.13, we obtain an isomorphism between the +lattice of all ideals in X and the lattice of all geometric ideals in C∗ +u(X). Direct +calculation shows (see also [12, Theorem 6.4]): +(4.2) +I(U(L)) = {T ∈ Cu[X] : supp(T) ⊆ Y × Y for some Y ∈ L}, +where the closure is taken in C∗ +u(X). +Now we consider a special class of geometric ideals coming from subspaces. +Given a subspace A ⊆ X, recall from [23, Section 5] that there is an associated +ideal IA in C∗ +u(X) whose K-theory is isomorphic to that of the uniform Roe algebra + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +19 +C∗(A). More precisely, recall that an operator T ∈ B(ℓ2(X)) is near A if there exists +R > 0 such that supp(T) ⊆ NR(A) × NR(A), and the ideal IA is defined to be the +operator norm closure of all operators in Cu[X] near A. +The ideal IA is called spatial in [13] since it is related to a subspace in X. However, +as shown in [13, Example 2.1], there exist non-spatial ideals in general. On the +other hand, spatial ideals play an important role in the computation of the K- +theory of Roe algebras via the Mayer-Vietoris sequence argument (see [23]). +To show that spatial ideals are geometric, we observe that the smallest ideal in +X containing A is LA := {Z ⊆ NR(A) : R > 0}. Hence applying Proposition 4.13, we +immediately obtain the following: +Lemma 4.14. Let A be a subset of X. Then the set +(4.3) +UA := U(LA) = +� +R>0 +NR(A) +is an invariant open subset of βX. Moreover, if U is an invariant open subset of βX +containing A, then UA ⊆ U. +Consequently, combining with (4.2) we reach the following: +Corollary 4.15. Let A be a subset of X, then I(UA) = IA. Hence the spatial ideal IA is +geometric. +For later use, we record the following result concerning the set UA defined in +(4.3). Recall from Corollary A.8 that for a subset Z ⊆ X, the closure Z in βX is +homeomorphic to βZ. +Lemma 4.16. Let A be a subset of X. Then we have: +G(X)UA = +� +R>0 +G(NR(A)). +Proof. By definition, we have G(X)UA = � +R>0 G(X)NR(A). Note that for each R > 0 +and (α, ω) ∈ G(X)NR(A), we have ω ∈ NR(A) and there exists S > 0 such that +(α, ω) ∈ ES due to Lemma 3.7. Hence the pair (α, ω) in NR+S(A) × NR+S(A) belongs +to G(NR+S(A)), which concludes the proof. +□ +To end this section, we remark that for a given ideal I in C∗ +u(X), it is usually hard +to compute the associated U(I) directly from definition. However, this is always +achievable for principal ideals: +Lemma 4.17. Let I = ⟨T⟩ be the principal ideal in C∗ +u(X) generated by T ∈ C∗ +u(X). Denote +U := +� +ε>0,R>0 +NR(r(suppε(T))). +Then U is an invariant open subset of βX, and we have U(I) = U. +Proof. By Lemma 4.14, it is clear that U is an invariant open subset of βX. By (4.1), +U(I) contains r(suppε(T)) for any ε > 0. Since U(I) is invariant, we obtain that U(I) +contains U again by Lemma 4.14. + +20 +QIN WANG AND JIAWEN ZHANG +On the other hand, we consider S = �n +i=1 aiTbi where ai, bi are non-zero with +supports being partial translations contained in ER for some R > 0. Hence for any +ε > 0, we have +r +� +suppε(S) +� +⊆ +n +� +i=1 +r +� +supp ε +n(aiTbi) +� +⊆ +n +� +i=1 +NR +� +r +� +supp +ε +n∥ai∥·∥bi∥(T) +�� +, +which implies that r +� +suppε(S) +� +⊆ U. Note that operators of the form �n +i=1 aiTbi as +above are dense in I. Hence for a general element ˜S ∈ I and ε > 0, there exists +S = �n +i=1 aiTbi where ai, bi are non-zero with supports being partial translations +such that ∥ ˜S − S∥ < ε/2. +Hence suppε( ˜S) ⊆ suppε/2(S), which concludes the +proof. +□ +5. Ghostly ideals +In the previous section, we observe that for an invariant open subset U of βX, +the associated geometric ideal I(U) is the smallest element in the lattice +IU = {I is an ideal in C∗ +u(X) : U(I) = U}. +In this section, we would like to explore the largest element in IU. As revealed in +Section 1, a natural idea is to include all operators T ∈ C∗ +u(X) sitting in some ideal +I with U(I) = U, which leads to the following: +Definition 5.1. For a space (X, d) and an invariant open subset U of βX, denote +˜I(U) := {T ∈ C∗ +u(X) : r(suppε(T)) ⊆ U for any ε > 0}. +We call ˜I(U) the ghostly ideal associated to U. +The terminology will become clear later (see Proposition 5.6 and Remark 5.8 +below). First let us verify that ˜I(U) is indeed an ideal in C∗ +u(X). +Lemma 5.2. For an invariant open subset U of βX, ˜I(U) is an ideal in C∗ +u(X). +Proof. For T, S ∈ C∗ +u(X) and ε > 0, we have suppε(T + S) ⊆ suppε/2(T) ∪ suppε/2(S). +It follows that ˜I(U) is a linear space in C∗ +u(X). Given T ∈ ˜I(U) and ε > 0, note that +r(suppε(T∗)) = s(suppε(T)) and hence T∗ ∈ ˜I(U) since U is invariant. Now given a +sequence {Tn}n in ˜I(U) converging to T ∈ C∗ +u(X) and an ε > 0, there exists n ∈ N +such that suppε(T) ⊆ suppε/2(Tn). Hence we obtain that T ∈ ˜I(U). +Finally, given T ∈ ˜I(U) and S ∈ C∗ +u(X) we need show that TS and ST belong to +˜I(U). Note that ST = (T∗S∗)∗ and ˜I(U) is closed under taking the ∗-operation, hence +it suffices to show that TS ∈ ˜I(U). Since ˜I(U) is closed in C∗ +u(X) with respect to +the operator norm, it suffices to consider the case that S ∈ Cu[X]. We can further +assume that S is a partial translation since ˜I(U) is closed under taking addition. +In this case, we have r(suppε(TS)) ⊆ r(suppε/∥S∥(T)) for any ε > 0. Hence we +conclude the proof. +□ +Using the language of ideals in X, we record that for an ideal L in X and +T ∈ C∗ +u(X), then T ∈ ˜I(U(L)) if and only if for any ε > 0 there exist R > 0 and Y ∈ L +such that for any (x, y) � ER ∩ (Y × Y) then |T(x, y)| < ε. + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +21 +The following result shows that the ghostly ideal ˜I(U) is indeed the largest +element in the lattice IU: +Lemma 5.3. Given an invariant open subset U of βX, we have the following: +(1) for an ideal I in C∗ +u(X) with U(I) = U, then I ⊆ ˜I(U). +(2) U(˜I(U)) = U. +Proof. (1). Given T ∈ I, the condition U(I) = U implies that r(suppε(T)) ⊆ U for +each ε > 0. Hence by definition, we have T ∈ ˜I(U). +(2). Note that +U(˜I(U)) = +� +T∈˜I(U),ε>0 +r(suppε(T)) ⊆ U. +On the other hand, Proposition 4.9 shows that U(I(U)) = U, which implies that +˜I(U) ⊇ I(U) thanks to (1). +Hence we have U(˜I(U)) ⊇ U(I(U)) = U again by +Proposition 4.9, which concludes the proof. +□ +Combining with Proposition 4.10, we reach the following desired result: +Theorem 5.4. Let (X, d) be a space as in Section 2.2, and U be an invariant open subset of +βX. Then any ideal I in C∗ +u(X) with U(I) = U sits between I(U) and ˜I(U). More precisely, +the geometric ideal I(U) is the smallest element while the ghostly ideal ˜I(U) is the largest +element in the lattice IU in (1.1). +As mentioned in Section 1, Theorem 5.4 draws the border of the lattice IU. +Therefore, once we can bust every ideal between I(U) and ˜I(U) for each invariant +open subset U ⊆ βX, then we will obtain a full description for the ideal structure +of the uniform Roe algebra C∗ +u(X) (see Question 9.1). +Now we aim to provide a geometric description for ghostly ideals, which helps +to explain the terminology. Let us start with an easy example. +Example 5.5. Taking U = X, then ˜I(X) is the ideal IG defined in Section 2.3. Indeed, +T ∈ ˜I(X) if and only if for any ε > 0, r(suppε(T)) is finite. This is equivalent to that +T ∈ C0(X × X) since T ∈ C∗ +u(X). On the other hand, it is clear that ˜I(βX) = C∗ +u(X). +More generally, we have the following result. Note that for any invariant open +subset U of βX, G(X)U is an open subset of β(X × X). Hence both C0(G(X)U) and +Ic(U) can be regarded as subalgebras in C(β(X × X)) � ℓ∞(X × X). +Proposition 5.6. For an invariant open subset U ⊆ βX and T ∈ C∗ +u(X), the following +are equivalent: +(1) T ∈ ˜I(U); +(2) T ∈ C0(G(X)U); +(3) T ∈ Ic(U) +∥·∥∞; +(4) T vanishes in the (βX \ U)-direction, i.e., Φω(T) = 0 for any ω ∈ βX \ U. +Proof. “(1) ⇒ (2)”: By definition, for any ε > 0 we have +r(supp(Tε)) = r(supp(Tε)) = r(suppε(T)) ⊆ U. + +22 +QIN WANG AND JIAWEN ZHANG +Consider the compact set K := { ˜ω ∈ β(X × X) : |T( ˜ω)| ≥ 2ε}. By Lemma 4.2, we +have r(K) ⊆ r(supp(Tε)) ⊆ U, which implies that K ⊆ G(X)U. Moreover, we have +|T( ˜ω)| < 2ε for any ˜ω ∈ β(X × X) \ K, which concludes that T ∈ C0(G(X)U). +“(2) ⇒ (1)”: Assume that T ∈ C0(G(X)U) ⊆ C(β(X × X)). Then for any ε > 0 there +exists a compact subset K ⊆ G(X)U such that for any ˜ω ∈ β(X × X) \ K, we have +|T( ˜ω)| < ε. This implies that { ˜ω ∈ β(X × X) : |T( ˜ω)| ≥ ε} ⊆ K. Using Lemma 4.2, we +obtain that supp(Tε) ⊆ K, which implies that r(suppε(T)) ⊆ U. Hence T ∈ ˜I(U). +“(2) ⇔ (3)”: This is due to the fact that +Ic(U) +∥·∥∞ = {f ∈ C0(G(X)) : f( ˜ω) = 0 for ˜ω � G(X)U}. +“(3) ⇔ (4)”: By Lemma 3.12, (4) holds if and only if T(α, γ) = 0 for any α, γ ∈ X(ω) +and ω ∈ βX \ U. Applying Lemma 2.9 and Lemma 3.4, this holds if and only if +T( ˜ω) = 0 whenever ˜ω � G(X)U, which describes elements in Ic(U) +∥·∥∞. Hence we +conclude the proof. +□ +Note that G(X)X = X × X. Hence as a direct corollary, we recover the following +characterisation for ghost operators: +Corollary 5.7 ([43, Proposition 8.2]). An operator T ∈ C∗ +u(X) is a ghost if and only if +Φω(T) = 0 for any non-principal ultrafilter ω on X. +Remark 5.8. In other words, Corollary 5.7 shows that a ghost in C∗ +u(X) is locally +invisible in all directions. Thissuggestsustoconsideroperatorsin ˜I(U)as“partial” +ghosts, which clarifies the terminology of “ghostly ideals”. +As an application of Proposition 5.6, we now provide another description for +ghostly ideals in terms of operator algebras, which will be used later in Section 7. +Let us start with the short exact sequences studied in [21] (see also [20, Section 2]). +Given an invariant open subset U ⊆ βX, notice that Uc = βX\U is also invariant. +Denote by G(X)Uc := G(X) ∩ s−1(Uc) and clearly, we have a decomposition: +G(X) = G(X)U ⊔ G(X)Uc. +Note that G(X)U is open in G(X), hence the above induces the following short +exact sequence of ∗-algebras: +(5.1) +0 −→ Cc(G(X)U) −→ Cc(G(X)) −→ Cc(G(X)Uc) −→ 0 +where the map Cc(G(X)U) −→ Cc(G(X)) is the inclusion and the map Cc(G(X)) −→ +Cc(G(X)Uc) is the restriction. +We may complete the sequence (5.1) with respect to the maximal groupoid +C∗-norms and obtain the following sequence: +(5.2) +0 −→ C∗ +max(G(X)U) −→ C∗ +max(G(X)) −→ C∗ +max(G(X)Uc) −→ 0, +which is easy to check by definition to be automatically exact (see, e.g., [26, Lemma +2.10]). We may also complete this sequence with respect to the reduced groupoid +C∗-norms and obtain the following sequence: +(5.3) +0 −→ C∗ +r(G(X)U) +iU +−→ C∗ +r(G(X)) +qU +−→ C∗ +r(G(X)Uc) −→ 0. +By construction, iU is injective, qU is surjective and qU ◦ iU = 0. +Also recall +from Lemma 4.7 that iU(C∗ +r(G(X)U)) = I(U), the geometric ideal associated to U. + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +23 +However in general, (5.3) fails to be exact at the middle item. This is crucial in [21] +to provide a counterexample to the Baum-Connes conjecture with coefficients. +More precisely when U = X, it is proved in [35, Proposition 2.11] for the group +case and [20, Proposition 4.4] for the Roe algebraic case (see also [43, Proposition +8.2]) that Ker(qX) = IG, i.e., the ideal consisting of all ghost operators in C∗ +u(X). +Hence from Example 4.8 and Example 5.5, the sequence (5.3) is exact for U = X if +and only if I(X) = ˜I(X). More generally, we have the following: +Proposition 5.9. Given an invariant open subset U ⊆ βX, the kernel of qU : C∗ +r(G(X)) → +C∗ +r(G(X)Uc) coincides with the ghostly ideal ˜I(U). Hence the sequence (5.3) is exact if and +only if I(U) = ˜I(U). +Proof. It is easy to see that an operator T ∈ C∗ +u(X) � C∗ +r(G(X)) belongs to the +kernel of qU if and only if λω(T) = 0 for any ω ∈ βX \ U, where λω is the left +regular representation from (2.1). Now Lemma 3.14 implies that Φω(T) = 0 for +any ω ∈ βX \ U. Finally, we conclude the proof thanks to Proposition 5.6. +□ +We end this section with an illuminating example from [44, Section 3] (see also +[20, Section 5]), which is important to construct counterexamples to Baum-Connes +type conjectures: +Example 5.10. Let {Xi}i∈N be a sequence of expander graphs or pertubed expander +graphs (see [44] for the precise definition). Let Yi,j = Xi for all j ∈ N and set +Y := � +i,j Yi,j. We endow Y with a metric d such that it is the graph metric on each +Yi,j and satisfies d(Yi,j, Yk,l) → ∞ as i + j + k + l → ∞. +Let Pi,j ∈ B(ℓ2(Yi,j)) be the orthogonal projection onto constant functions on Yi,j, +and we set P to be the direct sum of Pi,j in the strong operator topology. By the +assumption on the expansion of {Xi}i∈N, it is clear that P ∈ C∗ +u(Y). It is explained +in [44, Section 3] (see also [20, Lemma 5.1]) that P is not a ghost, i.e., P � ˜I(X). +However intuitively, P should vanish “in the i-direction”. We will make it more +precisely in the following. +Recall from [20, Section 5.1] that we have a surjective map βY → βX × βN +induced by the bijection of Y with X×N and the universal property of βY. Define: +f : βY −→ βX × βN −→ βX +where the second map is just the projection onto the first coordinate. Denote +U = f −1(X), which is open in βY. Note that U = � +i f −1(Xi), where each f −1(Xi) is +homeomorphic to Xi × βN. On the other hand, note that +U = +� +i +f −1(Xi) = +� +i +� +j +Yi,j = +� +ε>0 +r(suppε(P)) = +� +ε>0,R>0 +NR(r(suppε(P))). +Hence it follows from Lemma 4.17 that U is invariant (comparing with [20, Lemma +5.2]), and U(⟨P⟩) = U. (Note that P ∈ ˜I(U) was already implicitly proved in [20, +Theorem 5.5], thanks to Proposition 5.9.) Since U contains Y as a proper subset, +we reprove that P is not a ghost. +Moreover, it follows from Proposition 5.6 that P vanishes in the (∂βY \ U)- +direction. In particular, fixing an index j0 ∈ N and taking a sequence {xi ∈ Yi,j0}i∈N, +we choose a cluster point ω ∈ {xi}i. It is clear that ω � U. Intuitively speaking, this +means that P vanishes “in the i-direction”. + +24 +QIN WANG AND JIAWEN ZHANG +We remark that the first-named author proved in [44, Section 3] that the prin- +cipal ideal ⟨P⟩ cannot be decomposed into I(U) + (IG ∩ ⟨P⟩), which provided a +couterexample to the conjecture at the end of [12]. Our explanation above reveals +that the reason behind this counterexample is that the ghostly part of ⟨P⟩ could +not be “exhausted” merely by ghostly elements associated to X (rather than U). +Finally, we remark that the groupoid G(Y)U also plays a key role in constructing +a counterexample to the boundary coarse Baum-Connes conjecture introduced in +[20] (see Section 5.2 therein). +6. Maximal ideals +In this section, we would like to study maximal ideals in uniform Roe algebras +using the tools developed in Section 5. Throughout this section, let (X, d) be a +space as in Section 2.2. +6.1. Minimal points in the boundary. Recall from previous sections that ideals +are closely related to invariant open subsets of the unit space βX. +Hence we +introduce the following: +Definition 6.1. An invariant open subset U ⊆ βX is called maximal if U � βX and U +is not properly contained in any proper invariant open subset of βX. Similarly, an +invariant closed subset K ⊆ βX is called minimal if K � ∅ and K does not properly +contain any non-empty invariant closed subset of βX. +Proposition 6.2. For any maximal invariant open subset U ⊂ βX, the ghostly ideal ˜I(U) +is a maximal ideal in the uniform Roe algebra C∗ +u(X). Conversely for any maximal ideal I +in C∗ +u(X), the associated invariant open subset U(I) is maximal and we have I = ˜I(U(I)). +Proof. For any ideal J in C∗ +u(X) containing ˜I(U), we have U(J) ⊇ U(˜I(U)) = U by +Lemma 5.3(2). Since U is maximal, then either U(J) = βX or U(J) = U. If U(J) = βX, +it follows from Lemma 4.11 that J contains I(U(J)) = C∗ +u(X), which implies that +J = C∗ +u(X). If U(J) = U, then it follows from Lemma 5.3(1) that J ⊆ ˜I(U(J)) = ˜I(U), +which implies that J = ˜I(U). This concludes that ˜I(U) is maximal. +Conversely for any maximal ideal I in C∗ +u(X), we have U(I) � βX. For any open +invariant subset V � βX containing U, we have I ⊆ ˜I(U) ⊆ ˜I(V) by Theorem 5.4, +and ˜I(V) � C∗ +u(X). Hence due to the maximality of I, we obtain that I = ˜I(U) = ˜I(V). +This implies that U = V and also I = ˜I(U) as required. +□ +Taking complements, we obtain the following: +Corollary 6.3. For any minimal invariant closed subset K ⊆ βX, the ghostly ideal +˜I(βX \ K) is a maximal ideal in the uniform Roe algebra C∗ +u(X). Moreover, every maximal +ideal in C∗ +u(X) arises in this form. +Therefore, in order to describe maximal ideals in the uniform Roe algebra, it +suffices to study minimal invariant closed subsets of the unit space βX. Recall from +Lemma 3.4 that for each ω ∈ ∂βX, the limit space X(ω) is the smallest invariant +subset of βX containing ω. However, note that X(ω) might not be closed in general. +Definition 6.4. A point ω ∈ ∂βX is called minimal if the closure of the limit space +X(ω) in βX is minimal in the sense of Definition 6.1. + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +25 +The following result is straightforward, hence we omit the proof. It suggests us +to study minimal points in the boundary. +Lemma 6.5. For a minimal invariant closed subset K ⊆ βX, there exists a minimal point +ω ∈ ∂βX such that K = X(ω). Conversely for any minimal point ω ∈ ∂βX, the set X(ω) +is minimal. +One might wonder whether every ω ∈ ∂βX is minimal. However, things become +very complicated after taking closures and we will show later that this does not +hold even in the case of X = Z. Firstly, we notice the following: +Lemma 6.6. Let K be an invariant closed subset of βX. Then K contains a minimal point. +In particular, for any ω ∈ ∂βX there exists a minimal point ω′ such that ω′ ∈ X(ω). +Proof. This follows directly from the Zorn’s lemma together with the fact that βX +is compact. Details are left to readers. +□ +Consequently, we obtain: +Corollary 6.7. For ω ∈ ∂βX, ω is minimal if and only if for any α ∈ X(ω), we have +X(α) = X(ω). Writing X(ω) = � +λ∈Λ X(ωλ) for certain ωλ ∈ ∂βX, then ω is minimal if +and only if X(ωλ) = X(ω) for any λ ∈ Λ. +As promised, now we study the case of X = Z and show that it admits a number +of non-minimal points. The following is the main result: +Theorem 6.8. For the integer group Z with the usual metric, there exist non-minimal +points in the boundary ∂βZ. More precisely, for any sequence {hn}n∈N in Z tending to +infinity such that |hn − hm| → +∞ when n + m → ∞ and n � m, and any ω ∈ ∂βZ with +ω({hn}n∈N) = 1, then ω is not a minimal point. +To prove Theorem 6.8, we need some preparations. For future use, we record +the following result in the context of a countable discrete group Γ equipped with +a left-invariant word length metric. +Lemma 6.9. For ω, α ∈ ∂βΓ, then α ∈ Γ(ω) if and only if for any S ⊆ Γ with α(S) = 1, +there exists gS ∈ Γ such that ω(S · g−1 +S ) = 1. +Proof. Recall from Lemma 3.8 that Γ(ω) = {ρg(ω) : g ∈ Γ}, and from Appendix A +that {S : S ⊆ X} forms a basis for βX. Hence by definition, we obtain that α ∈ X(ω) +if and only if for any S ⊆ X with α ∈ S, there exists gS ∈ Γ such that ρgS(ω) ∈ S. +Equivalently, this means that for any S ⊆ X with α(S) = 1, there exists gS ∈ Γ such +that ρgS(ω)(S) = ω(S · g−1 +S ) = 1, which concludes the proof. +□ +Now we return to the case of Γ = Z and prove Theorem 6.8: +Proof of Theorem 6.8. Fix a subset H = {hn}n∈N ⊆ Z tending to infinity such that +|hn − hm| → +∞ when n + m → ∞ and n � m. For any non-zero g ∈ Z, note that +h ∈ (g + H) ∩ H if and only if there exists h′ ∈ H such that h − h′ = g. Since g is +fixed and distances between different points in H tend to infinity, we obtain that +(g + H) ∩ H is finite. Hence for any g1 � g2 in Z, (g1 + H) ∩ (g2 + H) is finite. + +26 +QIN WANG AND JIAWEN ZHANG +Fixing a non-principal ultrafilter ω ∈ ∂βZ with ω(H) = 1, we denote +U := {B ⊆ H : ω(B) = 1}. +We claim that for each n ∈ N, there exists gn ∈ Z and Bn ∈ U such that {Bn + gn}n∈N +are mutually disjoint. +Indeed, we take g0 = 0 and B0 = H. +Set g1 = 1 and +B1 := H \ (H − g1). Since H ∩ (H − g1) is finite by the previous paragraph, then +ω(B1) = 1, i.e., B1 ∈ U. Similarly for each n ∈ N, we take gn = n and Bn := +H \ +� +(H − g1) ∪ (H − g2) ∪ · · · ∪ (H − gn) +� +, which concludes the claim. +Consider � +H := � +n∈N(Bn + gn), and denote Un := {B ⊆ Bn : ω(B) = 1} for each +n ∈ N. By Lemma A.4, Un is an ultrafilter on Bn. Choose a non-principal ultrafilter +ω0 on N, and we consider: +� +U := +� � +n∈N +(An+gn) ⊆ +� +n∈N +(Bn+gn) = � +H : ∃ J ⊆ N with ω0(J) = 1 s.t. ∀n ∈ J, An ∈ Un +� +. +Lemma A.6 implies that � +U is an ultrafilter on � +H. We define a function α : P(Z) → +{0, 1} by setting α(S) = 1 if and only if S ∩ � +H ∈ � +U, which is indeed an ultrafilter on +Z thanks to Lemma A.5. Also note that α is non-principal and α(� +H) = 1. +Now we show that α ∈ Z(ω) while ω � Z(α), and hence conclude the proof. To +see that α ∈ Z(ω), we will consult Lemma 6.9. For any S ⊆ Z with α(S) = 1, by +definition we have S ∩ � +H ∈ � +U. Writing S ∩ � +H = � +n∈N(An + gn) with An ⊆ Bn, then +there exists J ∈ ω0 such that for any n ∈ J we have An ∈ Un. Hence for any n ∈ J, +we have S ⊇ S ∩ � +H ⊇ An + gn and ω(An) = 1, which implies that ω(S − gn) = 1. +Applying Lemma 6.9, we conclude that α ∈ Z(ω). +Finally, it remains to check that ω � Z(α). Assume the contrary, then Lemma +6.9 implies that there exists g ∈ Z and �B ⊆ � +H with α(�B) = 1 such that H ⊇ �B − g. +Writing �B = � +n∈N(An + gn) with An ⊆ Bn, then there exists J ∈ ω0 such that for any +n ∈ J, ω(An) = 1. This implies that +H + g ⊇ �B ⊇ +� +n∈J +(An + gn) ⊇ An0 + gn0 +for some n0 ∈ J with gn0 � g (this can be achieved since J is infinite). However, +it follows from the first paragraph that (H + g) ∩ (H + gn0) is finite. While this +intersection contains An0 + gn0, which is infinite since ω(An0) = 1. Therefore, we +reach a contradiction and conclude the proof. +□ +6.2. Maximal ideals via limit operators. We already see that maximal ideals in +the uniform Roe algebra correspond to minimal points in the Stone- ˇCech bound- +ary of the underlying space and in Section 6.1, we use topological methods to +show the existence of non-minimal points. Now we turn to a C∗-algebraic view- +point, and use the tool of limit operators to provide an alternative description for +these ideals. +First recall from Corollary 6.3 and Lemma 6.5 that maximal ideals in C∗ +u(X) +arise in the form of ˜I(βX \ X(ω)) for some boundary point ω ∈ ∂βX. Moreover +according to Proposition 5.9, ˜I(βX \ X(ω)) is the kernel of the following surjective +homomorphism: +qβX\X(ω) : C∗ +r(G(X)) −→ C∗ +r(G(X)X(ω)). + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +27 +Hence we obtain the following: +Corollary 6.10. A point ω ∈ ∂βX is minimal if and only if C∗ +r(G(X)X(ω)) is simple. +Example 6.11. Consider the case of a countable discrete group Γ. +For a point +ω ∈ ∂βΓ, it follows from Lemma 3.8 that the limit space Γ(ω) is identical to Γω, +and hence C∗ +r(G(Γ)Γ(ω)) is C∗-isomorphic to the reduced crossed product C(Γω) ⋊ Γ. +Thanks to Corollary 6.10, we obtain that ω is minimal if and only if C(Γω) ⋊ Γ +is simple. +Moreover, note that the action of Γ on βΓ is free (which is also a +consequence of Lemma 3.8). Hence it follows from [29, Corollary 4.6] that C(Γω)⋊Γ +is simple if and only if the action of Γ on Γω is minimal. In conclusion, we reach +the following: +Corollary 6.12. In the case of a countable discrete group Γ, a point ω ∈ ∂βΓ is minimal if +and only if the action of Γ on Γω is minimal. +Corollary 6.10 suggests an approach to distinguish minimal points via the +simplicity of the reduced groupoid C∗-algebra. However, this C∗-algebra is still +not easy to handle since it requires to consider all points in the X(ω)-direction (see +Proposition 5.6). Now we show that this can be simplified by merely considering +the ω-direction: +Lemma 6.13. For ω ∈ ∂βX, an operator T ∈ C∗ +u(X) belongs to the ideal ˜I(βX \ X(ω)) if +and only if T vanishes in the ω-direction, i.e., Φω(T) = 0. +Proof. We assume that Φω(T) = 0, and it suffices to show that Φα(T) = 0 for any +α ∈ X(ω). Fixing such an α, we take a net {ωλ}λ∈Λ in X(ω) such that ωλ → α and +it follows that Φωλ(T) = 0. For any γ1, γ2 ∈ X(α), Lemma 3.4 and the fact that the +coarse groupoid G(X) is ´etale imply that there exist γ1,λ and γ2,λ in X(ωλ) for each +λ ∈ Λ such that γ1,λ → γ1 and γ2,λ → γ2. Now Lemma 3.12 implies that +Φα(T)γ1γ2 = T((γ1, γ2)) = lim +λ∈Λ T((γ1,λ, γ2,λ)) = lim +λ∈Λ Φωλ(T)γ1,λγ2,λ = 0, +which concludes the proof. +□ +Hence for ω ∈ ∂βX, Lemma 6.13 implies that the associated ideal ˜I(β \ X(ω)) +coincides with the kernel of the following limit operator homomorphism (see also +[43, Theorem 4.10]): +(6.1) +Φω : C∗ +u(X) −→ C∗ +u(X(ω)), +T �→ Φω(T). +Consequently, we reach the following: +Corollary 6.14. A point ω ∈ ∂βX is minimal if and only if the image Im(Φω) is simple. +Hence when X(ω) is infinite and Φω is surjective, the point ω is not minimal. +Proof. For the last statement, it suffices to note that the ideal of compact operators +is always contained in C∗ +u(X(ω)), which concludes the proof. +□ +Thanks to Corollary 6.14, a special case of Theorem 6.8 can also be deduced +from a recent work by Roch [30]. More precisely, combining [43, Proposition B.6] +with [30, Lemma 2.1], we have the following: + +28 +QIN WANG AND JIAWEN ZHANG +Proposition 6.15. Let {hn}n∈N be a sequence in ZN tending to infinity such that +∥hn − hk∥∞ ≥ 3k +for any +k > n. +Then for any ω ∈ ∂βZN with ω({hn}n∈N) = 1, the map Φω : C∗ +u(ZN) −→ C∗ +u(ZN(ω)) is +surjective. +Note that the limit space ZN(ω) is bijective to ZN by Lemma 3.8, and hence +infinite. Therefore applying Corollary 6.14, we obtain the following (when N = 1, +it partially recovers Theorem 6.8). +Corollary 6.16. For any sequence {hn}n∈N in ZN tending to infinity such that ∥hn−hk∥∞ ≥ +3k for any k > n, and any ω ∈ ∂βZN with ω({hn}n∈N) = 1, then ω is not a minimal point. +Remark 6.17. We notice from the discussion above that for ω ∈ ∂βX, there is a +C∗-monomorphism: +(6.2) +C∗ +r(G(X)X(ω)) � C∗ +u(X)/˜I(βX \ X(ω)) −→ C∗ +u(X(ω)) +where the first comes from Proposition 5.9 and the second comes from (6.1) +together with Lemma 6.13. +Now we provide another explanation for this map in terms of groupoids. +Firstly, Corollary 3.5 implies that G(X)X(ω) = X(ω) × X(ω) and hence G(X)X(ω) = +X(ω) × X(ω) +G(X). Now we have +G(X)X(ω) = +� +S>0 +� +ES +G(X) ∩ X(ω) × X(ω) +G(X)� += +� +S>0 +ES +G(X) ∩ (X(ω) × X(ω)) +G(X) +, +where the last inequality is due to the fact that ES +G(X) is clopen in G(X). By Lemma +3.7, we have +ES +G(X) ∩ (X(ω) × X(ω)) = {(α, γ) ∈ X(ω) × X(ω) : dω(α, γ) ≤ S} =: ES(X(ω), dω) +i.e., the S-entourage in the limit space (X(ω), dω). Therefore, we conclude that +G(X)X(ω) = +� +S>0 +ES(X(ω), dω) +G(X). +On the other hand, we have (by definition) that +G(X(ω)) = +� +S>0 +ES(X(ω), dω) +β(X(ω)×X(ω)). +By the universal property of the Stone- ˇCech compactification, there is a surjective +continuous map ES(X(ω), dω) +β(X(ω)×X(ω)) −→ ES(X(ω), dω) +G(X), which induces an in- +jective map Cc(G(X)X(ω)) −→ Cc(G(X(ω))). Moreover, it is routine to check that it +induces a C∗-monomorphism +C∗ +r(G(X)X(ω)) −→ C∗ +r(G(X(ω))) � C∗ +u(X(ω)), +which can be verified to coincide with the map (6.2). Details are left to readers. +In particular, we consider a countable discrete group X = Γ. Fixing a point +ω ∈ ∂βΓ, we mentioned in Example 6.11 that C∗ +r(G(Γ)Γ(ω)) � C(Γω)⋊Γ. On the other +hand, we know from Lemma 3.8 that C∗ +u(Γ(ω)) � ℓ∞(Γω) ⋊ Γ. In this case, one can +check that the map (6.2) is induced by the natural embedding C(Γω) ֒→ ℓ∞(Γω). + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +29 +7. Geometric ideals vs ghostly ideals +In this section, we return to the lattice IU in (1.1). Recall from Theorem 5.4 that +for an invariant open subset U ⊆ βX, the geometric ideal I(U) and the ghostly +ideal ˜I(U) are the smallest and the largest elements in IU. Also as noticed before, +generally IU consists of more than one element. +Now we would like to study when I(U) = ˜I(U) or equivalently, when IU consists +of a single element. Moreover, we will discuss their K-theories and provide a +sufficient condition to ensure K∗(I(U)) = K∗(˜I(U)) for ∗ = 0, 1. +First recall from Proposition 5.9 that we already have a characterisation for +I(U) = ˜I(U) using short exact sequences, while the condition therein still seems +hard to check. Now we aim to search for a more practical criterion to ensure +I(U) = ˜I(U). We start with the following result, which combines [14, Theorem 4.4] +and [36, Theorem 1.3]. +Proposition 7.1. For a space (X, d), the following are equivalent: +(1) X has Property A; +(2) G(X) is amenable; +(3) G(X)∂βX is amenable; +(4) I(U) = ˜I(U) for any invariant open subset U ⊆ βX; +(5) I(X) = ˜I(X). +We remark that “(1) ⇒ (4)” was originally proved in [14] using approximations +by kernels. Here we will take a shortcut, and the idea will also be used later. +Proof of Proposition 7.1. “(1) ⇔ (2)” was proved in [39, Theorem 5.3]. “(2) ⇔ (3)” is +due to the permanence properties of amenability (see Section 2.6) together with +the fact that G(X)X � X × X is always amenable. +“(2) ⇒ (4)”: Let U ⊆ βX be an invariant open subset. As open/closed sub- +groupoids, both G(X)U and G(X)Uc are amenable as well. Consider the following +commutative diagram coming from (5.2) and (5.3): +0 +� C∗ +max(G(X)U) +� +� +C∗ +max(G(X)) +� +� +C∗ +max(G(X)Uc) +� +� +0 +0 +� C∗ +r(G(X)U) +� C∗ +r(G(X)) +� C∗ +r(G(X)Uc) +� 0. +By Proposition 2.12, all three vertical lines are isomorphisms. Hence the exactness +of the first row implies that the second row is exact as well. Therefore, we conclude +(4) thanks to Proposition 5.9. +“(4) ⇒ (5)” holds trivially, and “(5) ⇒ (1)” comes from [36, Theorem 1.3] together +with Example 4.8 and Example 5.5. Hence we conclude the proof. +□ +Proposition 7.1 provides a coarse geometric characterisation for I(X) = ˜I(X) +using Property A. However, we notice that assuming Property A is often too +strong to ensure that I(U) = ˜I(U) for merely a specific invariant open subset U ⊆ βX. +A trivial example is that I(βX) = ˜I(βX) holds for any space X. This suggests us to +explore a weaker criterion for I(U) = ˜I(U), and we reach the following: + +30 +QIN WANG AND JIAWEN ZHANG +Proposition 7.2. Let (X, d) be a space and U be an invariant open subset of βX. If the +canonical quotient map C∗ +max(G(X)Uc) → C∗ +r(G(X)Uc) is an isomorphism, then I(U) = +˜I(U). In particular, if the groupoid G(X)Uc is amenable then I(U) = ˜I(U). +Proof. We consider the following commutative diagram: +(7.1) +0 +� C∗ +max(G(X)U) +� +πU +� +C∗ +max(G(X)) +� +� +C∗ +max(G(X)Uc) +� +� +0 +0 +� ˜I(U) +� C∗ +r(G(X)) +� C∗ +r(G(X)Uc) +� 0. +Here the map πU is the composition: +(7.2) +C∗ +max(G(X)U) → C∗ +r(G(X)U) � I(U) ֒→ ˜I(U), +where the middle isomorphism comes from Lemma 4.7. Note that the top hor- +izontal line is automatically exact, while the bottom one is also exact thanks to +Proposition 5.9. Also note that the middle vertical map is always surjective and by +assumption, the right vertical map is an isomorphism. Hence via a diagram chas- +ing argument, we obtain that the left vertical map is surjective. This concludes +that I(U) = ˜I(U) thanks to (7.2). +□ +Remark 7.3. When U = X, Proposition 7.2 recovers “(3) ⇒ (5)” in Proposition 7.1. +Readers might wonder whether the converse of Proposition 7.2 holds as in the +case of U = X. We manage to provide a partial answer in Section 8.3 below. +Now we move to discuss the K-theory of geometric and ghostly ideals. Firstly, +we need an extra notion: +Definition 7.4. For a set S of subsets of X, denote L(S) the smallest ideal in X +containing S, and we say that L(S) is generated by S. An ideal L in X is called +countably generated if there exists a countable set S such that L = L(S). +An invariant open subset U ⊆ βX is called countably generated if the associated +ideal L(U) is countably generated. +Example 7.5. For a space X and a subspace A ⊆ X, it follows from Lemma 4.14 +that the spatial ideal IA is countably generated. On the other hand, it follows from +Lemma 4.17 that principal ideals are always countably generated. In particular, +the ideal ⟨P⟩ considered in [44, Section 3] (see also Example 5.10) is countably +generated. +The property of countable generatedness leads to the following: +Lemma 7.6. Let L be a countably generated ideal in X. Then there exists a countable +subset {Y1, Y2, · · · , Yn, · · · } in L such that +L = {Z ⊆ X : ∃ n ∈ N such that Z ⊆ Yn}. +Consequently for any countably generated open invariant subset U of βX, the subgroupoid +G(X)U is σ-compact. +To prove Lemma 7.6, we need an auxiliary result on the structure of L(S): +Lemma 7.7. For a set S of subsets of X, denote S(1) := {A1 ∪ · · · ∪ An : Ai ∈ S, n ∈ N} +and S(2) := {Nk( ˜A) : ˜A ∈ S(1), k ∈ N}. Then we have: +L(S) = {Z : ∃ Y ∈ S(2) such that Z ⊆ Y}. + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +31 +Proof. Denote S(3) := {Z : ∃ Y ∈ S(2) such that Z ⊆ Y}. It is clear that S(3) ⊆ L(S), +and any ideal containing S must contain S(3). Hence it remains to check that S(3) +is an ideal. +For Z ∈ S(3) and W ⊆ Z, there exists Y ∈ S(2) such that Z ⊆ Y. Hence W ⊆ Y, +which implies that W ∈ S(3). Also note that Y ∈ S(2) implies that there exists +˜A ∈ S(1) and k′ ∈ N such that Y = Nk′( ˜A). +Hence for any k ∈ N, we have +Nk(Z) ⊆ Nk(Y) ⊆ Nk+k′( ˜A), which implies that Nk(Z) ∈ S(3). +Finally for Z1, Z2 ∈ S(3), there exist ˜A1, ˜A2 ∈ S(1) and k1, k2 ∈ N such that +Zi ⊆ Nki( ˜Ai) for i = 1, 2. Hence +Z1 ∪ Z2 ⊆ Nk1( ˜A1) ∪ Nk2( ˜A2) ⊆ Nk1+k2( ˜A1 ∪ ˜A2) ∈ S(2), +which implies that Z1 ∪ Z2 ∈ S(3). Therefore, we conclude the proof. +□ +Proof of Lemma 7.6. By assumption, there exists a countable S such that L = L(S). +Using the notation of Lemma 7.7, the set S(2) is countable as well. Hence the first +statement follows directly from Lemma 7.7. +For the second, note that U = � +n∈N Yn and hence G(X)U = � +n∈N G(X)Yn. Since +each Yn is closed, we obtain that G(X)U is σ-compact as desired. +□ +Now we are in the position to discuss the K-theory of geometric and ghostly +ideals: +Proposition 7.8. Let X be a space which can be coarsely embedded into some Hilbert +space. Then for any countably generated invariant open subset U ⊆ βX, we have an +isomorphism +(ιU)∗ : K∗(I(U)) −→ K∗(˜I(U)) +for ∗ = 0, 1, where ιU is the inclusion map. Therefore for any ideal I in C∗ +u(X) with U(I) +countably generated, we have an injective homomorphism +(ιI)∗ : K∗(I ∩ Cu[X]) −→ K∗(I) +for ∗ = 0, 1, where ιI is the inclusion map. +Proof. Fixing such a U ⊆ βX, we consider the commutative diagram (7.1) from the +proof of Proposition 7.2, where both of the horizontal lines are exact. This implies +the following commutative diagram in K-theories: +· · · +� K∗(C∗ +max(G(X)U)) +� +(πU)∗ +� +K∗(C∗ +max(G(X))) +� +� +K∗(C∗ +max(G(X)Uc)) +� +� +K∗+1(C∗ +max(G(X)U)) +� +(πU)∗+1 +� +· · · +· · · +� K∗(˜I(U)) +� K∗(C∗ +r(G(X))) +� K∗(C∗ +r(G(X)Uc)) +� K∗+1(˜I(U)) +� · · · +where both horizontal lines are exact. Recall from [39, Theorem 5.4] that X is +coarsely embeddable if and only if G(X) is a-T-menable. It follows directly from +Definition 2.13 that as subgroupoids, both G(X)U and G(X)Uc are a-T-menable. +Moreover, it is clear that G(X) and G(X)Uc are σ-compact by definition. Hence +Proposition 2.14 implies that G(X) and G(X)Uc are K-amenable. This shows that +the middle two vertical maps in the above diagram are isomorphisms, which +implies that (πU)∗ is an isomorphism by the Five Lemma. Therefore from (7.2), +we obtain that the composition +K∗(C∗ +max(G(X)U)) −→ K∗(C∗ +r(G(X)U)) � K∗(I(U)) +(ιU)∗ +−→ K∗(˜I(U)) + +32 +QIN WANG AND JIAWEN ZHANG +is an isomorphism for ∗ = 0, 1. +On the other hand, it follows from Lemma 7.6 that G(X)U is σ-compact, which +implies that G(X)U is K-amenable again by Proposition 2.14. Hence we conclude +that the map (ιU)∗ is an isomorphism for ∗ = 0, 1. +For the last statement, we assume that U = U(I). By Lemma 4.11, we have that +I ∩ Cu[X] = I(U). Also Lemma 5.3(1) shows that I ⊆ ˜I(U). Hence the inclusion +map ιU can be decomposed as follows: +I(U) = I ∩ Cu[X] +ιI֒→ I ֒→ ˜I(U). +Therefore, (ιU)∗ being an isomorphism implies that (ιI)∗ is injective for ∗ = 0, 1. +□ +Applying Proposition 7.8 to the case of U = X, we partially recover the following +result by Finn-Sell (see [19, Proposition 35]), which is crucial for the counterex- +amples to the coarse Baum-Connes conjecture: +Corollary 7.9. Let X be a space which can be coarsely embedded into some Hilbert space. +Then the inclusion of K(ℓ2(X)) into IG induces an isomorphism on the K-theory level. +Remark 7.10. For a general ideal I in C∗ +u(X), our method in the proof of Proposition +7.8 only provides the injectivity of the induced map (ιI)∗. We wonder whether this +map is indeed an isomorphism under the same assumption (see Question 9.4). +We end this section with an example, which shows that not every ideal in a space +is countably generated. +Example 7.11. Let X = N × N, equipped with the metric induced from the Eu- +clidean metric dE on R2. For each θ ∈ [0, π +2] and k ∈ N, we define +ℓθ := {(x, y) ∈ R × R : y = tan(θ)x} +and +Sθ,k := {(x, y) ∈ X : dE((x, y), ℓθ) ≤ k} = Nk(ℓθ) ∩ X. +Consider S := {Sθ,k : θ ∈ [0, π +2], k ∈ N} and set S(1) := {A1 ∪ · · · ∪ An : Ai ∈ S, n ∈ N} +as in Lemma 7.7. For any R > 0 and A = A1 ∪ · · · ∪ An ∈ S(1) where Ai ∈ S, we +have NR(A) = NR(A1) ∪ · · · ∪ NR(An), which is contained in some element in S. +Hence applying Lemma 7.7, the ideal L(S) generated by S is: +(7.3) +L(S) = {Z : ∃ Y ∈ S(1) such that Z ⊆ Y}. +We claim that L(S) is not countably generated. Otherwise, there exists a count- +able subset S′ generating L(S). Moreover, according to (7.3) we can assume that +S′ = {Yn : n ∈ N} +where +Yn = Sθn,1,kn ∪ · · · ∪ Sθn,pn,kn ∈ S(1). +Choose θ ∈ [0, π +2] \ {θn,i : i = 1, 2, · · · , pn; n ∈ N}, and consider Y = Sθ,1. Since S′ +generates L(S), it follows from Lemma 7.7 that there exist R > 0 and Ym1, · · · , Yml ∈ +S′ such that +Sθ,1 = Y ⊆ NR(Ym1 ∪ · · · ∪ Yml). +Note thatthe righthand side iscontained in a finite union ofsome R′-neighbourhood +of lines (in R2 crossing the origin) with slopes in the set +� +tan(θn,i) : i = 1, 2, · · · , pn; n ∈ N +� +. +This leads to a contradiction due to the choice of θ, which concludes that L(S) +cannot be countably generated. + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +33 +8. Partial Property A and partial operator norm localisation property +In the previous section, we find a sufficient condition (Proposition 7.2) to ensure +˜I(U) = I(U) for a given invariant open subset U ⊆ βX. As promised in Remark 7.3, +now we study its converse and show that this is indeed an equivalent condition +under the assumption of countable generatedness. +Our strategy is to follow the outline of the case that U = X. More precisely, we +introduce a notion called partial Property A towards invariant subsets of the boundary +∂βX, and then consider its counterpart in the context of operator norm localisation +property to provide the desired characterisation. +8.1. Partial Property A. Recall from Proposition 7.1 that a space X has Property +A if and only if the groupoid G(X)∂βX is amenable, which characterises I(X) = ˜I(X). +Together with Proposition 7.2, this inspires us to introduce the following: +Definition 8.1. Let (X, d) be a space and U ⊆ βX be an invariant open subset. We +say that X has partial Property A towards ∂βX \ U if G(X)∂βX\U is amenable. +It is clear from definition that X has Property A if and only if it has partial +Property A towards the whole boundary ∂βX. On the other hand, it follows from +Proposition 7.2 that if X has partial Property A towards ∂βX \ U, then we have +I(U) = ˜I(U). The rest of this section is devoted to studying the converse. +Firstly, we aim to unpack the groupoid language and provide a concrete geomet- +ric description for partial Property A, which resembles the definition of Property +A (see Definition 2.1). The following is the main result: +Proposition 8.2. Let (X, d) be a space and U ⊆ βX an invariant open subset. Then X +has partial Property A towards Uc = ∂βX \ U if and only if for any ε, R > 0, there exist +S > 0, a subset D ⊆ X with D ⊇ Uc and a function f : X × X → [0, 1] satisfying: +(1) supp(f) ⊆ ES; +(2) for any x ∈ D, we have � +z∈X f(z, x) = 1; +(3) for any x, y ∈ D with d(x, y) ≤ R, we have � +z∈X | f(z, x) − f(z, y)| ≤ ε. +Comparing Proposition 8.2 with Definition 2.1, it is clear that Property A implies +partial Property A towards any invariant closed subset of ∂βX. +To prove Proposition 8.2, we start with the following lemma: +Lemma 8.3. With the same notation as above, X has partial Property A towards Uc if +and only if for any ε, R > 0, there exist S > 0 and a function f : X × X → [0, 1] +satisfying: +(1) supp(f) ⊆ ES; +(2) for any ω ∈ Uc, we have � +α∈X(ω) f(α, ω) = 1; +(3) for ω ∈ Uc and α ∈ X(ω) with dω(α, ω) ≤ R, then � +γ∈X(ω) | f(γ, α) − f(γ, ω)| ≤ ε, +where f ∈ C0(G(X)) is the continuous extension from Lemma 2.9. +Proof. By definition, X has partial Property A towards Uc if and only if for any +ε > 0 and compact K ⊆ G(X)Uc, there exists g ∈ Cc(G(X)Uc) with range in [0, 1] such + +34 +QIN WANG AND JIAWEN ZHANG +that for any γ ∈ K we have +� +α∈Gr(γ) +g(α) = 1 +and +� +α∈Gr(γ) +|g(α) − g(αγ)| < ε. +Recall from (5.1) that the restriction map Cc(G(X)) → Cc(G(X)Uc) is surjective, and +hence g can be regarded as a function in Cc(G(X)). Taking f to be the restriction +of g on X × X, then f ∈ ℓ∞(X × X) and there exists S > 0 such that supp(f) ⊆ ES +for some S > 0. Using the notation from Lemma 2.9, we have g = f. Note that +G(X)Uc = � +R>0(ER ∩ G(X)Uc), and hence compact subsets of G(X)Uc are always +contained in those of the form ER ∩ G(X)Uc. Furthermore, Lemma 3.7 implies +ER ∩ G(X)Uc = +� +ω∈Uc +{(α, γ) ∈ X(ω) × X(ω) : dω(α, γ) ≤ R}. +Combining with Lemma 3.4, we conclude the proof. +□ +As a direct corollary (together with Lemma 3.7), we obtain: +Corollary 8.4. Assume that X has partial Property A towards Uc. Then the family of +metric spaces {(X(ω), dω)}ω∈Uc has uniform Property A in the sense that the parameters +in Definition 2.1 can be chosen uniformly. +Remark 8.5. It is unclear to us whether the converse of Corollary 8.4 holds. Note +that X(ω) might contain points outside X(ω) as discussed in Theorem 6.8, hence we +do not know whether functions on X(ω) × X(ω) can be glued together to provide +a continuous function on G(X)Uc. +Proof of Proposition 8.2. Sufficiency: For any ε, R > 0, choose S > 0, D ⊆ X and a +function g : X × X → [0, 1] satisfying the conditions (1)-(3) for ε and 3R. Take a +map p : NR(D) → D such that the restriction of p on D is the identity map and +d(p(x), x) ≤ R. Now we define: +f(x, y) = +� g(x, p(y)), +y ∈ NR(D); +g(x, y), +otherwise. +It is clear that supp(f) ⊆ ER+S. Moreover for any y1, y2 ∈ NR(D) with d(y1, y2) ≤ R, +we have d(p(y1), p(y2)) ≤ 3R and hence +� +x∈X +| f(x, y1) − f(x, y2)| = +� +x∈X +|g(x, p(y1)) − g(x, p(y2))| ≤ ε. +Therefore (enlarging S to S + R) we obtain that for any ε, R > 0, there exist S > 0, +a subset D ⊆ X with D ⊇ Uc and a function f : X × X → [0, 1] satisfying: +(1) supp(f) ⊆ ES; +(2) for any x ∈ D, we have � +z∈X f(z, x) = 1; +(3) for any x ∈ D and y ∈ X with d(x, y) ≤ R, we have � +z∈X | f(z, x) − f(z, y)| ≤ ε. +Now we fix ε, R > 0 and take such S, D and function f. +Given ω ∈ Uc, we have ω(D) = 1. Choose {tα : Dα → Rα} to be a compatible +family for ω. Applying Proposition 3.3, there exists Yω ⊆ X with ω(Y) = 1 and a +local coordinate system {hy : B(ω, R + S) → B(y, R + S)}y∈Yω such that the map +hy : B(ω, R + S) → B(y, R + S), +α �→ tα(y) + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +35 +is a surjective isometry for each y ∈ Yω. Replacing Y by Yω ∩ D, we assume that +Yω ⊆ D. Note that supp(f) ⊆ ES, and hence applying Lemma 3.7 we have +� +α∈X(ω) +f(α, ω) = +� +α∈B(ω,S) +f(α, ω) = +� +α∈B(ω,S) +lim +x→ω f(tα(x), x) = +lim +x→ω,x∈Yω +� +α∈B(ω,S) +f(tα(x), x) += +lim +x→ω,x∈Yω +� +z∈B(x,S) +f(z, x) = +lim +x→ω,x∈Yω +� +z∈X +f(z, x), +(8.1) +where the last item equals 1 thanks to the assumption. +On the other hand, for any α ∈ B(ω, R) we have limx→ω d(tα(x), x) ≤ R. Hence +shrinking Yω if necessary, we can assume that d(tα(x), x) ≤ R for any x ∈ Yω. +Therefore, we have +� +γ∈X(ω) +| f(γ, α) − f(γ, ω)| = +� +γ∈B(ω,R+S) +| f(γ, α) − f(γ, ω)| +(8.2) += +lim +x→ω,x∈Yω +� +γ∈B(ω,R+S) +| f(tγ(x), tα(x)) − f(tγ(x), x)| += +lim +x→ω,x∈Yω +� +z∈B(x,R+S) +| f(z, tα(x)) − f(z, x)| += +lim +x→ω,x∈Yω +� +z∈X +| f(z, tα(x)) − f(z, x)|, +where the last item is no more than ε by assumption. Therefore applying Lemma +8.3, we conclude the sufficiency. +Necessity: Given ε, R > 0, Lemma 8.3 provides S > 0 and a function f : X × X → +[0, 1] satisfying the conditions (1)-(3) therein. +Fix an ω ∈ Uc and we choose +{tα : Dα → Rα} to be a compatible family for ω. Applying Proposition 3.3, there +exists Yω ⊆ X with ω(Y) = 1 and a local coordinate system {hy : B(ω, R + 2S) → +B(y, R + S)}y∈Yω such that the map +hy : B(ω, R + S) → B(y, R + S), +α �→ tα(y) +is a surjective isometry for each y ∈ Yω. By the calculations in (8.1), we obtain that +lim +x→ω,x∈Yω +� +z∈X +f(z, x) = 1. +Hence for the given ε, there exists Y′ +ω ⊆ Yω with ω(Y′ +ω) = 1 such that for any x ∈ Y′ +ω +we have +� +z∈X +f(z, x) ∈ (1 − ε, 1 + ε). +On the other hand, for any α ∈ B(ω, R) we apply the calculations in (8.2) and +obtain: +lim +x→ω,x∈Y′ω +� +z∈X +| f(z, tα(x)) − f(z, x)| = +� +γ∈X(ω) +| f(γ, α) − f(γ, ω)| ≤ ε. +Note that for x ∈ Y′ +ω, Proposition 3.3 implies that {tα(x) : α ∈ B(ω, R)} = B(x, R). +Hence there exists Y′′ +ω ⊆ Y′ +ω with ω(Y′′ +ω) = 1 such that for any x ∈ Y′′ +ω and y ∈ B(x, R), +we have +� +z∈X +| f(z, y) − f(z, x)| < 2ε. + +36 +QIN WANG AND JIAWEN ZHANG +Taking D := � +ω∈Uc Y′′ +ω, then it is clear that D ⊇ Uc. Moreover, the analysis above +shows that: +• for any x ∈ D we have � +z∈X f(z, x) ∈ (1 − ε, 1 + ε); +• for any x ∈ D and y ∈ B(x, R), we have � +z∈X | f(z, y) − f(z, x)| < 2ε. +Finallyusinga standard normalisation argument(orequivalently, applyingLemma +2.11 and modifying Lemma 8.3 accordingly), we conclude the proof. +□ +Setting ξy(x) := f(x, y) for the function f in Proposition 8.2, we can rewrite +Proposition 8.2 as follows: +Proposition 8.2′. Let (X, d) be a space and U ⊆ βX be an invariant open subset. Then +X has partial Property A towards Uc if and only if for any ε, R > 0, there exist S > 0, a +subset D ⊆ X with D ⊇ Uc and a function ξ : D → ℓ1(X)1,+, x �→ ξx satisfying: +(1) supp(ξx) ⊆ B(x, S) for any x ∈ D; +(2) for any x, y ∈ D with d(x, y) ≤ R, we have ∥ξx − ξy∥1 ≤ ε. +Remark 8.6. We remark that the function ξ in Proposition 8.2′ can be made such +that ξx ∈ ℓ1(D)1,+. In fact, this is the same trick as in the case of Property A (see, +e.g., [27, Proposition 4.2.5]). Moreover, we can further replace ℓ1(D)1,+ by ℓ2(D)1,+ +using the Mazur map (see, e.g., [45, Proposition 1.2.4] for the same trick). +Now we provide an alternative picture for Proposition 8.2 using the notion of +ideals in spaces (see Definition 4.12). Recall that for an ideal L in X, we denote +U(L) := � +Y∈L Y. We need the following auxiliary lemma: +Lemma 8.7. Let L be an ideal in X and D ⊆ X. Then D ⊇ U(L)c if and only if there +exists Y ∈ L such that D ⊇ Yc. +Proof. Assume Y ∈ L such that D ⊇ Yc. Note that Y ∩ Yc = ∅ and Y ∪ Yc = βX. +Hence we have D ⊇ Yc = βX \ Y ⊇ βX \ U(L) = U(L)c. +Conversely, assume that D ⊇ U(L)c. Then (D)c ⊆ U(L) = � +Y∈L Y. Since D is +clopen in the compact space βX, the set (D)c is compact as well. Hence there exists +Y1, · · · , Yn ∈ L such that (D)c ⊆ Y1 ∪· · ·∪Yn = Y1 ∪ · · · ∪ Yn. Since L is an ideal, the +set Y := Y1∪· · ·∪Yn ∈ L. Then we have (D)c ⊆ Y, which implies that D ⊇ (Y)c = Yc. +Finally we obtain D = D ∩ X ⊇ Yc ∩ X = Yc, which conclude the proof. +□ +Thanks to Lemma 8.7, now we can rewrite Proposition 8.2 (combining with +Remark 8.6) as follows: +Proposition 8.2′′. Let (X, d) be a space, U ⊆ βX an invariant open subset and L = L(U) +the associated ideal in X. Then X has partial Property A towards Uc if and only if for any +ε, R > 0, there exist S > 0, a subset Y ∈ L(U) and a function ξ : Yc → ℓ2(Yc)1,+, x �→ ξx +satisfying: +(1) supp(ξx) ⊆ B(x, S) for any x ∈ Yc; +(2) for any x, y ∈ Yc with d(x, y) ≤ R, we have ∥ξx − ξy∥1 ≤ ε. +Similar to the proof in the case of Property A, we also have the following: + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +37 +Corollary 8.8. Let (X, d) be a space, U ⊆ βX an invariant open subset and L = L(U) the +associated ideal in X. Then X has partial Property A towards Uc if and only if for any +R > 0 and ε > 0, there exist S > 0, a subset Y ∈ L(U) and a kernel k : Yc × Yc → R of +positive type satisfying: +(1) for x, y ∈ Yc, we have k(x, y) = k(y, x) and k(x, x) = 1; +(2) for x, y ∈ Yc with d(x, y) ≥ S, we have k(x, y) = 0; +(3) for x, y ∈ Yc with d(x, y) ≤ R, we have |1 − k(x, y)| ≤ ε. +8.2. Partial operator norm localisation property. Recall that the notion of oper- +ator norm localisation property (ONL) was introduced by Chen, Tessera, Wang +and Yu in [10], and proved by Sako in [37] that ONL is equivalent to Property A. +Here we introduce a partial version of ONL, parallel to Definition 8.1. +Let ν be a positive locally finite Borel measure on X and H be a separable +infinite-dimensional Hilbert space. For an operator T ∈ B(L2(X, ν) ⊗ H), we can +also define its propagation as in Section 2.3. We introduce the following: +Definition 8.9. Let (X, d) be a space and U ⊆ βX an invariant open subset. We +say that X has partial operator norm localisation property (partial ONL) towards Uc = +∂βX \ U if there exists c ∈ (0, 1] such that for any R > 0 there exist S > 0 and +D ⊆ X with D ⊇ Uc satisfying the following: for any positive locally finite Borel +measure ν on X with supp(ν) ⊆ D and any a ∈ B(L2(X, ν) ⊗ H) with propagation +at most R, there exists a non-zero ζ ∈ L2(X, ν) ⊗ H with diam(supp(ζ)) ≤ S such +that c∥a∥ · ∥ζ∥ ≤ ∥aζ∥. +The aim of the rest of this subsection is to show that partial ONL is equivalent +to partial Property A. We will follow the outline of [37]. +To simplify the statement, denote CR +u[X; H] := {T ∈ B(ℓ2(X) ⊗ H) : prop(a) ≤ R} +for R ≥ 0. +For a subspace Y ⊆ X, it is clear that CR +u[Y] � χYCR +u[X]χY (resp. +CR +u[Y; H] � χYCR +u[X; H]χY), and hence can be regarded as a subset of CR +u[X] (resp. +CR +u[X; H]) with support in Y × Y. Similarly, C∗ +u(Y) � χYC∗ +u(X)χY can be regarded +as a C∗-subalgebra in C∗ +u(X). Hence we will not tell the difference in the sequel. +For S > 0, denote +BY +S := +� +x∈X +B(ℓ2(B(x, S) ∩ Y)), +whose elements will be written as b = ([bx(y, z)]y,z∈B(x,S)∩Y)x∈X. We also consider the +map +ψY +S : C∗ +u(Y) � χYC∗ +u(X)χY −→ BY +S +by +a �→ ([a(y, z)]y,z∈B(x,S)∩Y)x∈X. +Recall the notions of completely positive map and completely bounded map: +• A self-adjoint closed subspace F of a unital C∗-algebra B such that 1B ∈ F is +called an operator system. +• A linear map φ from F to a C∗-algebra C is said to be completely positive if +the map φ(n) = φ ⊗ id : F ⊗ Mn(C) → C ⊗ Mn(C) is positive for every n. +• A linear map θ : F → C is said to be completely bounded if the sequence +{∥θ(n) : F⊗Mn(C) → C⊗Mn(C)∥} is bounded. Denote ∥θ∥cb := supn∈N ∥θ(n)∥. +We have the following characterisation for partial ONL, which is analogous to +[37, Proposition 3.1]. The proof is almost identical, hence omitted. + +38 +QIN WANG AND JIAWEN ZHANG +Lemma 8.10. Let (X, d) be a space and U ⊆ βX an invariant open subset. Then the +following are equivalent: +(1) X has partial ONL towards Uc; +(2) there exists c ∈ (0, 1] such that for any R > 0 there exist S > 0 and D ⊆ X with +D ⊇ Uc satisfying condition (α): for any a ∈ CR +u[D; H] there exists a non-zero +ζ ∈ ℓ2(X) ⊗ H with diam(supp(ζ)) ≤ S and c∥a∥ · ∥ζ∥ ≤ ∥aζ∥; +(3) for any c ∈ (0, 1) and R > 0, there exist S > 0 and D ⊆ X with D ⊇ Uc satisfying +condition (α); +(4) for any c ∈ (0, 1) and R > 0, there exist S > 0 and D ⊆ X with D ⊇ Uc +satisfying condition (β): for any a ∈ CR +u[D] there exists a non-zero ξ ∈ ℓ2(X) with +diam(supp(ξ)) ≤ S and c∥a∥ · ∥ξ∥ ≤ ∥aξ∥; +(5) for any ε, R > 0 there exist S > R and D ⊆ X with D ⊇ Uc such that +∥(ψD +S |CR +u[D])−1 : ψD +S (CR +u[D]) −→ CR +u[D]∥ < 1 + ε; +(6) for any ε, R > 0 there exist S > R and D ⊆ X with D ⊇ Uc such that +∥(ψD +S |CR +u[D])−1 : ψD +S (CR +u[D]) −→ CR +u[D]∥cb < 1 + ε; +We record the following, which comes directly from Lemma 8.7 and 8.10. +Lemma 8.11. Let (X, d) be a space, U ⊆ βX an invariant open subset and L = L(U) the +associated ideal in X. Then X has partial ONL towards Uc if and only if for any c ∈ (0, 1) +and R > 0 there exist S > 0 and Y ∈ L(U) satisfying the following: for any a ∈ CR +u[Yc] +there exists a non-zero ξ ∈ ℓ2(X) with diam(supp(ξ)) ≤ S and c∥a∥ · ∥ξ∥ ≤ ∥aξ∥. +Finally, we can mimic the proof of [37, Theorem 4.1] using Proposition 8.2′, +Remark 8.6, Corollary 8.8 and Lemma 8.10 instead, and reach the following. The +proof is almost identical, and hence omitted. +Proposition 8.12. Let (X, d) be a space and U ⊆ βX be an invariant open subset. Then +X has partial Property A towards Uc if and only if X has partial ONL towards Uc. +To end this subsection, we study a permanence property of partial ONL, which +will help to prove the main result in the next subsection. +Let X be a space and L an ideal in X. Assume that X can be decomposed into +X = X1 ∪ X2. Consider +(8.3) +Li := {Y ∩ Xi : Y ∈ L} +for +i = 1, 2. +Then it is routine to check that Li is an ideal in Xi for i = 1, 2, and +L = {Y1 ∪ Y2 : Yi ∈ Li, i = 1, 2}. +With respect to the decomposition above, we now show that partial ONL is +preserved under finite unions. +Proposition 8.13. With the notation as above, assume that Xi has partial ONL towards +βXi \ U(Li) for i = 1, 2. Then X has partial ONL towards βX \ U(L). +One way to prove Proposition 8.13 is to follow the proof of [16, Lemma 3.3] +with minor changes. Here we choose another approach using Proposition 8.12. +Recall from Corollary A.8 that for a subset Z ⊆ X, the closure Z in βX is +homeomorphic to βZ. Hence we will regard them as the same object in the sequel. + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +39 +Now we can easily transfer the restriction of ideals in (8.3) to that of invariant +open subsets (the proof is straightforward, hence omitted): +Lemma 8.14. Let X = X1 ∪ X2, L be an ideal in X and U = U(L) ⊆ βX be the associated +invariant open subset of βX. Then U ∩ Xi is an invariant open subset of Xi = βXi, which +corresponds to the ideal Li in (8.3) for i = 1, 2. +Although U ∩ Xi is invariant in βXi, generally it is not invariant in βX. This +coincides with the fact that χXiC∗ +u(X)χXi � C∗ +u(Xi) is a just subalgebra in C∗ +u(X) +rather than an ideal. Instead, we consider the spatial ideal IXi recalled in Section +4, and prove the following permanence property for partial Property A: +Lemma 8.15. Let X = X1 ∪ X2, U ⊆ βX be an invariant open subset and Ui = U ∩ Xi +for i = 1, 2. If Xi has partial Property A towards βXi \ Ui for i = 1, 2, then X has partial +Property A towards βX \ U. +Proof. For any R > 0 and i = 1, 2, set Ui(R) := � +Y∈L(Ui) NR(Y). +Clearly, Ui(R) +is an invariant open subset of NR(Xi) = β(NR(Xi)). Moreover, it follows from +Proposition 8.2 that NR(Xi) has partial Property A towards NR(Xi) \ Ui(R). By +definition, this means that the groupoid G(NR(Xi))NR(Xi)\Ui(R) is amenable. Hence +as a subgroupoid, G(NR(Xi))NR(Xi)\U is amenable, which further implies that the +groupoid � +R>0 G(NR(Xi))NR(Xi)\U is amenable. +Note from Lemma 4.14 that � +R NR(Xi) \ U is invariant in βX and Lemma 4.16 +implies that +G(X)� +R NR(Xi)\U = +� +R>0 +G(NR(Xi))NR(Xi)\U, +which is hence amenable. Note that � +R NR(X1) ∪ � +R NR(X2) = βX, and hence due +to the extension property we obtain that +G(X)βX\U = G(X)� +R NR(X1)\U ∪ G(X)� +R NR(X2)\U +is amenable as required. +□ +Combining Proposition 8.12 and Lemma 8.15, we conclude Proposition 8.13. +8.3. Characterisation for I(U) = ˜I(U). Having established all the necessary ingre- +dients above, now we present the main result of this section: +Theorem 8.16. Let (X, d) be a space as in Section 2.2 and U ⊆ βX be a countably +generated invariant open subset. Then the following are equivalent: +(1) X has partial Property A towards βX \ U; +(2) ˜I(U) = I(U); +(3) the ideal IG of all ghost operators is contained in I(U). +Note that U = X is countably generated, and hence Theorem 8.16 recovers [36, +Theorem 1.3] (see Example 4.8 and Example 5.5). Borrowing the language of +[36], condition (3) in Theorem 8.16 says that all the ghosts can be busted in the +geometric ideal I(U). +We follow the outline of the proof for [36, Theorem 1.3]. Firstly, we need a +modified version of [36, Lemma 4.2]: + +40 +QIN WANG AND JIAWEN ZHANG +Lemma 8.17. Let (X, d) be a space, U ⊆ βX be a countably generated invariant open +subset and L = L(U) the associated ideal in X. Assume that X does not have partial ONL +towards Uc. Then there exist κ ∈ (0, 1), R > 0, a sequence (Tn) in Cu[X], a sequence (Bn) +of finite subsets of X and a sequence (Sn) of positive real numbers such that: +(a) (Sn) is an increasing sequence tending to infinity as n → ∞; +(b) each Tn is positive and has norm 1; +(c) for n � m, then Bn ∩ Bm = ∅; +(d) each Tn is supported in Bn × Bn; +(e) for each n and ξ ∈ ℓ2(X) with ∥ξ∥ = 1 and diam(suppξ) ≤ Sn, then ∥Tnξ∥ ≤ κ; +(f) for each Y ∈ L(U), there exists n such that Bn ∩ Y = ∅. +Proof. Fixing a basepoint x0 ∈ X, consider the decomposition X = X(1) ∪ X(2) with +X(1) := +� +m even +{x ∈ X : m2 ≤ d(x, x0) < (m + 1)2} +and +X(2) := +� +m odd +{x ∈ X : m2 ≤ d(x, x0) < (m + 1)2}. +Set Li := {Y ∩ X(i) : Y ∈ L} for i = 1, 2. By assumption, X does not have partial +Property A towards Uc. Hence without loss of generality, we can assume that +X(1) does not have partial Property A towards βX(1) \ U(L1) thanks to Proposition +8.13. Note that this implies that U(L1) � βX(1). It is clear that L1 is also countably +generated, and hence according to Lemma 7.6 there exists a countable subset +{Y1, Y2, · · · , Yn, · · · } of L1 such that +L1 = {Z ⊆ X(1) : ∃ n ∈ N such that Z ⊆ Yn}. +In the sequel, we fix such a sequence {Y1, Y2, · · · , Yn, · · · }. +Due to Lemma 8.11, we know that there exist c ∈ (0, 1) and R > 0 such that for +any Y ∈ L1 and S > 0, there exists T ∈ CR +u[X(1) \ Y] with ∥T∥ = 1 satisfying: for any +ξ ∈ ℓ2(X(1)) with diam(suppξ) ≤ S and ∥ξ∥ = 1, then ∥Tξ∥ < c. We call such an +operator (R, c, S, Y)-localised. Replacing T by T∗T (and R by 2R and c by √c), we +see that there exist c ∈ (0, 1) and R > 0 such that for any Y ∈ L1 and S > 0, there +exists a positive (R, c, S, Y)-localised operator of norm one. Let us fix such c and R +in the rest of the proof, and set κ := +2c +1+c < 1. +Note that X(1) can be decomposed into: +X(1) := +� +m∈N +Xm +where each Xm is finite and d(Xm, Xn) > R for any n � m. Hence each T ∈ CR +u[X(1)] +splits as a block diagonal sum of finite rank operators T = +� +m T(m) where T(m) ∈ +B(ℓ2(Xm)), with respect to this decomposition. +Take S1 = 1. By assumption, there exists a positive (R, c, S1, Y1)-localised op- +erator T ∈ CR +u[X(1) \ Y1] with norm 1. +Note that ∥T∥ = supm ∥T(m)∥, and then +there exists m1 ∈ N such that ∥T(m1)∥ > 1+c +2 . We set T1 := T(m1)/∥T(m1)∥ and denote +B1 := Xm1 ∩ (X(1) \ Y1), which is nonempty since T has support in B1 × B1 by as- +sumption. Then for any ξ ∈ ℓ2(X(1)) with ∥ξ∥ = 1 and diam(supp(ξ)) ≤ S1, we + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +41 +have +∥T1ξ∥ ≤ +2c +1 + c = κ < 1. +Now we take S2 > max +� +diam +� � +k≤m1 Xk +� +, 2 +� +. +By assumption, there exists a +positive (R, c, S2, Y2)-localised operator T ∈ CR +u[X(1) \ Y2] with norm 1. Again there +exists m2 such that ∥T(m2)∥ > 1+c +2 , which forces m2 > m1. We set T2 := T(m2)/∥T(m2)∥ +and denote B2 := Xm2 ∩(X(1)\Y2), which is nonempty since T has support in B2×B2 +by assumption. Similarly for any ξ ∈ ℓ2(X(1)) with ∥ξ∥ = 1 and diam(supp(ξ)) ≤ S2, +we have ∥T2ξ∥ ≤ κ. +Inductively, we can construct a sequence (Tn) in Cu[X(1)] ⊆ Cu[X], a sequence +(Bn) of finite subsets of X(1) ⊆ X and a sequence (Sn) of positive real numbers +satisfying condition (a)-(e) in the statement. Furthermore for each Z ∈ L(U), there +exists Yn containing Z∩X(1) for some n. By construction, we know that Bn ∩Yn = ∅ +and Bn ⊆ X(1). Hence we have: +Bn ∩ Z = Bn ∩ X(1) ∩ Z ⊆ Bn ∩ Yn = ∅, +which provides condition (f) and concludes the proof. +□ +Remark 8.18. Comparing Lemma 8.17 with [36, Lemma 4.2], we note that condition +(f) is the only extra condition added in Lemma 8.17. +It seems hard to write +condition (f) in the language of the invariant open subset U instead of the ideal +L(U), which indicates the importance of using the notion of ideals in spaces as +mentioned in Section 1. +Now we are in the position to prove Theorem 8.16. +Proof of Theorem 8.16. “(1) ⇒ (2)” is contained in Proposition 7.2, and “(2) ⇒ (3)” +holds trivially since IG = ˜I(X) ⊆ ˜I(U). Hence it suffices to show “(3) ⇒ (1)”, and +we follow the outline of the proof for [36, Theorem 1.3]. +Assume that X does not have partial Property A towards Uc, then it follows +from Proposition 8.12 that X does not have partial ONL towards Uc. Then from +Lemma 8.17, there exist κ ∈ (0, 1), R > 0, a sequence (Tn) in Cu[X], a sequence +(Bn) of finite subsets of X and a sequence (Sn) of positive real numbers satisfying +condition (a)-(f) therein. Now we consider the operator +T := +� +n +Tn, +which is a positive operator in Cu[X] of norm one. +Now we take a continuous function f : [0, 1] → [0, 1] such that supp f ⊆ [1+κ +2 , 1] +and f(1) = 1. Consider the operator f(T) ∈ C∗ +u(X), which is positive, norm one, +and admits a decomposition +f(T) = +� +n +f(Tn), +where each f(Tn) ∈ B(ℓ2(Bn)). We will show that f(T) ∈ ˜I(X) \ I(U), and hence +conclude a contradiction. +First we show that f(T) � I(U). Recall from (4.2) that +I(U) = {T′ ∈ Cu[X] : supp(T′) ⊆ Y × Y for some Y ∈ L(U)}, + +42 +QIN WANG AND JIAWEN ZHANG +Now for any T′ ∈ Cu[X] with supp(T′) ⊆ Y × Y for some Y ∈ L(U), condition (f) in +Lemma 8.17 implies that there exists n such that Bn ∩ Y = ∅. Hence we have: +∥ f(T) − T′∥ ≥ ∥χBn f(T)χBn − χBnT′χBn∥ = ∥ f(Tn) − 0∥ = 1, +which implies that f(T) � I(U). +On the other hand, using the same argument as for [36, Theorem 1.3] (since +Lemma 8.17 provides all the conditions required in [36, Lemma 4.2]), we obtain +that f(T) is a ghost operator. Hence according to Example 5.5, we have f(T) ∈ ˜I(X). +Therefore, we conclude the proof. +□ +9. Open questions +Here we collect several open questions around this topic. +First recall from Theorem 5.4 that for a space (X, d), any ideal I in the uniform +Roe algebra C∗ +u(X) must lie between I(U) and ˜I(U) for U = U(I). However, the +structure of the lattice +IU = {I is an ideal in C∗ +u(X) : U(I) = U} +in (1.1) is still unclear. Note that for any invariant open subset V ⊇ U of βX, the +ideal I(V)∩ ˜I(U) belongs to the lattice IU. Unfortunately, we do not know whether +these ideals can bust every element in IU. Hence we pose the following: +Question 9.1. Let (X, d) be a space and U ⊆ βX be an invariant open subset. Can we +describe elements in the lattice IU = {I is an ideal in C∗ +u(X) : U(I) = U} in details? For +I ∈ IU, can we find an invariant open subset V ⊇ U such that I = I(V) ∩ ˜I(U)? +Note that an answer to the above question together with Theorem 5.4 will +provide a full description for the ideal structure of the uniform Roe algebra. +Our next question concerns minimal points discussed in Section 6.1. Recall that +minimal points in the Stone- ˇCech boundary correspond to maximal ideals in the +uniform Roe algebra. However as shown in Theorem 6.8, there exist a number of +non-minimal points in the boundary. Hence it would be interesting to explore a +practical approach to distinguish minimal points. +Question 9.2. Given a space (X, d), can we find a practical approach to distinguish +minimal points in the Stone- ˇCech boundary ∂βX? +Note from Theorem 6.8 that the answer might not be easy even in the elementary +case that X = Z. +Our last questions concern the assumption of countably generatedness used +in Section 7 and 8. Recall that in Proposition 7.8 we prove that the inclusion +ιU : I(U) ֒→ ˜I(U) induces an isomorphism in K-theory when the space is coarsely +embeddable and U is countably generated. We ask the following: +Question 9.3. Does the inclusion ιU : I(U) ֒→ ˜I(U) induce an isomorphism in K-theory +for coarsely embeddable X without the assumption that U is countably generated? +Also note that even under the assumption of countable generatedness, we are +merely able to show that (ιI)∗ : K∗(I ∩ Cu[X]) −→ K∗(I) is injective in Proposition +7.8. Hence we also pose the following: + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +43 +Question 9.4. For an ideal I in C∗ +u(X) with U(I) countably generated and X coarsely +embeddable, is the map (ιI)∗ : K∗(I ∩ Cu[X]) −→ K∗(I) surjective for ∗ = 0, 1? +Our final question is designed for Theorem 8.16. Recall that the assumption of +countably generatedness plays an important role in Lemma 8.17, which is in turn +crucial in the proof of Theorem 8.16. Hence we ask the following: +Question 9.5. Let (X, d) be a space and U ⊆ βX be an invariant open subset. +If +˜I(U) = I(U), can we deduce that X has partial Property A towards βX \ U? +Appendix A. Ultrafilters +Here we collect some basic knowledge on ultrafilters, which is used throughout +the paper. The material should be fairly well-known (see, e.g., [9, Appendix A], +[34, Chapter 7.4] or [43, Appendix A]). While some of the results might not be +standard and we have not dug into the reference, we include the proofs for +convenience to readers. +Definition A.1. Let X be a set and P(X) be its power set. An ultrafilter on X is a +family U ⊆ P(X) satisfying the following: +(I.1) ∅ � U; +(I.2) for A, B ∈ U, then A ∩ B ∈ U; +(I.3) for A ∈ U and A ⊆ B, then B ∈ U; +(I.4) for any A ⊆ X, either A ∈ U or X \ A ∈ U. +For an ultrafilter U on X, we can associate a function ω : P(X) → {0, 1} by +setting ω(A) = 1 if and only if A ∈ U. It follows from (I.1)-(I.4) above that ω is a +finitely additive {0, 1}-valued probability measure on P(X). Conversely for such +a function ω on P(X), we can associate a family Uω := {A ⊆ X : ω(A) = 1}. It is +clear that Uω is an ultrafilter on X, and these two procedures are inverse to each +other. Therefore throughout the paper, we slide between these two notions freely +without further explanation. +For a ∈ X, it is clear that the family {A ∈ P(X) : a ∈ A} is an ultrafilter on +X. Such an ultrafiler is called principal. An ultrafilter which is not principal is +called non-principal. An argument using Zorn’s lemma shows that non-principal +ultrafilters always exist whenever X is infinite. +The following is well-known (see, e.g., [43, Lemma A.2]): +Lemma A.2. Let ω be an ultrafilter on a set X, and D ⊆ X with ω(D) = 1. Let f : D → Y +be a function from D to a compact Hausdorff topological space Y. Then there exists a +unique point y ∈ Y such that for any open neighbourhood U of y, we have ω(f −1(U)) = 1. +Definition A.3. The unique point in Lemma A.2 is called the ultralimit of f along +ω or the ω-limit of f, denoted by limω f or lima→ω f(a). +We record the following localisation result: +Lemma A.4. Let U be an ultrafilter on a set X, and A ⊆ X with A ∈ U. Then we have: +(1) {S ∩ A : S ∈ U} = {S ⊆ A : S ∈ U} is an ultrafilter on A, denoted by UA. +(2) U = {S ⊆ X : S ∩ A ∈ UA} = {S ⊆ X : ∃ S′ ∈ UA such that S′ ⊆ S}. + +44 +QIN WANG AND JIAWEN ZHANG +Proof. (1). It follows from Definition A.1 that {S ∩ A : S ∈ U} = {S ⊆ A : S ∈ U}, +and hence (I.1)-(I.3) hold for UA. Concerning (I.4): given B ⊆ A, if B ∈ U then +B ∈ UA as well; if B � U then X \ B ∈ U, and hence A \ B = A ∩ (X \ B) ∈ UA. +(2). This is straightforward, hence omitted. +□ +We can also extend an ultrafilter on a subset to the whole space. The proof is +straightforward, hence omitted. +Lemma A.5. Let Y be a subset of a set X, and U0 an ultrafilter on Y. Define +U := {S ⊆ X : S ∩ Y ∈ U0}. +Then U is an ultrafilter on X. +The following result provides an approach to combine a family of ultrafilters +into a single one: +Lemma A.6. Let {Xi}i∈I be a family of sets, and Ui be an ultrafilter on Xi for each i ∈ I. +Let ω0 be an ultrafilter on I. Consider the set X := � +i∈I Xi and define: +U := +� � +i∈I +Ai ⊆ +� +i∈I +Xi : ∃ J ⊆ I with ω0(J) = 1 such that ∀i ∈ J, Ai ∈ Ui +� +. +Then U is an ultrafilter on X. +Proof. Firstly, it is clear that ∅ � U. Assume that � +i∈I Ai and � +i∈I Bi ∈ U, i.e., there +exist JA, JB ⊆ I with ω0(JA) = ω0(JB) = 1 such that Ai ∈ Ui for any i ∈ JA and Bi ∈ Ui +for any i ∈ JB. Consider (� +i∈I Ai) ∩ (� +i∈I Bi) = � +i∈I(Ai ∩ Bi) and J = JA ∩ JB. Then +ω0(J) = 1 and for each i ∈ J, Ai and Bi are in Ui. This implies that Ai ∩ Bi ∈ Ui, +which concludes (I.2). +It is clear that (I.3) holds for U and finally, we consider (I.4). +Given A = +� +i∈I Ai ⊆ X, denote J := {i ∈ I : Ai ∈ Ui}. If ω0(J) = 1, then it follows that A ∈ U. +Otherwise, assume that ω0(J) = 0. Then we consider X \ A = � +i∈I(Xi \ Ai). Then +I \ J = {i ∈ I : Xi \ Ai ∈ Ui} and ω0(I \ J) = 1, which implies that X \ A ∈ U and +concludes the proof. +□ +Recall that ultrafilters can also be characterised by the Stone- ˇCech compactifi- +cation. More precisely, we have the following (see, e.g., [34, Chapter 7.4]): +Lemma A.7. Let X be a set and βX be the Stone- ˇCech compactification of X. +(1) Given ω ∈ βX, the family {A ⊆ X : ω ∈ A} is an ultrafilter on X. +(2) Given an ultrafilter U on X, the intersection � +A∈U A consists of a single point. +The procedures above are inverse to each other, and hence βX can be characterised by +ultrafilters on X. Moreover, points in ∂βX correspond to non-principal ultrafilters. +Thanks to Lemma A.7, we will also use ultrafilters and points in the Stone- ˇCech +compactification freely without further explanation throughout the paper. +For convenience, we also record that for D ⊆ X, its closure D in βX satisfies: +D = {ω ∈ βX : ω(D) = 1} +and D is clopen in βX. The topology of βX is generated by {D : D ⊆ X}. + +GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS +45 +Finally we recall the following, which can be proved either directly using the +universal property of the Stone- ˇCech compactification or deduced directly from +Lemma A.4, A.5 and A.7: +Corollary A.8. 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EPFL Press, Lausanne, 2009. +[46] Rufus Willett and Guoliang Yu. Higher index theory for certain expanders and Gromov +monster groups, I. Adv. Math., 229(3):1380–1416, 2012. +[47] Guoliang Yu. The coarse Baum-Connes conjecture for spaces which admit a uniform embed- +ding into Hilbert space. Invent. Math., 139(1):201–240, 2000. +(Q. Wang) Research Center for Operator Algebras, and Shanghai Key Laboratory of Pure +Mathematics and Mathematical Practice, School of Mathematical Sciences, East China Nor- +mal University, Shanghai, 200241, China. +Email address: qwang@math.ecnu.edu.cn +(J. Zhang) School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai, +200433, China. +Email address: jiawenzhang@fudan.edu.cn + diff --git a/K9E4T4oBgHgl3EQfJgxB/content/tmp_files/load_file.txt b/K9E4T4oBgHgl3EQfJgxB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..110c5076ded40a999ed8cc614b3d22f14ddbbf9c --- /dev/null +++ b/K9E4T4oBgHgl3EQfJgxB/content/tmp_files/load_file.txt @@ -0,0 +1,2112 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf,len=2111 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='04921v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='OA] 12 Jan 2023 GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS QIN WANG AND JIAWEN ZHANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In this paper, we investigate the ideal structure of uniform Roe algebras for general metric spaces beyond the scope of Yu’s property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Inspired by the ideal of ghost operators coming from expander graphs and in contrast to the notion of geometric ideal, we introduce a notion of ghostly ideal in a uniform Roe algebra, whose elements are locally invisible in certain directions at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We show that the geometric ideal and the ghostly ideal are respectively the smallest and the largest element in the lattice of ideals with a common invariant open subset of the unit space of the coarse groupoid by Skandalis-Tu-Yu, and hence the study of ideal structure can be reduced to classifying ideals between the geometric and the ghostly ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As an application, we provide a concrete description for the maximal ideals in a uniform Roe algebra in terms of the minimal points in the Stone- ˇCech boundary of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We also provide a criterion to ensure that the geometric and the ghostly ideals have the same K-theory, which helps to recover counterexamples to the Baum-Connes type conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, we introduce a notion of partial Property A for a metric space to characterise the situation in which the geometric ideal coincides with the ghostly ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Mathematics Subject Classification (2020): 47L20, 46L80, 51F30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Keywords: Uniform Roe algebras, Coarse groupoids, Geometric and ghostly ideals, Max- imal ideals, Partial Property A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Introduction Roe algebras are C∗-algebras associated to metric spaces, which encode the coarse geometry of the underlying spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' They were introduced by Roe in his pi- oneering work on higher index theory [31], where he discovered that the K-theory of Roe algebras serves as a receptacle for higher indices of elliptic differential oper- ators on open manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence the computation for the K-theory of Roe algebras becomes crucial in the study of higher index theory, and a pragmatic and prac- tical approach is to consult the Baum-Connes type conjectures [3, 4, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' There is also a uniform version of the Roe algebra, which equally plays a key role in higher index theory (see [39, 41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Over the last four decades, there have been a number of excellent works around this topic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', [17, 21, 24, 46, 47]), which lead to significant progresses in topology, geometry and analysis (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', [32, 33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, the analytic structure of (uniform) Roe algebras reflects the coarse geometry of the underlying spaces, and the rigidity problem asks whether the coarse geometry of a metric space can be fully determined by the associated (uniform) Roe algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This problem was initially studied by ˇSpakula and Willett in [42], followed by a series of works in the last decade [5, 6, 7, 8, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recently this problem is completely solved in the uniform case by the profound work Date: January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' QW is partially supported by NSFC (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 11831006, 12171156), and the Science and Technology Commission of Shanghai Municipality (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 22DZ2229014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' JZ is supported by NSFC11871342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 1 2 QIN WANG AND JIAWEN ZHANG [2], which again highlights the importance of uniform Roe algebras in coarse geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Meanwhile, uniform Roe algebras have also attained rapidly-growing interest from researchers in mathematical physics, especially in the theory of topological materials and topological insulators (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', [18] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Due to their importance, Chen and the first-named author initiated the study of the ideal structure for (uniform) Roe algebras [11, 12, 13, 14, 15, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' They succeeded in obtaining a full description for the ideal structure of the uniform Roe algebra when the underlying space has Yu’s Property A (see [12, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' However, the general picture is far from clear beyond the scope of Property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In the present paper, we aim to provide a systematic study on the ideal structure of uniform Roe algebras for general discrete metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' To outline our main results, let us first explain some notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a discrete metric space of bounded geometry (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 for precise definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Thinking of operators on ℓ2(X) as X-by-X matrices, we say that such an operator has finite propagation if the non-zero entries appear only in an entourage of finite width (measured by the metric on X) around the main diagonal (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3 for full details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The set of all finite propagation operators forms a ∗-subalgebra of B(ℓ2(X)), and its norm closure is called the uniform Roe algebra of X and denoted by C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' There is another viewpoint on the uniform Roe algebra based on groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from [39] that Skandalis, Tu and Yu introduced a notion of coarse groupoid G(X) associated to a discrete metric space X, and they succeeded in relating coarse geometry to the theory of groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The coarse groupoid G(X) is a lo- cally compact, Hausdorff, ´etale and principal groupoid (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5 for precise definitions), and the unit space of G(X) coincides with the Stone- ˇCech compacti- fication βX of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, the uniform Roe algebra C∗ u(X) can be interpreted as the reduced groupoid C∗-algebra of G(X) (see also [34, Chapter 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In [12], Chen and the first-named author concentrated on a class of ideals in the uniform Roe algebra in which finite propagation operators therein are dense, and they showed that these ideals can be described geometrically using the coarse groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More precisely, recall that a subset U ⊆ βX is invariant if any element γ in G(X) with source in U also has its range in U (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As shown in [12] (see also Section 4), for any ideal I in C∗ u(X) one can associate an invariant open subset U(I) of βX, and conversely for any invariant open subset U ⊆ βX one can associate an ideal I(U) in C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Furthermore, these two procedures provide a one-to-one correspondence between invariant open subsets of βX and ideals in C∗ u(X) in which finite propagation operators therein are dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Based on [12], the first-named author introduced the following notion in [44, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4]: Definition A (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a discrete metric space of bounded geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' An ideal I in the uniform Roe algebra C∗ u(X) is called geometric if the set of all finite propagation operators in I is dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As explained above, [12, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3] (see also Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9) indicates that the geometric ideals in C∗ u(X) can be fully determined by invariant open subsets of βX, which explains the terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consequently, the geometric ideals in GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 3 C∗ u(X) are easy to handle and they must have the form of I(U) for some invariant open subset U ⊆ βX, called the geometric ideal associated to U (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, it follows from [14, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4] that all ideals in C∗ u(X) are geometric when X has Yu’s Property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' However, things get complicated beyond the context of Property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As noticed in [12, Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5], when X comes from a sequence of expander graphs then the ideal IG consisting of all ghost operators are not geometric (see also [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall that an operator T ∈ B(ℓ2(X)) is a ghost if T ∈ C0(X × X) when regarding T as a function on X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Ghost operators are introduced by Yu, and they are crucial to provide counterexamples to the coarse Baum-Connes conjecture ([21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Direct calculations show that the associated invariant open subsets for IG and for the ideal of compact operators in B(ℓ2(X)) are the same, both of which equal X (see also Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence for a general metric space X and an invariant open subset U ⊆ βX, there might be more than one ideal I in the uniform Roe algebra C∗ u(X) satisfying U(I) = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore, the study of the ideal structure for C∗ u(X) can be reduced to analyse the lattice (where the order is given by inclusion) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1) IU := {I is an ideal in C∗ u(X) : U(I) = U} for each invariant open subset U ⊆ βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' One of the main contributions of the present paper is to find the smallest and the largest elements in the lattice IU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Following the discussions in [12], it is easy to see that I(U(I)) ⊆ I for any ideal I in C∗ u(X), which implies that the geometric ideal I(U) is the smallest element in IU (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' To explore the largest element, we have to include every ideal I in C∗ u(X) with U(I) = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Inspired by the definition of U(I) (see Equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1)), we introduce the following key notion: Definition B (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a discrete metric space of bounded geometry and U be an invariant open subset of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The ghostly ideal associated to U is defined to be ˜I(U) := {T ∈ C∗ u(X) : r(suppε(T)) ⊆ U for any ε > 0}, where suppε(T) := {(x, y) ∈ X ×X : |T(x, y)| ≥ ε} and r : X ×X → X is the projection onto the first coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We show that ˜I(U) is indeed an ideal in the uniform Roe algebra C∗ u(X) (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2) and moreover, we obtain the following desired result: Theorem C (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a discrete metric space of bounded geometry and U be an invariant open subset of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then any ideal I in C∗ u(X) with U(I) = U sits between I(U) and ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More precisely, the geometric ideal I(U) is the smallest element while the ghostly ideal ˜I(U) is the largest element in the lattice IU in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Theorem C draws the border of the lattice IU in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1), as an important step to study the ideal structure of uniform Roe algebras for general metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More precisely, once we can bust every ideal between I(U) and ˜I(U) for each invariant open subset U ⊆ βX, then we will obtain a full description for the ideal structure of the uniform Roe algebra C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We pose it as an open question in Section 9 and hope this will be done in some future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Concerning the ghostly ideal ˜I(U), we also provide an alternative picture in terms of limit operators developed in [43], showing that ˜I(U) consists of operators 4 QIN WANG AND JIAWEN ZHANG which vanish in the (βX \\ U)-direction (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that ghost operators vanish in all directions (see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7), and hence operators in ˜I(U) can be regarded as “partial” ghosts, which clarifies its terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Thanks to this viewpoint, we discover the deep reason behind the counterexample to the conjecture in [12], constructed by the first-named author in [44, Section 3] (see Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As an application, we manage to describe maximal ideals in the uniform Roe algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More precisely, it follows directly from Theorem C that maximal ideals correspond to minimal invariant closed subsets of the Stone- ˇCech boundary ∂βX := βX \\ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover using the theory of limit spaces1 developed in [43], we prove the following: Proposition D (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a strongly discrete metric space of bounded geometry and I be a maximal ideal in the uniform Roe algebra C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then there exists a point ω ∈ ∂βX such that I coincides with the ghostly ideal ˜I(βX \\ X(ω)), where X(ω) is the limit space of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A point ω ∈ ∂βX satisfying the condition in Proposition D is called a minimal point (see Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We show that there exist a number of non-minimal points in the boundary even for the simple case of X = Z: Theorem E (Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For the integer group Z with the usual metric, there exist non-minimal points in the boundary ∂βZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More precisely, for any sequence {hn}n∈N in Z tending to infinity such that |hn − hm| → +∞ when n + m → ∞ and n � m, and any ω ∈ ∂βZ with ω({hn}n∈N) = 1, then ω is not a minimal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We provide two approaches to prove Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' One is topological, which makes use of several constructions and properties of ultrafilters (recalled in Ap- pendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The other is C∗-algebraic, which replies on a description of maximal ideals in terms of limit operators (see Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='13) together with a recent result by Roch [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Returning to the lattice IU defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1), we already notice that generally IU consists of more than one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence it will be interesting and important to explore when IU has only a single element, or equivalently (thanks to Theorem C), when the geometric ideal I(U) coincides with the ghostly ideal ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' To study this problem, we start with an extra picture for geometric and ghostly ideals using the associated groupoid C∗-algebras (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Based on these descriptions, we show that the amenability of the restriction G(X)∂βX\\U of the coarse groupoid ensures that I(U) = ˜I(U) (see Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Meanwhile, we also discuss the K-theory of the geometric and ghostly ideals and provide a criterion to ensure that K∗(I(U)) = K∗(˜I(U)) for ∗ = 0, 1: Proposition F (Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let X be a discrete metric space of bounded geometry which can be coarsely embedded into some Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then for any countably generated invariant open subset U ⊆ βX, we have an isomorphism (ιU)∗ : K∗(I(U)) −→ K∗(˜I(U)) 1Note that the theory of limit spaces and limit operators developed in [43] only concerns strongly discrete metric spaces of bounded geometry (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 for precise definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Although as noticed in [43] this will not lose any generality, we put this assumption to simplify proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 5 for ∗ = 0, 1, where ιU is the inclusion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that there is a technical condition of countable generatedness (see Def- inition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4) used in Proposition F, which holds for a number of examples (see Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5) including X itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' However as shown in Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='11, there does exist an invariant open subset which is not countably generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Applying Propo- sition F to the case of U = X, we partially recover [19, Proposition 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This is crucial in the constructions of counterexamples to the Baum-Connes type conjec- tures (see [39] for the coarse version and [20, Section 5] for the boundary version, which is based on the example considered in [44, Section 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence presumably Proposition F will find further applications in higher index theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Conversely, it is natural to ask whether I(U) = ˜I(U) implies that the restriction groupoid G(X)∂βX\\U is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that when U = X, [36, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3] implies that I(X) = ˜I(X) is equivalent to that X has Property A (see also Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5), which is further equivalent to that the coarse groupoid G(X) is amenable thanks to [39, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Inspired by these works, we introduce the following partial version of Property A: Definition G (Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a discrete metric space of bounded geometry and U ⊆ βX be an invariant open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We say that X has partial Property A towards ∂βX \\ U if G(X)∂βX\\U is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Finally we reach the following, which recovers [36, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3] when U = X: Theorem H (Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a strongly discrete metric space of bounded geometry and U ⊆ βX be a countably generated invariant open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then the following are equivalent: (1) X has partial Property A towards βX \\ U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) ˜I(U) = I(U);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (3) the ideal IG of all ghost operators is contained in I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The proof of Theorem H follows the outline of the case that U = X (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' [36, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3]), and is divided into several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Firstly, we unpack the groupoid language of Definition G and provide a concrete geometric description similar to the definition of Property A (see Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then we introduce a notion of partial operator norm localisation property (Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9), which is a partial version of the operator norm localisation property (ONL) introduced in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Parallel to Sako’s result that Property A is equivalent to ONL ([37]), we show that partial Property A is equivalent to partial ONL (see Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Finally thanks to the assumption of countable generatedness, we conclude Theorem H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We also remark that in the proof of Theorem H, we make use of the notion of ideals in spaces introduced by Chen and the first-named author in [12] (see also Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='12) instead of using invariant open subsets of βX directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This has the advantage of playing within the given space rather than going to the mysterious Stone- ˇCech boundary, which allows us to step over several technical gaps (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In Section 2, we recall necessary background knowledge in coarse geometry and groupoid theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In Section 3, we recall the theory of limit spaces and limit operators developed in [43], which will be an important tool used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Section 4 is devoted to the notion 6 QIN WANG AND JIAWEN ZHANG of geometric ideals (Definition A) studied in [12, 44], and we also discuss their minimality in the lattice of ideals IU from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We introduce the key notion of ghostly ideals (Definition B) in Section 5, prove Theorem C and provide several characterisations for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then we discuss maximal ideals in uniform Roe algebras in Section 6, and prove Proposition D and Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In Section 7, we study the problem when the geometric ideal coincides with the ghostly ideal, discuss their K-theories and prove Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then in Section 8, we introduce the notion of partial Property A (Definition G) and prove Theorem H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Finally, we list some open questions in Section 9, and provide Appendix A to record the notion of ultrafilters and their properties used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We would like to thank Baojie Jiang and J´an ˇSpakula for some helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Standard notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Here we collect the notation used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a set X, denote by |X| the cardinality of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a subset A ⊆ X, denote by χA the characteristic function of A, and set δx := χ{x} for x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' When X is a locally compact Hausdorff space, we denote by C(X) the set of complex-valued continuous functions on X, and by Cb(X) the subset of bounded continuous functions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall that the support of a function f ∈ C(X) is the closure of {x ∈ X : f(x) � 0}, written as supp(f), and denote by Cc(X) the set of continuous functions with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We also denote by C0(X) the set of continuous functions vanishing at infinity, which is the closure of Cc(X) with respect to the supremum norm ∥ f∥∞ := sup{| f(x)| : x ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' When X is discrete, denote ℓ∞(X) := Cb(X) and ℓ2(X) the Hilbert space of complex-valued square-summable functions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Denote by B(ℓ2(X)) the C∗- algebra of all bounded linear operators on ℓ2(X), and by K(ℓ2(X)) the C∗-subalgebra of all compact operators on ℓ2(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a discrete space X, denote by βX its Stone- ˇCech compactification and ∂βX := βX \\ X the Stone- ˇCech boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Notions from coarse geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Here we collect necessary notions from coarse geometry, and guide readers to [27, 34] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a discrete metric space (X, d), denote the closed ball by BX(x, r) := {y ∈ X : d(x, y) ≤ r} for x ∈ X and r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a subset A ⊆ X and r > 0, denote the r-neighbourhood of A in X by Nr(A) := {x ∈ X : dX(x, A) ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For R > 0, denote the R-entourage by ER := {(x, y) ∈ X × X : d(x, y) ≤ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We saythat(X, d)hasbounded geometry ifforanyr > 0, the numbersupx∈X |BX(x, r)| is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Also say that (X, d) is strongly discrete if the set {d(x, y) : x, y ∈ X} is a dis- crete subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We say that “X is a space” as shorthand for “X is a strongly discrete metric space of bounded geometry” (as in [43]) throughout the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We remark that although our results hold without the assumption of strong discreteness, we choose to add it so as to simplify the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As discussed in [43, Section 2], this will not lose any generality since one can always modify a discrete metric space (using a coarse equivalence) to satisfy this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 7 Now we recall the notion of Property A introduced by Yu (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', [45, Propo- sition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4] for the equivalence to Yu’s original definition): Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1 ([47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A space (X, d) is said to have Property A if for any ε, R > 0 there exist an S > 0 and a function f : X × X → [0, +∞) satisfying: (1) supp(f) ⊆ ES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) for any x ∈ X, we have � z∈X f(z, x) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (3) for any x, y ∈ X with d(x, y) ≤ R, then � z∈X | f(z, x) − f(z, y)| ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Using a standard normalisation argument, we have the following: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A space (X, d) has Property A if and only if for any ε, R > 0 there exist an S > 0 and a function f : X × X → [0, +∞) satisfying: (1) supp(f) ⊆ ES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) for any x ∈ X, we have | � z∈X f(z, x) − 1| ≤ ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (3) for any x, y ∈ X with d(x, y) ≤ R, then � z∈X | f(z, x) − f(z, y)| ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We also need a characterisation for Property A using kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall that a kernel on X is a function k: X × X → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We say that k is of positive type if for any n ∈ N, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' , xn ∈ X and λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' , λn ∈ R, we have: n � i,j=1 λiλjk(xi, xj) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The following is well-known (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', [45, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4]): Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A space (X, d) has Property A if and only if for any R > 0 and ε > 0, there exist S > 0 and a kernel k : X × X → R of positive type satisfying the following: (1) for x, y ∈ X, we have k(x, y) = k(y, x) and k(x, x) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) for x, y ∈ X with d(x, y) ≥ S, we have k(x, y) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (3) for x, y ∈ X with d(x, y) ≤ R, we have |1 − k(x, y)| ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We also recall the notion of coarse embedding: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, dX) and (Y, dY) be metric spaces and f : X → Y be a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We say that f is a coarse embedding if there exist functions ρ± : [0, ∞) → [0, ∞) with limt→+∞ ρ±(t) = +∞ such that for any x, y ∈ X we have ρ−(dX(x, y)) ≤ dY(f(x), f(y)) ≤ ρ+(dX(x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' If additionally there exists C > 0 such that Y = NC(f(X)), then we say that f is a coarse equivalence and (X, dX), (Y, dY) are coarsely equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Uniform Roe algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a discrete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Each operator T ∈ B(ℓ2(X)) can be written in the matrix form T = (T(x, y))x,y∈X, where T(x, y) = ⟨Tδy, δx⟩ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We also regard T ∈ B(ℓ2(X)) as a bounded function on X × X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', an element in ℓ∞(X × X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Denote by ∥T∥ the operator norm of T in B(ℓ2(X)), and ∥T∥∞ the supremum norm when regarding T as a function in ℓ∞(X × X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is clear that ∥T∥∞ ≤ ∥T∥ for any T ∈ B(ℓ2(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Given an operator T ∈ B(ℓ2(X)), we define the support of T to be supp(T) := {(x, y) ∈ X × X : T(x, y) � 0}, 8 QIN WANG AND JIAWEN ZHANG and the propagation of T to be prop(T) := sup{d(x, y) : (x, y) ∈ supp(T)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (1) The set of all finite propagation operators in B(ℓ2(X)) forms a ∗-algebra, called the algebraic uniform Roe algebra of X and denoted by Cu[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For each R ≥ 0, denote the subset CR u[X] := {T ∈ B(ℓ2(X)) : prop(T) ≤ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is clear that Cu[X] = � R≥0 CR u[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) The uniform Roe algebra of X is defined to be the operator norm closure of Cu[X] in B(ℓ2(X)), which forms a C∗-algebra and is denoted by C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The following notion was originally introduced by Yu: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' An operator T ∈ C∗ u(X) is called a ghost if T ∈ C0(X × X) when regarding T as a function in ℓ∞(X × X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In other words, for any ε there exists a finite subset F ⊆ X such that for any (x, y) � F × F, we have |T(x, y)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is easy to see that all the ghost operators in C∗ u(X) form an ideal in C∗ u(X), denoted by IG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Intuitively speaking, a ghost operator is locally invisible at infinity in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This will be made more precise in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Groupoids and C∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We collect here some basic notions and termi- nology on groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Details can be found in [28], or [38] in the ´etale case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall that a groupoid is a small category, in which every morphism is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More precisely, a groupoid consists of a set G, a subset G(0) called the unit space, two maps s, r : G → G(0) called the source and range maps respectively, a composition law: G(2) := {(γ1, γ2) ∈ G × G : s(γ1) = r(γ2)} ∋ (γ1, γ2) �→ γ1γ2 ∈ G, and an inverse map γ �→ γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' These operationssatisfya couple ofaxioms, including associativity law and the fact that elements in G(0) act as units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For x ∈ G(0), denote Gx := r−1(x) and Gx := s−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For Y ⊆ G(0), denote GY Y := r−1(Y) ∩ s−1(Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that GY Y is a subgroupoid of G (in the sense that it is stable under multiplication and inverse), called the reduction of G by Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A subset Y is said to be invariant if r−1(Y) = s−1(Y), and we write GY instead of GY Y in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A locally compact Hausdorff groupoid is a groupoid equipped with a locally compact and Hausdorff topology such that the structure maps (composition and inverse) are continuous with respect to the induced topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Such a groupoid is called ´etale (also called r-discrete) if the range (hence the source) map is a local homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Clearly in this case, each fibre Gx (and Gx) is discrete with the induced topology, and G(0) is clopen in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The notion of ´etaleness for a groupoid can be regarded as an analogue of discreteness in the group case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let X be a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The pair groupoid of X is X×X as a set, whose unit space is {(x, x) ∈ X × X : x ∈ X} and identified with X for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The source map is the projection onto the second coordinate and the range map is the projection onto the first coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The composition is given by (x, y)· (y, z) = (x, z) for x, y, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' When X is a discrete Hausdorff space, then X × X is a locally compact Hausdorff ´etale groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 9 Now we introduce the algebras associated to groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Here we only focus on the case of ´etaleness, and guide readers to [28] for the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let G be a locally compact, Hausdorff and ´etale groupoid with unit space G(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that the space Cc(G) is a ∗-involutive algebra with respect to the following operations: for f, g ∈ Cc(G), (f ∗ g)(γ) = � α∈Gs(γ) f(γα−1)g(α), f ∗(γ) = f(γ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consider the following algebraic norm on Cc(G) defined by: ∥ f∥I := max \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3sup x∈G(0) � γ∈Gx | f(γ)|, sup x∈G(0) � γ∈Gx | f ∗(γ)| \uf8fc\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The completion of Cc(G) with respect to the norm ∥ · ∥I is denoted by L1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The maximal (full) groupoid C∗-algebra C∗ max(G) is defined to be the completion of Cc(G) with respect to the norm: ∥ f∥max := sup ∥π(f)∥, where the supremum is taken over all ∗-representations π of L1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In order to define the reduced counterpart, we recall that for each x ∈ G(0) the regular representation at x, denoted by λx : Cc(G) → B(ℓ2(Gx)), is defined as follows: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1) � λx(f)ξ � (γ) := � α∈Gx f(γα−1)ξ(α), where f ∈ Cc(G) and ξ ∈ ℓ2(Gx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is routine work to check that λx is a well-defined ∗-homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The reduced norm on Cc(G) is ∥ f∥r := sup x∈G(0) ∥λx(f)∥, and the reduced groupoid C∗-algebra C∗ r(G) is defined to be the completion of the ∗-algebra Cc(G) with respect to this norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Clearly, each regular representation λx can be extended to a homomorphism λx : C∗ r(G) → B(ℓ2(Gx)) automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is also routine to check that there is a canonical surjective homomorphism from C∗ max(G) to C∗ r(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Coarse groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The coarse groupoid G(X) on X was introduced by Skandalis, Tu and Yu in [39] (see also [34, Chapter 10]) to relate coarse geometry to the theory of groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As a topological space, G(X) := � r>0 Er β(X×X) ⊆ β(X × X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7 that X × X is the pair groupoid with source and range maps s(x, y) = y and r(x, y) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' These maps extend to maps G(X) → βX, still denoted by r and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now consider (r, s) : G(X) → βX × βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It was shown in [39, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7] that the map (r, s) is injective, and hence G(X) can be endowed with a groupoid structure 10 QIN WANG AND JIAWEN ZHANG induced by the pair groupoid βX × βX, called the coarse groupoid of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore, G(X) can also be equivalently defined by G(X) := � r>0 Er βX×βX ⊆ βX × βX, with the weak topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It was also shown in [39, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2] that the coarse groupoid G(X) is locally compact, Hausdorff, ´etale and principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Clearly, the unit space of G(X) can be identified with βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Given f ∈ Cc(G(X)), then f is a continuous function supported on Er for some r > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' equivalently, we can interpret f as a bounded function on Er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we define an operator θ(f) on ℓ2(X) by setting its matrix coefficients to be θ(f)(x, y) := f(x, y) for x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We have the following: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 ([34, Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The map θ provides a ∗-isomorphism from Cc(G(X)) to Cu[X], and extends to a C∗-isomorphism Θ : C∗ r(G(X)) → C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that Θ maps the C∗-subalgebra C∗ r(X × X) onto the compact operators K(ℓ2(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3 that an operator T ∈ B(ℓ2(X)) can be regarded as an element in ℓ∞(X × X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Following the notation from [12], we denote by T the continuous extension of T on β(X × X) when regarding T ∈ ℓ∞(X × X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then supp(T) = supp(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that G(X) is open in β(X × X), hence C0(G(X)) is a subalgebra in C(β(X × X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Restricting to G(X), we also regard T as a function on G(X) and hence we can talk of the value T(α, γ) for (α, γ) ∈ G(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, we have the following: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For T ∈ Cu[X], we have T ∈ Cc(G(X)) and θ(T) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For T ∈ C∗ u(X), we have T ∈ C0(G(X)) and Θ(T) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The first statement is a direct corollary of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Since ∥T∥∞ ≤ ∥T∥ for any T ∈ B(ℓ2(X)), the second follows from the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Amenability and a-T-menability for groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Amenable groupoids com- prise a large class of groupoids with relatively nice properties, which are literally the analogue of amenable groups in the world of groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Here we only focus on the case of ´etaleness, in which the notion of amenability behaves quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A standard reference is [1] and another reference for just ´etale groupoids is [9, Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10 ([1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A locally compact, Hausdorff and ´etale groupoid G is said to be (topologically) amenable if for any ε > 0 and compact K ⊆ G, there exists f ∈ Cc(G) with range in [0, 1] such that for any γ ∈ K we have � α∈Gr(γ) f(α) = 1 and � α∈Gr(γ) | f(α) − f(αγ)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Similar to the case of Property A, we have the following by a standard normal- isation argument: GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 11 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A locally compact, Hausdorff and ´etale groupoid G is amenable if and only if for any ε > 0 and compact K ⊆ G, there exists f ∈ Cc(G) with range in [0, +∞) such that for any γ ∈ K we have ��� � α∈Gr(γ) f(α) − 1 ��� < ε and � α∈Gr(γ) | f(α) − f(αγ)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Amenability for ´etale groupoids enjoy similar permanence properties as in the case of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For example, open or closed subgroupoids of amenable ´etale groupoids are amenable, and amenability is preserved under taking groupoid extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' See [1, Section 5] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Also recall that we have the following: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='12 ([9, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let G be a locally compact, Hausdorff, ´etale and amenable groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then the natural quotient C∗ max(G) → C∗ r(G) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we recall the notion of a-T-menability for groupoids introduced by Tu [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let G be a locally compact, Hausdorff and ´etale groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A continuous function f : G → R is said to be of negative type if (1) f|G(0) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) for any γ ∈ G, f(γ) = f(γ−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (3) Given γ1, · · · , γn ∈ G with the same range and λ1, · · · , λn ∈ R with �n i=1 λi = 0, we have � i,j λiλj f(γ−1 i γj) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A continuous function f : G → R is called locally proper if for any compact subset K ⊆ G(0), the restriction of f on GK K is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='13 ([40, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A locallycompact, Hausdorff and ´etale groupoid G is said to be a-T-menable if there exists a continuous locally proper function f : G → R of negative type on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Analogous to the case of groups, Tu [40] proved that a locally compact, σ- compact, Hausdorff and ´etale groupoid G is a-T-menable if and only if there exists a continuous field of Hilbert spaces over G(0) with a proper affine action of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We also need the following significant result by Tu: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14 ([40, Th´eor`eme 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let G be a locally compact, σ-compact, Haus- dorff, ´etale and a-T-menable groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then G is K-amenable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', the quotient map induces an isomorphism K∗(C∗ max(G)) → K∗(C∗ r(G)) for ∗ = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Finally, we record the following result for coarse groupoids: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='15 ([39, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space and G(X) be the associated coarse groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then: (1) X has Property A if and only if G(X) is amenable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) X can be coarsely embedded into Hilbert space if and only if G(X) is a-T-menable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Limit spaces and limit operators In this section, we recall the theory of limit spaces and limit operators for metric spaces developed by ˇSpakula and Willett in [43], which becomes an important tool for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 12 QIN WANG AND JIAWEN ZHANG Throughout the section, we always assume that (X, d) is a space (see “Conven- tion” in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2) and G(X) is its coarse groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We will freely use the notion of ultrafilters on X, and related materials are recalled in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Limit spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' First recall that a function t : D → R with D, R ⊆ X is called a partial translation if t is a bijection from D to R, and supx∈X d(x, t(x)) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The graph of t is {(t(x), x) : x ∈ D}, denoted by gr(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is well-known that each entourage E on X can be decomposed into finitely many graphs of partial translations (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', [34, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10]) thanks to the bounded geometry of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1 ([43, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Fix an ultrafilter ω ∈ βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A partial translation t : D → R on X is compatible with ω if ω(D) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In this case, regarding t as a function from D to βX, we define the following thanks to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2: t(ω) := lim ω t ∈ βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In other words, consider the extension t : D → R then t(ω) = t(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' An ultrafilter α ∈ βX is compatible with ω if there exists a partial translation t compatible with ω and t(ω) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Denote by X(ω) the collection of all ultrafilters on X compatible with ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A compatible family for ω is a collection of partial translations {tα}α∈X(ω) such that each tα is compatible with ω and tα(ω) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Fix an ultrafilter ω on X, and a compatible family {tα}α∈X(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Define a function dω : X(ω) × X(ω) → [0, ∞) by dω(α, β) := lim x→ω d(tα(x), tβ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is shown in [43, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7] that dω is a uniformly discrete metric of bounded geometry on X(ω) which does not depend on the choice of {tα}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This leads to the following: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 ([43, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For each non-principal ultrafilter ω on X, the metric space (X(ω), dω) is called the limit space of X at ω, which is a space in the sense of “Convention” in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is shown in [43, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9] that for any α ∈ X(ω), we have X(α) = X(ω) as metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Also note that when ω is principal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', ω ∈ X, then it is clear that (X(ω), dω) = (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We recall the following result, which reveals that the local geometry of X can be recaptured by those of the limit spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3 ([43, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let ω be a non-principal ultrafilter on X, and {tα : Dα → Rα} a compatible family for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then for each finite F ⊆ X(ω), there exists a subset Y ⊆ X with ω(Y) = 1 such that for each y ∈ Y, there is a finite subset G(y) ⊆ X such that the map fy : F → G(y), α �→ tα(y) is a surjective isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Such a collection {fy}y∈Y is called a local coordinate systerm for F, and the maps fy are called local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Furthermore, if F is a metric ball B(ω, r), then there exist Y ⊆ X with ω(Y) = 1 and a local coordinate system {fy : F → G(y)}y∈Y such that each G(y) is the ball B(y, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As shown in [43, Appendix C], limit spaces can be described in terms of the coarse groupoid G(X): GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 13 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4 ([43, Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Given a non-principal ultrafilter ω ∈ βX, the map F : X(ω) → G(X)ω, α �→ (α, ω) is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence X(ω) is the smallest invariant subset of βX containing ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Here we consider G(X) as a subset of βX × βX, as explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consequently, we obtain the following: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As a set, we have G(X) = � X × X � ⊔ � ω∈∂βX � X(ω) × X(ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we would like to provide a quantitative version for Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' First we record the following observation, whose proof is straightforward and almost identical to that of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4 (originally from [34, discussion in 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='18-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='24]): Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let t : D → R be a partial translation on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then we have: gr(t) βX×βX = gr(t) ⊔ � ω∈∂βX � (α, ω) : ω(D) = 1 and α = t(ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In general, we have the following: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For any S ≥ 0, we have: ES βX×βX = ES ⊔ � ω∈∂βX � (α, γ) ∈ X(ω) × X(ω) : dω(α, γ) ≤ S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As explained at the beginning of this subsection, we can decompose ES = gr(t1) ⊔ · · · ⊔ gr(tN) where each ti : Di → Ri is a partial translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we have ES = gr(t1) ∪ · · · ∪ gr(tN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6, we obtain that ES is contained in the right hand side in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, given ω ∈ ∂βX and α, γ ∈ X(ω) with dω(α, γ) ≤ S, then we have (X(ω), dω) = (X(γ), dγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Take a partial translation t : D → R such that γ(D) = 1 and α = t(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that dγ(α, γ) = lim x→γ d(t(x), x) ≤ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence for D′ := {x ∈ D : d(t(x), x) ≤ S}, we have γ(D′) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consider the restriction of t on D′, denoted by t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then t′ is also a partial translation and α = t′(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6, we obtain that (α, γ) ∈ gr(t′), which is contained in ES as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Now we compute concrete examples of limit spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' First we recall the case of groups from [43, Appendix B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let Γ be a countable discrete group, equipped with a left-invariant bounded geometry and strongly discrete metric d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For each g ∈ Γ, denote ρg : Γ → Γ, h �→ hg the right translation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Each ρg is a partial translation with full domain, and hence is compatible with every ω ∈ βΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, we have the following: 14 QIN WANG AND JIAWEN ZHANG Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 ([43, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For each non-principal ultrafilter ω ∈ βΓ, the map bω : Γ −→ Γ(ω), g �→ ρg(ω) is an isometric bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Inspired by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8, we provide the following general method: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let {tλ : Dλ → Rλ}λ∈Λ be a family of partial translations on X satisfying the following: for each S > 0, there exists a finite subset ΛS ⊆ Λ such that gr(tλ) ∩ gr(tµ) is finite for λ � µ in ΛS and ES \\ � � λ∈ΛS gr(tλ) � is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then for any non-principal ultrafilter ω on X and α ∈ X(ω), there exists λ ∈ Λ such that ω(Dλ) = 1 and α = tλ(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By definition, we assume that α = t(ω) for some partial translation t : D → R with ω(D) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For λ ∈ Λ, set �Dλ := {x ∈ D ∩ Dλ : t(x) = tλ(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Choose S > 0 such that gr(t) ⊆ ES, and hence there exists a finite subset F ⊆ ES such that gr(t) ⊆ � � λ∈ΛS gr(tλ) � ⊔ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This implies that D ⊆ � � λ∈ΛS �Dλ � ⊔ F′ for some finite F′ ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Since ω is non- principal, ΛS is finite and �Dλ ∩ �Dµ is finite for any λ, µ ∈ ΛS, there exists a unique λ ∈ ΛS such that ω(�Dλ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This implies that ω(Dλ) = 1 and α = tλ(ω), which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Back to the case of the group Γ, the set {ρg : Γ → Γ}g∈Γ satisfies the condition in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9, and hence the map bω in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is straightforward to check that bω is isometric, which recovers the proof for Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consider X = N with the usual metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For each k ∈ Z with k ≥ 0, define a partial translation ρk : N −→ N, n �→ n + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For k ∈ Z with k < 0, define a partial translation ρk : [−k, ∞) ∩ N −→ N, n �→ n + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then it is clear that the set {ρk}k∈Z satisfies the condition in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now for a non-principal ultrafilter ω on N, consider the map bω : Z −→ N(ω), k �→ ρk(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that for k, l ∈ Z, we have dω(ρk(ω), ρl(ω)) = lim n→ω |(k + n) − (l + n)| = |k − l|, which implies that bω is isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9 shows that bω is surjec- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore, every limit space of N is isometric to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This provides a detailed proof for [43, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14(2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Similar to the analysis in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10, we can also apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9 to obtain proofs for [43, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14(3)-(5)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Details are left to readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Limit operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we recall the notion of limit operators for metric spaces introduced by ˇSpakula and Willett: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='11 ([43, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a non-principal ultrafilter ω on X, fix a compatible family {tα}α∈X(ω) for ω and let T ∈ C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The limit operator of T at ω, denoted by Φω(T), is an X(ω)-by-X(ω) indexed matrix defined by Φω(T)αγ := lim x→ω Ttα(x)tγ(x) for α, γ ∈ X(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It was studied in [43, Chapter 4] that the above definition does not depend on the choice of the compatible family {tα}α∈X(ω) for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Furthermore, the limit operator Φω(T) is indeed a bounded operator on ℓ2(X(ω)), and belongs to the uniform Roe algebra C∗ u(X(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9 that for an operator T ∈ C∗ u(X), the continuous extension T ∈ C0(G(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We have the following, which was implicitly mentioned in the proof of [43, Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a non-principal ultrafilter ω on X and T ∈ C∗ u(X), we have Φω(T)αγ = T(α, γ) for α, γ ∈ X(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Choose partial translations tα, tγ compatible with α, γ such that tα(ω) = α, tγ(ω) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By definition, we have Φω(T)αγ = lim x→ω Ttα(x)tγ(x) = lim x→ω T(tα(x), tγ(x)) = T(α, γ) where the last equality comes from the discussion before Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Since the limit operator Φω(T) contains the information of the asymptotic be- haviour of T “in the ω-direction”, we introduce the following: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For an ω ∈ ∂βX, we say that an operator T ∈ C∗ u(X) is locally invisible (or vanishes) in the ω-direction if Φω(T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a subset V ⊆ ∂βX, we say that T is locally invisible (or vanishes) in the V-direction if Φω(T) = 0 for any ω ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Finally, we recall from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 that there is a C∗-isomorphism Θ : C∗ r(G(X)) → C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This allows us to relate limit operators to left regular rep- resentations of C∗ r(G(X)): Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14 ([43, Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a non-principal ultrafilter ω on X, let Wω : ℓ2(G(X)ω) → ℓ2(X(ω)) be the unitary representation induced by F in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then we have the following commutative diagram: C∗ r(G(X)) λω � Θ � � B(ℓ2(G(X)ω)) AdWω � � C∗ u(X) Φω � B(ℓ2(X(ω))), where λω is the left regular representation from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 16 QIN WANG AND JIAWEN ZHANG 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Geometric ideals In this section, we recall the notion of geometric ideals, which was originally introduced by the first-named author in [44] (see also [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Throughout the section, let X be a space in the sense of “Convention” in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1 ([12, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For an operator T ∈ C∗ u(X) and ε > 0, the ε-support of T is defined to be suppε(T) := {(x, y) ∈ X × X : |T(x, y)| ≥ ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Also define the ε-truncation of T to be Tε(x, y) := � T(x, y), if |T(x, y)| ≥ ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is clear that supp(Tε) = suppε(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We also record the following elementary result for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The proof is straightforward, hence omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Given T ∈ C∗ u(X) and ε > 0, we have supp(Tε) ⊆ { ˜ω ∈ β(X × X) : |T( ˜ω)| ≥ ε} ⊆ supp(Tε/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The following is a key result in [12]: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3 ([12, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let I be an ideal in the uniform Roe algebra C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For each T ∈ I and ε > 0, we have Tε ∈ I ∩ Cu[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, we have I ∩ Cu[X] = {Tε : T ∈ I, ε > 0} where the closures are taken with respect to the operator norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we recall the notion of geometric ideals from [44]: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' An ideal I in the uniform Roe algebra C∗ u(X) is called geometric if I ∩ Cu[X] is dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In [12], Chen and the first-named author provide a full description for geometric ideals in C∗ u(X) in terms of invariant open subsets of G(X)(0) = βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' To outline their work, let us start with the following elementary observation: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let U be a non-empty invariant open subset of βX, then X ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Since X is dense in βX and U is open and non-empty, we obtain that U ∩ X � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Take an x ∈ U ∩ X, then the pair (x, y) ∈ G(X) for any y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Thanks to the invariance of U, we obtain that y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This implies that X ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Given an invariant open subset U ⊆ βX, denote G(X)U := G(X) ∩ s−1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Fol- lowing [12], we define Ic(U) : = {f ∈ Cc(G(X)) : f( ˜ω) = 0 for any ˜ω � G(X)U} = {T ∈ Cu[X] : T( ˜ω) = 0 for any ˜ω � G(X)U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Obviously, Ic(U) is a two-sided ideal in Cc(G(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Denote its closure in C∗ r(G(X)) by I(U), which is a geometric ideal in C∗ r(G(X)) � C∗ u(X) from the definition (see also [12, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This leads to the following: GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 17 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For an invariant open subset U ⊆ βX, the ideal I(U) is called the geometric ideal associated to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For later use, we record the following alternative description for the geometric ideal I(U): Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let U be an invariant open subset of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then the ideal I(U) is isomorphic to the reduced groupoid C∗-algebra C∗ r(G(X)U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This was implicitly contained in the proof of [12, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For convenience to the readers, we include a proof here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By definition, C∗ r(G(X)U) is isomorphic to the norm closure of Cc(G(X)U) in C∗ r(G(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that Cc(G(X)U) = {T ∈ Cu[X] : supp(T) ⊆ G(X)U} ⊆ Ic(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, for T ∈ Ic(U) we have T = limε→0 Tε since T has finite propaga- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that supp(Tε) ⊆ G(X)U, which implies Tε ∈ Cc(G(X)U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we obtain that Cc(G(X)U) and Ic(U) have the same closure in C∗ r(G(X)), which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For U = X, it follows directly from definition that G(X)X = X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence combining Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7, we obtain that the geometric ideal associated to X is I(X) = K(ℓ2(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, for U = βX it is clear that I(βX) = C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Conversely, following [12, Section 4 and 5] we can associate an invariant open subset of βX to any ideal in the uniform Roe algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More precisely, let I be an ideal in the uniform Roe algebra C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Define: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1) U(I) := � T∈I,ε>0 r(suppε(T)) = � T∈I∩Cu[X],ε>0 r(suppε(T)), where the second equality follows directly from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Also [12, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2] implies that U(I) is an invariant open subset of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Furthermore, as a special case of [12, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3], we have the following: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a space (X, d), the map I �→ U(I) provides an isomorphism between the lattice of all geometric ideals in C∗ u(X) and the lattice of all invariant open subsets of βX, with the inverse map given by U �→ I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9 shows that geometric ideals in C∗ u(X) can be fully determined by invariant open subsets of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In contrast, general ideals in C∗ u(X) cannot be characterised merely by the associated subsets of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For example, direct calcu- lations show that the associated invariant open subsets for the ideal IG defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3 and for the ideal of compact operators in B(ℓ2(X)) are the same, both of which equal X (see also Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence as pointed out in Section 1, the study of the ideal structure for the uniform Roe algebra can be reduced to analyse the lattice (where the order is given by inclusion) IU = {I is an ideal in C∗ u(X) : U(I) = U} in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1) for each invariant open subset U ⊆ βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The following result busts the smallest element in IU: 18 QIN WANG AND JIAWEN ZHANG Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let U be an invariant open subset of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then the geometric ideal I(U) is the smallest element in the lattice IU in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10 follows directly from the following lemma: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space and I an ideal in C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then we have I(U(I)) = I ∩ Cu[X], where the closure is taken in C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we have I(U(I)) ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Denoting ˚I := I ∩ Cu[X], it is clear that ˚I ∩ Cu[X] = I ∩ Cu[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This implies that ˚I is a geometric ideal in C∗ u(X), and hence ˚I = I(U(˚I)) by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By definition, we have U(˚I) = � T∈˚I∩Cu[X],ε>0 r(suppε(T)) = � T∈I∩Cu[X],ε>0 r(suppε(T)) = U(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore, we obtain that I(U(I)) = I(U(˚I)) = ˚I = I ∩ Cu[X] as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ We would like to recall another description for geometric ideals based on the notion of ideals in spaces introduced in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It has the advantage of playing within the given metric space, rather than going to the mysterious Stone- ˇCech boundary, and hence will help us to step over several technical gaps in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='12 ([12, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' An ideal in a space (X, d) is a collection L of subsets of X satisfying the following: (1) if Y ∈ L and Z ⊆ Y, then Z ∈ L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) if R ≥ 0 and Y ∈ L, then NR(Y) ∈ L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (3) if Y, Z ∈ L, then Y ∪ Z ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For an ideal L in X, we define U(L) := � Y∈L Y βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Conversely, given an invariant open subset U of βX, we define L(U) := {Y ⊆ X : Y βX ⊆ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As a special case of [12, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3], we have the following: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a space (X, d), the map L �→ U(L) provides an isomorphism between the lattice of all ideals in X and the lattice of all invariant open subsets of βX, with the inverse map given by U �→ L(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Combining Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='13, we obtain an isomorphism between the lattice of all ideals in X and the lattice of all geometric ideals in C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Direct calculation shows (see also [12, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4]): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2) I(U(L)) = {T ∈ Cu[X] : supp(T) ⊆ Y × Y for some Y ∈ L}, where the closure is taken in C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we consider a special class of geometric ideals coming from subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Given a subspace A ⊆ X, recall from [23, Section 5] that there is an associated ideal IA in C∗ u(X) whose K-theory is isomorphic to that of the uniform Roe algebra GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 19 C∗(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More precisely, recall that an operator T ∈ B(ℓ2(X)) is near A if there exists R > 0 such that supp(T) ⊆ NR(A) × NR(A), and the ideal IA is defined to be the operator norm closure of all operators in Cu[X] near A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The ideal IA is called spatial in [13] since it is related to a subspace in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' However, as shown in [13, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1], there exist non-spatial ideals in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, spatial ideals play an important role in the computation of the K- theory of Roe algebras via the Mayer-Vietoris sequence argument (see [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' To show that spatial ideals are geometric, we observe that the smallest ideal in X containing A is LA := {Z ⊆ NR(A) : R > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence applying Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='13, we immediately obtain the following: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let A be a subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then the set (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3) UA := U(LA) = � R>0 NR(A) is an invariant open subset of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, if U is an invariant open subset of βX containing A, then UA ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consequently, combining with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2) we reach the following: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let A be a subset of X, then I(UA) = IA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence the spatial ideal IA is geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For later use, we record the following result concerning the set UA defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 that for a subset Z ⊆ X, the closure Z in βX is homeomorphic to βZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let A be a subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then we have: G(X)UA = � R>0 G(NR(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By definition, we have G(X)UA = � R>0 G(X)NR(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that for each R > 0 and (α, ω) ∈ G(X)NR(A), we have ω ∈ NR(A) and there exists S > 0 such that (α, ω) ∈ ES due to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence the pair (α, ω) in NR+S(A) × NR+S(A) belongs to G(NR+S(A)), which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ To end this section, we remark that for a given ideal I in C∗ u(X), it is usually hard to compute the associated U(I) directly from definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' However, this is always achievable for principal ideals: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let I = ⟨T⟩ be the principal ideal in C∗ u(X) generated by T ∈ C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Denote U := � ε>0,R>0 NR(r(suppε(T))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then U is an invariant open subset of βX, and we have U(I) = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14, it is clear that U is an invariant open subset of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1), U(I) contains r(suppε(T)) for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Since U(I) is invariant, we obtain that U(I) contains U again by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 20 QIN WANG AND JIAWEN ZHANG On the other hand, we consider S = �n i=1 aiTbi where ai, bi are non-zero with supports being partial translations contained in ER for some R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence for any ε > 0, we have r � suppε(S) � ⊆ n � i=1 r � supp ε n(aiTbi) � ⊆ n � i=1 NR � r � supp ε n∥ai∥·∥bi∥(T) �� , which implies that r � suppε(S) � ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that operators of the form �n i=1 aiTbi as above are dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence for a general element ˜S ∈ I and ε > 0, there exists S = �n i=1 aiTbi where ai, bi are non-zero with supports being partial translations such that ∥ ˜S − S∥ < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence suppε( ˜S) ⊆ suppε/2(S), which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Ghostly ideals In the previous section, we observe that for an invariant open subset U of βX, the associated geometric ideal I(U) is the smallest element in the lattice IU = {I is an ideal in C∗ u(X) : U(I) = U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In this section, we would like to explore the largest element in IU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As revealed in Section 1, a natural idea is to include all operators T ∈ C∗ u(X) sitting in some ideal I with U(I) = U, which leads to the following: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a space (X, d) and an invariant open subset U of βX, denote ˜I(U) := {T ∈ C∗ u(X) : r(suppε(T)) ⊆ U for any ε > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We call ˜I(U) the ghostly ideal associated to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The terminology will become clear later (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6 and Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' First let us verify that ˜I(U) is indeed an ideal in C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For an invariant open subset U of βX, ˜I(U) is an ideal in C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For T, S ∈ C∗ u(X) and ε > 0, we have suppε(T + S) ⊆ suppε/2(T) ∪ suppε/2(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It follows that ˜I(U) is a linear space in C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Given T ∈ ˜I(U) and ε > 0, note that r(suppε(T∗)) = s(suppε(T)) and hence T∗ ∈ ˜I(U) since U is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now given a sequence {Tn}n in ˜I(U) converging to T ∈ C∗ u(X) and an ε > 0, there exists n ∈ N such that suppε(T) ⊆ suppε/2(Tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we obtain that T ∈ ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Finally, given T ∈ ˜I(U) and S ∈ C∗ u(X) we need show that TS and ST belong to ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that ST = (T∗S∗)∗ and ˜I(U) is closed under taking the ∗-operation, hence it suffices to show that TS ∈ ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Since ˜I(U) is closed in C∗ u(X) with respect to the operator norm, it suffices to consider the case that S ∈ Cu[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We can further assume that S is a partial translation since ˜I(U) is closed under taking addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In this case, we have r(suppε(TS)) ⊆ r(suppε/∥S∥(T)) for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Using the language of ideals in X, we record that for an ideal L in X and T ∈ C∗ u(X), then T ∈ ˜I(U(L)) if and only if for any ε > 0 there exist R > 0 and Y ∈ L such that for any (x, y) � ER ∩ (Y × Y) then |T(x, y)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 21 The following result shows that the ghostly ideal ˜I(U) is indeed the largest element in the lattice IU: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Given an invariant open subset U of βX, we have the following: (1) for an ideal I in C∗ u(X) with U(I) = U, then I ⊆ ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) U(˜I(U)) = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Given T ∈ I, the condition U(I) = U implies that r(suppε(T)) ⊆ U for each ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence by definition, we have T ∈ ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that U(˜I(U)) = � T∈˜I(U),ε>0 r(suppε(T)) ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9 shows that U(I(U)) = U, which implies that ˜I(U) ⊇ I(U) thanks to (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we have U(˜I(U)) ⊇ U(I(U)) = U again by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Combining with Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10, we reach the following desired result: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2, and U be an invariant open subset of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then any ideal I in C∗ u(X) with U(I) = U sits between I(U) and ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More precisely, the geometric ideal I(U) is the smallest element while the ghostly ideal ˜I(U) is the largest element in the lattice IU in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As mentioned in Section 1, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4 draws the border of the lattice IU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore, once we can bust every ideal between I(U) and ˜I(U) for each invariant open subset U ⊆ βX, then we will obtain a full description for the ideal structure of the uniform Roe algebra C∗ u(X) (see Question 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we aim to provide a geometric description for ghostly ideals, which helps to explain the terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let us start with an easy example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Taking U = X, then ˜I(X) is the ideal IG defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Indeed, T ∈ ˜I(X) if and only if for any ε > 0, r(suppε(T)) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This is equivalent to that T ∈ C0(X × X) since T ∈ C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, it is clear that ˜I(βX) = C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More generally, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that for any invariant open subset U of βX, G(X)U is an open subset of β(X × X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence both C0(G(X)U) and Ic(U) can be regarded as subalgebras in C(β(X × X)) � ℓ∞(X × X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For an invariant open subset U ⊆ βX and T ∈ C∗ u(X), the following are equivalent: (1) T ∈ ˜I(U);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) T ∈ C0(G(X)U);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (3) T ∈ Ic(U) ∥·∥∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (4) T vanishes in the (βX \\ U)-direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', Φω(T) = 0 for any ω ∈ βX \\ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' “(1) ⇒ (2)”: By definition, for any ε > 0 we have r(supp(Tε)) = r(supp(Tε)) = r(suppε(T)) ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 22 QIN WANG AND JIAWEN ZHANG Consider the compact set K := { ˜ω ∈ β(X × X) : |T( ˜ω)| ≥ 2ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2, we have r(K) ⊆ r(supp(Tε)) ⊆ U, which implies that K ⊆ G(X)U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, we have |T( ˜ω)| < 2ε for any ˜ω ∈ β(X × X) \\ K, which concludes that T ∈ C0(G(X)U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' “(2) ⇒ (1)”: Assume that T ∈ C0(G(X)U) ⊆ C(β(X × X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then for any ε > 0 there exists a compact subset K ⊆ G(X)U such that for any ˜ω ∈ β(X × X) \\ K, we have |T( ˜ω)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This implies that { ˜ω ∈ β(X × X) : |T( ˜ω)| ≥ ε} ⊆ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2, we obtain that supp(Tε) ⊆ K, which implies that r(suppε(T)) ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence T ∈ ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' “(2) ⇔ (3)”: This is due to the fact that Ic(U) ∥·∥∞ = {f ∈ C0(G(X)) : f( ˜ω) = 0 for ˜ω � G(X)U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' “(3) ⇔ (4)”: By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='12, (4) holds if and only if T(α, γ) = 0 for any α, γ ∈ X(ω) and ω ∈ βX \\ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4, this holds if and only if T( ˜ω) = 0 whenever ˜ω � G(X)U, which describes elements in Ic(U) ∥·∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Note that G(X)X = X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence as a direct corollary, we recover the following characterisation for ghost operators: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7 ([43, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' An operator T ∈ C∗ u(X) is a ghost if and only if Φω(T) = 0 for any non-principal ultrafilter ω on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In other words, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7 shows that a ghost in C∗ u(X) is locally invisible in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Thissuggestsustoconsideroperatorsin ˜I(U)as“partial” ghosts, which clarifies the terminology of “ghostly ideals”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As an application of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6, we now provide another description for ghostly ideals in terms of operator algebras, which will be used later in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let us start with the short exact sequences studied in [21] (see also [20, Section 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Given an invariant open subset U ⊆ βX, notice that Uc = βX\\U is also invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Denote by G(X)Uc := G(X) ∩ s−1(Uc) and clearly, we have a decomposition: G(X) = G(X)U ⊔ G(X)Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that G(X)U is open in G(X), hence the above induces the following short exact sequence of ∗-algebras: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1) 0 −→ Cc(G(X)U) −→ Cc(G(X)) −→ Cc(G(X)Uc) −→ 0 where the map Cc(G(X)U) −→ Cc(G(X)) is the inclusion and the map Cc(G(X)) −→ Cc(G(X)Uc) is the restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We may complete the sequence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1) with respect to the maximal groupoid C∗-norms and obtain the following sequence: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2) 0 −→ C∗ max(G(X)U) −→ C∗ max(G(X)) −→ C∗ max(G(X)Uc) −→ 0, which is easy to check by definition to be automatically exact (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', [26, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We may also complete this sequence with respect to the reduced groupoid C∗-norms and obtain the following sequence: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3) 0 −→ C∗ r(G(X)U) iU −→ C∗ r(G(X)) qU −→ C∗ r(G(X)Uc) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By construction, iU is injective, qU is surjective and qU ◦ iU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Also recall from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7 that iU(C∗ r(G(X)U)) = I(U), the geometric ideal associated to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 23 However in general, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3) fails to be exact at the middle item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This is crucial in [21] to provide a counterexample to the Baum-Connes conjecture with coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More precisely when U = X, it is proved in [35, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='11] for the group case and [20, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4] for the Roe algebraic case (see also [43, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2]) that Ker(qX) = IG, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', the ideal consisting of all ghost operators in C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 and Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5, the sequence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3) is exact for U = X if and only if I(X) = ˜I(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More generally, we have the following: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Given an invariant open subset U ⊆ βX, the kernel of qU : C∗ r(G(X)) → C∗ r(G(X)Uc) coincides with the ghostly ideal ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence the sequence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3) is exact if and only if I(U) = ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is easy to see that an operator T ∈ C∗ u(X) � C∗ r(G(X)) belongs to the kernel of qU if and only if λω(T) = 0 for any ω ∈ βX \\ U, where λω is the left regular representation from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14 implies that Φω(T) = 0 for any ω ∈ βX \\ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Finally, we conclude the proof thanks to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ We end this section with an illuminating example from [44, Section 3] (see also [20, Section 5]), which is important to construct counterexamples to Baum-Connes type conjectures: Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let {Xi}i∈N be a sequence of expander graphs or pertubed expander graphs (see [44] for the precise definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let Yi,j = Xi for all j ∈ N and set Y := � i,j Yi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We endow Y with a metric d such that it is the graph metric on each Yi,j and satisfies d(Yi,j, Yk,l) → ∞ as i + j + k + l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let Pi,j ∈ B(ℓ2(Yi,j)) be the orthogonal projection onto constant functions on Yi,j, and we set P to be the direct sum of Pi,j in the strong operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By the assumption on the expansion of {Xi}i∈N, it is clear that P ∈ C∗ u(Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is explained in [44, Section 3] (see also [20, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1]) that P is not a ghost, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', P � ˜I(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' However intuitively, P should vanish “in the i-direction”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We will make it more precisely in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from [20, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1] that we have a surjective map βY → βX × βN induced by the bijection of Y with X×N and the universal property of βY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Define: f : βY −→ βX × βN −→ βX where the second map is just the projection onto the first coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Denote U = f −1(X), which is open in βY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that U = � i f −1(Xi), where each f −1(Xi) is homeomorphic to Xi × βN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, note that U = � i f −1(Xi) = � i � j Yi,j = � ε>0 r(suppε(P)) = � ε>0,R>0 NR(r(suppε(P))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence it follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='17 that U is invariant (comparing with [20, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2]), and U(⟨P⟩) = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (Note that P ∈ ˜I(U) was already implicitly proved in [20, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5], thanks to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=') Since U contains Y as a proper subset, we reprove that P is not a ghost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, it follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6 that P vanishes in the (∂βY \\ U)- direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In particular, fixing an index j0 ∈ N and taking a sequence {xi ∈ Yi,j0}i∈N, we choose a cluster point ω ∈ {xi}i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is clear that ω � U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Intuitively speaking, this means that P vanishes “in the i-direction”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 24 QIN WANG AND JIAWEN ZHANG We remark that the first-named author proved in [44, Section 3] that the prin- cipal ideal ⟨P⟩ cannot be decomposed into I(U) + (IG ∩ ⟨P⟩), which provided a couterexample to the conjecture at the end of [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Our explanation above reveals that the reason behind this counterexample is that the ghostly part of ⟨P⟩ could not be “exhausted” merely by ghostly elements associated to X (rather than U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Finally, we remark that the groupoid G(Y)U also plays a key role in constructing a counterexample to the boundary coarse Baum-Connes conjecture introduced in [20] (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Maximal ideals In this section, we would like to study maximal ideals in uniform Roe algebras using the tools developed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Throughout this section, let (X, d) be a space as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Minimal points in the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from previous sections that ideals are closely related to invariant open subsets of the unit space βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we introduce the following: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' An invariant open subset U ⊆ βX is called maximal if U � βX and U is not properly contained in any proper invariant open subset of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Similarly, an invariant closed subset K ⊆ βX is called minimal if K � ∅ and K does not properly contain any non-empty invariant closed subset of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For any maximal invariant open subset U ⊂ βX, the ghostly ideal ˜I(U) is a maximal ideal in the uniform Roe algebra C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Conversely for any maximal ideal I in C∗ u(X), the associated invariant open subset U(I) is maximal and we have I = ˜I(U(I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For any ideal J in C∗ u(X) containing ˜I(U), we have U(J) ⊇ U(˜I(U)) = U by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Since U is maximal, then either U(J) = βX or U(J) = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' If U(J) = βX, it follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='11 that J contains I(U(J)) = C∗ u(X), which implies that J = C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' If U(J) = U, then it follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3(1) that J ⊆ ˜I(U(J)) = ˜I(U), which implies that J = ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This concludes that ˜I(U) is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Conversely for any maximal ideal I in C∗ u(X), we have U(I) � βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For any open invariant subset V � βX containing U, we have I ⊆ ˜I(U) ⊆ ˜I(V) by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4, and ˜I(V) � C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence due to the maximality of I, we obtain that I = ˜I(U) = ˜I(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This implies that U = V and also I = ˜I(U) as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Taking complements, we obtain the following: Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For any minimal invariant closed subset K ⊆ βX, the ghostly ideal ˜I(βX \\ K) is a maximal ideal in the uniform Roe algebra C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, every maximal ideal in C∗ u(X) arises in this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore, in order to describe maximal ideals in the uniform Roe algebra, it suffices to study minimal invariant closed subsets of the unit space βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4 that for each ω ∈ ∂βX, the limit space X(ω) is the smallest invariant subset of βX containing ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' However, note that X(ω) might not be closed in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A point ω ∈ ∂βX is called minimal if the closure of the limit space X(ω) in βX is minimal in the sense of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 25 The following result is straightforward, hence we omit the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It suggests us to study minimal points in the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a minimal invariant closed subset K ⊆ βX, there exists a minimal point ω ∈ ∂βX such that K = X(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Conversely for any minimal point ω ∈ ∂βX, the set X(ω) is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' One might wonder whether every ω ∈ ∂βX is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' However, things become very complicated after taking closures and we will show later that this does not hold even in the case of X = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Firstly, we notice the following: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let K be an invariant closed subset of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then K contains a minimal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In particular, for any ω ∈ ∂βX there exists a minimal point ω′ such that ω′ ∈ X(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This follows directly from the Zorn’s lemma together with the fact that βX is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Details are left to readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Consequently, we obtain: Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For ω ∈ ∂βX, ω is minimal if and only if for any α ∈ X(ω), we have X(α) = X(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Writing X(ω) = � λ∈Λ X(ωλ) for certain ωλ ∈ ∂βX, then ω is minimal if and only if X(ωλ) = X(ω) for any λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As promised, now we study the case of X = Z and show that it admits a number of non-minimal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The following is the main result: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For the integer group Z with the usual metric, there exist non-minimal points in the boundary ∂βZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More precisely, for any sequence {hn}n∈N in Z tending to infinity such that |hn − hm| → +∞ when n + m → ∞ and n � m, and any ω ∈ ∂βZ with ω({hn}n∈N) = 1, then ω is not a minimal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' To prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8, we need some preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For future use, we record the following result in the context of a countable discrete group Γ equipped with a left-invariant word length metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For ω, α ∈ ∂βΓ, then α ∈ Γ(ω) if and only if for any S ⊆ Γ with α(S) = 1, there exists gS ∈ Γ such that ω(S · g−1 S ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 that Γ(ω) = {ρg(ω) : g ∈ Γ}, and from Appendix A that {S : S ⊆ X} forms a basis for βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence by definition, we obtain that α ∈ X(ω) if and only if for any S ⊆ X with α ∈ S, there exists gS ∈ Γ such that ρgS(ω) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Equivalently, this means that for any S ⊆ X with α(S) = 1, there exists gS ∈ Γ such that ρgS(ω)(S) = ω(S · g−1 S ) = 1, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Now we return to the case of Γ = Z and prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8: Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Fix a subset H = {hn}n∈N ⊆ Z tending to infinity such that |hn − hm| → +∞ when n + m → ∞ and n � m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For any non-zero g ∈ Z, note that h ∈ (g + H) ∩ H if and only if there exists h′ ∈ H such that h − h′ = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Since g is fixed and distances between different points in H tend to infinity, we obtain that (g + H) ∩ H is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence for any g1 � g2 in Z, (g1 + H) ∩ (g2 + H) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 26 QIN WANG AND JIAWEN ZHANG Fixing a non-principal ultrafilter ω ∈ ∂βZ with ω(H) = 1, we denote U := {B ⊆ H : ω(B) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We claim that for each n ∈ N, there exists gn ∈ Z and Bn ∈ U such that {Bn + gn}n∈N are mutually disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Indeed, we take g0 = 0 and B0 = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Set g1 = 1 and B1 := H \\ (H − g1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Since H ∩ (H − g1) is finite by the previous paragraph, then ω(B1) = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', B1 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Similarly for each n ∈ N, we take gn = n and Bn := H \\ � (H − g1) ∪ (H − g2) ∪ · · · ∪ (H − gn) � , which concludes the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consider � H := � n∈N(Bn + gn), and denote Un := {B ⊆ Bn : ω(B) = 1} for each n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4, Un is an ultrafilter on Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Choose a non-principal ultrafilter ω0 on N, and we consider: � U := � � n∈N (An+gn) ⊆ � n∈N (Bn+gn) = � H : ∃ J ⊆ N with ω0(J) = 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' ∀n ∈ J, An ∈ Un � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6 implies that � U is an ultrafilter on � H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We define a function α : P(Z) → {0, 1} by setting α(S) = 1 if and only if S ∩ � H ∈ � U, which is indeed an ultrafilter on Z thanks to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Also note that α is non-principal and α(� H) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we show that α ∈ Z(ω) while ω � Z(α), and hence conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' To see that α ∈ Z(ω), we will consult Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For any S ⊆ Z with α(S) = 1, by definition we have S ∩ � H ∈ � U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Writing S ∩ � H = � n∈N(An + gn) with An ⊆ Bn, then there exists J ∈ ω0 such that for any n ∈ J we have An ∈ Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence for any n ∈ J, we have S ⊇ S ∩ � H ⊇ An + gn and ω(An) = 1, which implies that ω(S − gn) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Applying Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9, we conclude that α ∈ Z(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Finally, it remains to check that ω � Z(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Assume the contrary, then Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9 implies that there exists g ∈ Z and �B ⊆ � H with α(�B) = 1 such that H ⊇ �B − g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Writing �B = � n∈N(An + gn) with An ⊆ Bn, then there exists J ∈ ω0 such that for any n ∈ J, ω(An) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This implies that H + g ⊇ �B ⊇ � n∈J (An + gn) ⊇ An0 + gn0 for some n0 ∈ J with gn0 � g (this can be achieved since J is infinite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' However, it follows from the first paragraph that (H + g) ∩ (H + gn0) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' While this intersection contains An0 + gn0, which is infinite since ω(An0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore, we reach a contradiction and conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Maximal ideals via limit operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We already see that maximal ideals in the uniform Roe algebra correspond to minimal points in the Stone- ˇCech bound- ary of the underlying space and in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1, we use topological methods to show the existence of non-minimal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we turn to a C∗-algebraic view- point, and use the tool of limit operators to provide an alternative description for these ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' First recall from Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5 that maximal ideals in C∗ u(X) arise in the form of ˜I(βX \\ X(ω)) for some boundary point ω ∈ ∂βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover according to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9, ˜I(βX \\ X(ω)) is the kernel of the following surjective homomorphism: qβX\\X(ω) : C∗ r(G(X)) −→ C∗ r(G(X)X(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 27 Hence we obtain the following: Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A point ω ∈ ∂βX is minimal if and only if C∗ r(G(X)X(ω)) is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consider the case of a countable discrete group Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a point ω ∈ ∂βΓ, it follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 that the limit space Γ(ω) is identical to Γω, and hence C∗ r(G(Γ)Γ(ω)) is C∗-isomorphic to the reduced crossed product C(Γω) ⋊ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Thanks to Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10, we obtain that ω is minimal if and only if C(Γω) ⋊ Γ is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, note that the action of Γ on βΓ is free (which is also a consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence it follows from [29, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6] that C(Γω)⋊Γ is simple if and only if the action of Γ on Γω is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In conclusion, we reach the following: Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In the case of a countable discrete group Γ, a point ω ∈ ∂βΓ is minimal if and only if the action of Γ on Γω is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10 suggests an approach to distinguish minimal points via the simplicity of the reduced groupoid C∗-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' However, this C∗-algebra is still not easy to handle since it requires to consider all points in the X(ω)-direction (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we show that this can be simplified by merely considering the ω-direction: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For ω ∈ ∂βX, an operator T ∈ C∗ u(X) belongs to the ideal ˜I(βX \\ X(ω)) if and only if T vanishes in the ω-direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', Φω(T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We assume that Φω(T) = 0, and it suffices to show that Φα(T) = 0 for any α ∈ X(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Fixing such an α, we take a net {ωλ}λ∈Λ in X(ω) such that ωλ → α and it follows that Φωλ(T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For any γ1, γ2 ∈ X(α), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4 and the fact that the coarse groupoid G(X) is ´etale imply that there exist γ1,λ and γ2,λ in X(ωλ) for each λ ∈ Λ such that γ1,λ → γ1 and γ2,λ → γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='12 implies that Φα(T)γ1γ2 = T((γ1, γ2)) = lim λ∈Λ T((γ1,λ, γ2,λ)) = lim λ∈Λ Φωλ(T)γ1,λγ2,λ = 0, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Hence for ω ∈ ∂βX, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='13 implies that the associated ideal ˜I(β \\ X(ω)) coincides with the kernel of the following limit operator homomorphism (see also [43, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10]): (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1) Φω : C∗ u(X) −→ C∗ u(X(ω)), T �→ Φω(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consequently, we reach the following: Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A point ω ∈ ∂βX is minimal if and only if the image Im(Φω) is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence when X(ω) is infinite and Φω is surjective, the point ω is not minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For the last statement, it suffices to note that the ideal of compact operators is always contained in C∗ u(X(ω)), which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Thanks to Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14, a special case of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 can also be deduced from a recent work by Roch [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More precisely, combining [43, Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6] with [30, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1], we have the following: 28 QIN WANG AND JIAWEN ZHANG Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let {hn}n∈N be a sequence in ZN tending to infinity such that ∥hn − hk∥∞ ≥ 3k for any k > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then for any ω ∈ ∂βZN with ω({hn}n∈N) = 1, the map Φω : C∗ u(ZN) −→ C∗ u(ZN(ω)) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that the limit space ZN(ω) is bijective to ZN by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8, and hence infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore applying Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14, we obtain the following (when N = 1, it partially recovers Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For any sequence {hn}n∈N in ZN tending to infinity such that ∥hn−hk∥∞ ≥ 3k for any k > n, and any ω ∈ ∂βZN with ω({hn}n∈N) = 1, then ω is not a minimal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We notice from the discussion above that for ω ∈ ∂βX, there is a C∗-monomorphism: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2) C∗ r(G(X)X(ω)) � C∗ u(X)/˜I(βX \\ X(ω)) −→ C∗ u(X(ω)) where the first comes from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9 and the second comes from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1) together with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we provide another explanation for this map in terms of groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Firstly, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5 implies that G(X)X(ω) = X(ω) × X(ω) and hence G(X)X(ω) = X(ω) × X(ω) G(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we have G(X)X(ω) = � S>0 � ES G(X) ∩ X(ω) × X(ω) G(X)� = � S>0 ES G(X) ∩ (X(ω) × X(ω)) G(X) , where the last inequality is due to the fact that ES G(X) is clopen in G(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7, we have ES G(X) ∩ (X(ω) × X(ω)) = {(α, γ) ∈ X(ω) × X(ω) : dω(α, γ) ≤ S} =: ES(X(ω), dω) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', the S-entourage in the limit space (X(ω), dω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore, we conclude that G(X)X(ω) = � S>0 ES(X(ω), dω) G(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, we have (by definition) that G(X(ω)) = � S>0 ES(X(ω), dω) β(X(ω)×X(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By the universal property of the Stone- ˇCech compactification, there is a surjective continuous map ES(X(ω), dω) β(X(ω)×X(ω)) −→ ES(X(ω), dω) G(X), which induces an in- jective map Cc(G(X)X(ω)) −→ Cc(G(X(ω))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, it is routine to check that it induces a C∗-monomorphism C∗ r(G(X)X(ω)) −→ C∗ r(G(X(ω))) � C∗ u(X(ω)), which can be verified to coincide with the map (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Details are left to readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In particular, we consider a countable discrete group X = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Fixing a point ω ∈ ∂βΓ, we mentioned in Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='11 that C∗ r(G(Γ)Γ(ω)) � C(Γω)⋊Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, we know from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 that C∗ u(Γ(ω)) � ℓ∞(Γω) ⋊ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In this case, one can check that the map (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2) is induced by the natural embedding C(Γω) ֒→ ℓ∞(Γω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 29 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Geometric ideals vs ghostly ideals In this section, we return to the lattice IU in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4 that for an invariant open subset U ⊆ βX, the geometric ideal I(U) and the ghostly ideal ˜I(U) are the smallest and the largest elements in IU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Also as noticed before, generally IU consists of more than one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we would like to study when I(U) = ˜I(U) or equivalently, when IU consists of a single element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, we will discuss their K-theories and provide a sufficient condition to ensure K∗(I(U)) = K∗(˜I(U)) for ∗ = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' First recall from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9 that we already have a characterisation for I(U) = ˜I(U) using short exact sequences, while the condition therein still seems hard to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we aim to search for a more practical criterion to ensure I(U) = ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We start with the following result, which combines [14, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4] and [36, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a space (X, d), the following are equivalent: (1) X has Property A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) G(X) is amenable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (3) G(X)∂βX is amenable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (4) I(U) = ˜I(U) for any invariant open subset U ⊆ βX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (5) I(X) = ˜I(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We remark that “(1) ⇒ (4)” was originally proved in [14] using approximations by kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Here we will take a shortcut, and the idea will also be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' “(1) ⇔ (2)” was proved in [39, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' “(2) ⇔ (3)” is due to the permanence properties of amenability (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6) together with the fact that G(X)X � X × X is always amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' “(2) ⇒ (4)”: Let U ⊆ βX be an invariant open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As open/closed sub- groupoids, both G(X)U and G(X)Uc are amenable as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consider the following commutative diagram coming from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3): 0 � C∗ max(G(X)U) � � C∗ max(G(X)) � � C∗ max(G(X)Uc) � � 0 0 � C∗ r(G(X)U) � C∗ r(G(X)) � C∗ r(G(X)Uc) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='12, all three vertical lines are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence the exactness of the first row implies that the second row is exact as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore, we conclude (4) thanks to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' “(4) ⇒ (5)” holds trivially, and “(5) ⇒ (1)” comes from [36, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3] together with Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 and Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1 provides a coarse geometric characterisation for I(X) = ˜I(X) using Property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' However, we notice that assuming Property A is often too strong to ensure that I(U) = ˜I(U) for merely a specific invariant open subset U ⊆ βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A trivial example is that I(βX) = ˜I(βX) holds for any space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This suggests us to explore a weaker criterion for I(U) = ˜I(U), and we reach the following: 30 QIN WANG AND JIAWEN ZHANG Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space and U be an invariant open subset of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' If the canonical quotient map C∗ max(G(X)Uc) → C∗ r(G(X)Uc) is an isomorphism, then I(U) = ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In particular, if the groupoid G(X)Uc is amenable then I(U) = ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We consider the following commutative diagram: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1) 0 � C∗ max(G(X)U) � πU � C∗ max(G(X)) � � C∗ max(G(X)Uc) � � 0 0 � ˜I(U) � C∗ r(G(X)) � C∗ r(G(X)Uc) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Here the map πU is the composition: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2) C∗ max(G(X)U) → C∗ r(G(X)U) � I(U) ֒→ ˜I(U), where the middle isomorphism comes from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that the top hor- izontal line is automatically exact, while the bottom one is also exact thanks to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Also note that the middle vertical map is always surjective and by assumption, the right vertical map is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence via a diagram chas- ing argument, we obtain that the left vertical map is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This concludes that I(U) = ˜I(U) thanks to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' When U = X, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 recovers “(3) ⇒ (5)” in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Readers might wonder whether the converse of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 holds as in the case of U = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We manage to provide a partial answer in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we move to discuss the K-theory of geometric and ghostly ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Firstly, we need an extra notion: Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a set S of subsets of X, denote L(S) the smallest ideal in X containing S, and we say that L(S) is generated by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' An ideal L in X is called countably generated if there exists a countable set S such that L = L(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' An invariant open subset U ⊆ βX is called countably generated if the associated ideal L(U) is countably generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a space X and a subspace A ⊆ X, it follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14 that the spatial ideal IA is countably generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, it follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='17 that principal ideals are always countably generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In particular, the ideal ⟨P⟩ considered in [44, Section 3] (see also Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10) is countably generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The property of countable generatedness leads to the following: Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let L be a countably generated ideal in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then there exists a countable subset {Y1, Y2, · · · , Yn, · · · } in L such that L = {Z ⊆ X : ∃ n ∈ N such that Z ⊆ Yn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consequently for any countably generated open invariant subset U of βX, the subgroupoid G(X)U is σ-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' To prove Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6, we need an auxiliary result on the structure of L(S): Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a set S of subsets of X, denote S(1) := {A1 ∪ · · · ∪ An : Ai ∈ S, n ∈ N} and S(2) := {Nk( ˜A) : ˜A ∈ S(1), k ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then we have: L(S) = {Z : ∃ Y ∈ S(2) such that Z ⊆ Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 31 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Denote S(3) := {Z : ∃ Y ∈ S(2) such that Z ⊆ Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is clear that S(3) ⊆ L(S), and any ideal containing S must contain S(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence it remains to check that S(3) is an ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For Z ∈ S(3) and W ⊆ Z, there exists Y ∈ S(2) such that Z ⊆ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence W ⊆ Y, which implies that W ∈ S(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Also note that Y ∈ S(2) implies that there exists ˜A ∈ S(1) and k′ ∈ N such that Y = Nk′( ˜A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence for any k ∈ N, we have Nk(Z) ⊆ Nk(Y) ⊆ Nk+k′( ˜A), which implies that Nk(Z) ∈ S(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Finally for Z1, Z2 ∈ S(3), there exist ˜A1, ˜A2 ∈ S(1) and k1, k2 ∈ N such that Zi ⊆ Nki( ˜Ai) for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence Z1 ∪ Z2 ⊆ Nk1( ˜A1) ∪ Nk2( ˜A2) ⊆ Nk1+k2( ˜A1 ∪ ˜A2) ∈ S(2), which implies that Z1 ∪ Z2 ∈ S(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore, we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By assumption, there exists a countable S such that L = L(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Using the notation of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7, the set S(2) is countable as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence the first statement follows directly from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For the second, note that U = � n∈N Yn and hence G(X)U = � n∈N G(X)Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Since each Yn is closed, we obtain that G(X)U is σ-compact as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Now we are in the position to discuss the K-theory of geometric and ghostly ideals: Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let X be a space which can be coarsely embedded into some Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then for any countably generated invariant open subset U ⊆ βX, we have an isomorphism (ιU)∗ : K∗(I(U)) −→ K∗(˜I(U)) for ∗ = 0, 1, where ιU is the inclusion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore for any ideal I in C∗ u(X) with U(I) countably generated, we have an injective homomorphism (ιI)∗ : K∗(I ∩ Cu[X]) −→ K∗(I) for ∗ = 0, 1, where ιI is the inclusion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Fixing such a U ⊆ βX, we consider the commutative diagram (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1) from the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2, where both of the horizontal lines are exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This implies the following commutative diagram in K-theories: · · � K∗(C∗ max(G(X)U)) � (πU)∗ � K∗(C∗ max(G(X))) � � K∗(C∗ max(G(X)Uc)) � � K∗+1(C∗ max(G(X)U)) � (πU)∗+1 � · · · · � K∗(˜I(U)) � K∗(C∗ r(G(X))) � K∗(C∗ r(G(X)Uc)) � K∗+1(˜I(U)) � · · · where both horizontal lines are exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from [39, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4] that X is coarsely embeddable if and only if G(X) is a-T-menable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It follows directly from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='13 that as subgroupoids, both G(X)U and G(X)Uc are a-T-menable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, it is clear that G(X) and G(X)Uc are σ-compact by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14 implies that G(X) and G(X)Uc are K-amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This shows that the middle two vertical maps in the above diagram are isomorphisms, which implies that (πU)∗ is an isomorphism by the Five Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2), we obtain that the composition K∗(C∗ max(G(X)U)) −→ K∗(C∗ r(G(X)U)) � K∗(I(U)) (ιU)∗ −→ K∗(˜I(U)) 32 QIN WANG AND JIAWEN ZHANG is an isomorphism for ∗ = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, it follows from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6 that G(X)U is σ-compact, which implies that G(X)U is K-amenable again by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we conclude that the map (ιU)∗ is an isomorphism for ∗ = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For the last statement, we assume that U = U(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='11, we have that I ∩ Cu[X] = I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Also Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3(1) shows that I ⊆ ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence the inclusion map ιU can be decomposed as follows: I(U) = I ∩ Cu[X] ιI֒→ I ֒→ ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore, (ιU)∗ being an isomorphism implies that (ιI)∗ is injective for ∗ = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Applying Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 to the case of U = X, we partially recover the following result by Finn-Sell (see [19, Proposition 35]), which is crucial for the counterex- amples to the coarse Baum-Connes conjecture: Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let X be a space which can be coarsely embedded into some Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then the inclusion of K(ℓ2(X)) into IG induces an isomorphism on the K-theory level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a general ideal I in C∗ u(X), our method in the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 only provides the injectivity of the induced map (ιI)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We wonder whether this map is indeed an isomorphism under the same assumption (see Question 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We end this section with an example, which shows that not every ideal in a space is countably generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let X = N × N, equipped with the metric induced from the Eu- clidean metric dE on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For each θ ∈ [0, π 2] and k ∈ N, we define ℓθ := {(x, y) ∈ R × R : y = tan(θ)x} and Sθ,k := {(x, y) ∈ X : dE((x, y), ℓθ) ≤ k} = Nk(ℓθ) ∩ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consider S := {Sθ,k : θ ∈ [0, π 2], k ∈ N} and set S(1) := {A1 ∪ · · · ∪ An : Ai ∈ S, n ∈ N} as in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For any R > 0 and A = A1 ∪ · · · ∪ An ∈ S(1) where Ai ∈ S, we have NR(A) = NR(A1) ∪ · · · ∪ NR(An), which is contained in some element in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence applying Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7, the ideal L(S) generated by S is: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3) L(S) = {Z : ∃ Y ∈ S(1) such that Z ⊆ Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We claim that L(S) is not countably generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Otherwise, there exists a count- able subset S′ generating L(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, according to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3) we can assume that S′ = {Yn : n ∈ N} where Yn = Sθn,1,kn ∪ · · · ∪ Sθn,pn,kn ∈ S(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Choose θ ∈ [0, π 2] \\ {θn,i : i = 1, 2, · · · , pn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' n ∈ N}, and consider Y = Sθ,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Since S′ generates L(S), it follows from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7 that there exist R > 0 and Ym1, · · · , Yml ∈ S′ such that Sθ,1 = Y ⊆ NR(Ym1 ∪ · · · ∪ Yml).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note thatthe righthand side iscontained in a finite union ofsome R′-neighbourhood of lines (in R2 crossing the origin) with slopes in the set � tan(θn,i) : i = 1, 2, · · · , pn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' n ∈ N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This leads to a contradiction due to the choice of θ, which concludes that L(S) cannot be countably generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 33 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Partial Property A and partial operator norm localisation property In the previous section, we find a sufficient condition (Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2) to ensure ˜I(U) = I(U) for a given invariant open subset U ⊆ βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' As promised in Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3, now we study its converse and show that this is indeed an equivalent condition under the assumption of countable generatedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Our strategy is to follow the outline of the case that U = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More precisely, we introduce a notion called partial Property A towards invariant subsets of the boundary ∂βX, and then consider its counterpart in the context of operator norm localisation property to provide the desired characterisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Partial Property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1 that a space X has Property A if and only if the groupoid G(X)∂βX is amenable, which characterises I(X) = ˜I(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Together with Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2, this inspires us to introduce the following: Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space and U ⊆ βX be an invariant open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We say that X has partial Property A towards ∂βX \\ U if G(X)∂βX\\U is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is clear from definition that X has Property A if and only if it has partial Property A towards the whole boundary ∂βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, it follows from Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 that if X has partial Property A towards ∂βX \\ U, then we have I(U) = ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The rest of this section is devoted to studying the converse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Firstly, we aim to unpack the groupoid language and provide a concrete geomet- ric description for partial Property A, which resembles the definition of Property A (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The following is the main result: Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space and U ⊆ βX an invariant open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then X has partial Property A towards Uc = ∂βX \\ U if and only if for any ε, R > 0, there exist S > 0, a subset D ⊆ X with D ⊇ Uc and a function f : X × X → [0, 1] satisfying: (1) supp(f) ⊆ ES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) for any x ∈ D, we have � z∈X f(z, x) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (3) for any x, y ∈ D with d(x, y) ≤ R, we have � z∈X | f(z, x) − f(z, y)| ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Comparing Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 with Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1, it is clear that Property A implies partial Property A towards any invariant closed subset of ∂βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' To prove Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2, we start with the following lemma: Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' With the same notation as above, X has partial Property A towards Uc if and only if for any ε, R > 0, there exist S > 0 and a function f : X × X → [0, 1] satisfying: (1) supp(f) ⊆ ES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) for any ω ∈ Uc, we have � α∈X(ω) f(α, ω) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (3) for ω ∈ Uc and α ∈ X(ω) with dω(α, ω) ≤ R, then � γ∈X(ω) | f(γ, α) − f(γ, ω)| ≤ ε, where f ∈ C0(G(X)) is the continuous extension from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By definition, X has partial Property A towards Uc if and only if for any ε > 0 and compact K ⊆ G(X)Uc, there exists g ∈ Cc(G(X)Uc) with range in [0, 1] such 34 QIN WANG AND JIAWEN ZHANG that for any γ ∈ K we have � α∈Gr(γ) g(α) = 1 and � α∈Gr(γ) |g(α) − g(αγ)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1) that the restriction map Cc(G(X)) → Cc(G(X)Uc) is surjective, and hence g can be regarded as a function in Cc(G(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Taking f to be the restriction of g on X × X, then f ∈ ℓ∞(X × X) and there exists S > 0 such that supp(f) ⊆ ES for some S > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Using the notation from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9, we have g = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that G(X)Uc = � R>0(ER ∩ G(X)Uc), and hence compact subsets of G(X)Uc are always contained in those of the form ER ∩ G(X)Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Furthermore, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7 implies ER ∩ G(X)Uc = � ω∈Uc {(α, γ) ∈ X(ω) × X(ω) : dω(α, γ) ≤ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Combining with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4, we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ As a direct corollary (together with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7), we obtain: Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Assume that X has partial Property A towards Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then the family of metric spaces {(X(ω), dω)}ω∈Uc has uniform Property A in the sense that the parameters in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1 can be chosen uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is unclear to us whether the converse of Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that X(ω) might contain points outside X(ω) as discussed in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8, hence we do not know whether functions on X(ω) × X(ω) can be glued together to provide a continuous function on G(X)Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Sufficiency: For any ε, R > 0, choose S > 0, D ⊆ X and a function g : X × X → [0, 1] satisfying the conditions (1)-(3) for ε and 3R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Take a map p : NR(D) → D such that the restriction of p on D is the identity map and d(p(x), x) ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we define: f(x, y) = � g(x, p(y)), y ∈ NR(D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' g(x, y), otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is clear that supp(f) ⊆ ER+S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover for any y1, y2 ∈ NR(D) with d(y1, y2) ≤ R, we have d(p(y1), p(y2)) ≤ 3R and hence � x∈X | f(x, y1) − f(x, y2)| = � x∈X |g(x, p(y1)) − g(x, p(y2))| ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore (enlarging S to S + R) we obtain that for any ε, R > 0, there exist S > 0, a subset D ⊆ X with D ⊇ Uc and a function f : X × X → [0, 1] satisfying: (1) supp(f) ⊆ ES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) for any x ∈ D, we have � z∈X f(z, x) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (3) for any x ∈ D and y ∈ X with d(x, y) ≤ R, we have � z∈X | f(z, x) − f(z, y)| ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we fix ε, R > 0 and take such S, D and function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Given ω ∈ Uc, we have ω(D) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Choose {tα : Dα → Rα} to be a compatible family for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3, there exists Yω ⊆ X with ω(Y) = 1 and a local coordinate system {hy : B(ω, R + S) → B(y, R + S)}y∈Yω such that the map hy : B(ω, R + S) → B(y, R + S), α �→ tα(y) GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 35 is a surjective isometry for each y ∈ Yω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Replacing Y by Yω ∩ D, we assume that Yω ⊆ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that supp(f) ⊆ ES, and hence applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7 we have � α∈X(ω) f(α, ω) = � α∈B(ω,S) f(α, ω) = � α∈B(ω,S) lim x→ω f(tα(x), x) = lim x→ω,x∈Yω � α∈B(ω,S) f(tα(x), x) = lim x→ω,x∈Yω � z∈B(x,S) f(z, x) = lim x→ω,x∈Yω � z∈X f(z, x), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1) where the last item equals 1 thanks to the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, for any α ∈ B(ω, R) we have limx→ω d(tα(x), x) ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence shrinking Yω if necessary, we can assume that d(tα(x), x) ≤ R for any x ∈ Yω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore, we have � γ∈X(ω) | f(γ, α) − f(γ, ω)| = � γ∈B(ω,R+S) | f(γ, α) − f(γ, ω)| (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2) = lim x→ω,x∈Yω � γ∈B(ω,R+S) | f(tγ(x), tα(x)) − f(tγ(x), x)| = lim x→ω,x∈Yω � z∈B(x,R+S) | f(z, tα(x)) − f(z, x)| = lim x→ω,x∈Yω � z∈X | f(z, tα(x)) − f(z, x)|, where the last item is no more than ε by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore applying Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3, we conclude the sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Necessity: Given ε, R > 0, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3 provides S > 0 and a function f : X × X → [0, 1] satisfying the conditions (1)-(3) therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Fix an ω ∈ Uc and we choose {tα : Dα → Rα} to be a compatible family for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3, there exists Yω ⊆ X with ω(Y) = 1 and a local coordinate system {hy : B(ω, R + 2S) → B(y, R + S)}y∈Yω such that the map hy : B(ω, R + S) → B(y, R + S), α �→ tα(y) is a surjective isometry for each y ∈ Yω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By the calculations in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1), we obtain that lim x→ω,x∈Yω � z∈X f(z, x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence for the given ε, there exists Y′ ω ⊆ Yω with ω(Y′ ω) = 1 such that for any x ∈ Y′ ω we have � z∈X f(z, x) ∈ (1 − ε, 1 + ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, for any α ∈ B(ω, R) we apply the calculations in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2) and obtain: lim x→ω,x∈Y′ω � z∈X | f(z, tα(x)) − f(z, x)| = � γ∈X(ω) | f(γ, α) − f(γ, ω)| ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that for x ∈ Y′ ω, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3 implies that {tα(x) : α ∈ B(ω, R)} = B(x, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence there exists Y′′ ω ⊆ Y′ ω with ω(Y′′ ω) = 1 such that for any x ∈ Y′′ ω and y ∈ B(x, R), we have � z∈X | f(z, y) − f(z, x)| < 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 36 QIN WANG AND JIAWEN ZHANG Taking D := � ω∈Uc Y′′ ω, then it is clear that D ⊇ Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, the analysis above shows that: for any x ∈ D we have � z∈X f(z, x) ∈ (1 − ε, 1 + ε);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' for any x ∈ D and y ∈ B(x, R), we have � z∈X | f(z, y) − f(z, x)| < 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Finallyusinga standard normalisation argument(orequivalently, applyingLemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='11 and modifying Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3 accordingly), we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Setting ξy(x) := f(x, y) for the function f in Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2, we can rewrite Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 as follows: Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space and U ⊆ βX be an invariant open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then X has partial Property A towards Uc if and only if for any ε, R > 0, there exist S > 0, a subset D ⊆ X with D ⊇ Uc and a function ξ : D → ℓ1(X)1,+, x �→ ξx satisfying: (1) supp(ξx) ⊆ B(x, S) for any x ∈ D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) for any x, y ∈ D with d(x, y) ≤ R, we have ∥ξx − ξy∥1 ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We remark that the function ξ in Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2′ can be made such that ξx ∈ ℓ1(D)1,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In fact, this is the same trick as in the case of Property A (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', [27, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, we can further replace ℓ1(D)1,+ by ℓ2(D)1,+ using the Mazur map (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', [45, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4] for the same trick).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we provide an alternative picture for Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 using the notion of ideals in spaces (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall that for an ideal L in X, we denote U(L) := � Y∈L Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We need the following auxiliary lemma: Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let L be an ideal in X and D ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then D ⊇ U(L)c if and only if there exists Y ∈ L such that D ⊇ Yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Assume Y ∈ L such that D ⊇ Yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that Y ∩ Yc = ∅ and Y ∪ Yc = βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we have D ⊇ Yc = βX \\ Y ⊇ βX \\ U(L) = U(L)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Conversely, assume that D ⊇ U(L)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then (D)c ⊆ U(L) = � Y∈L Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Since D is clopen in the compact space βX, the set (D)c is compact as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence there exists Y1, · · · , Yn ∈ L such that (D)c ⊆ Y1 ∪· · ·∪Yn = Y1 ∪ · · · ∪ Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Since L is an ideal, the set Y := Y1∪· · ·∪Yn ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then we have (D)c ⊆ Y, which implies that D ⊇ (Y)c = Yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Finally we obtain D = D ∩ X ⊇ Yc ∩ X = Yc, which conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Thanks to Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7, now we can rewrite Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 (combining with Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6) as follows: Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space, U ⊆ βX an invariant open subset and L = L(U) the associated ideal in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then X has partial Property A towards Uc if and only if for any ε, R > 0, there exist S > 0, a subset Y ∈ L(U) and a function ξ : Yc → ℓ2(Yc)1,+, x �→ ξx satisfying: (1) supp(ξx) ⊆ B(x, S) for any x ∈ Yc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) for any x, y ∈ Yc with d(x, y) ≤ R, we have ∥ξx − ξy∥1 ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Similar to the proof in the case of Property A, we also have the following: GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 37 Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space, U ⊆ βX an invariant open subset and L = L(U) the associated ideal in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then X has partial Property A towards Uc if and only if for any R > 0 and ε > 0, there exist S > 0, a subset Y ∈ L(U) and a kernel k : Yc × Yc → R of positive type satisfying: (1) for x, y ∈ Yc, we have k(x, y) = k(y, x) and k(x, x) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) for x, y ∈ Yc with d(x, y) ≥ S, we have k(x, y) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (3) for x, y ∈ Yc with d(x, y) ≤ R, we have |1 − k(x, y)| ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Partial operator norm localisation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall that the notion of oper- ator norm localisation property (ONL) was introduced by Chen, Tessera, Wang and Yu in [10], and proved by Sako in [37] that ONL is equivalent to Property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Here we introduce a partial version of ONL, parallel to Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let ν be a positive locally finite Borel measure on X and H be a separable infinite-dimensional Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For an operator T ∈ B(L2(X, ν) ⊗ H), we can also define its propagation as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We introduce the following: Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space and U ⊆ βX an invariant open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We say that X has partial operator norm localisation property (partial ONL) towards Uc = ∂βX \\ U if there exists c ∈ (0, 1] such that for any R > 0 there exist S > 0 and D ⊆ X with D ⊇ Uc satisfying the following: for any positive locally finite Borel measure ν on X with supp(ν) ⊆ D and any a ∈ B(L2(X, ν) ⊗ H) with propagation at most R, there exists a non-zero ζ ∈ L2(X, ν) ⊗ H with diam(supp(ζ)) ≤ S such that c∥a∥ · ∥ζ∥ ≤ ∥aζ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The aim of the rest of this subsection is to show that partial ONL is equivalent to partial Property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We will follow the outline of [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' To simplify the statement, denote CR u[X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' H] := {T ∈ B(ℓ2(X) ⊗ H) : prop(a) ≤ R} for R ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a subspace Y ⊆ X, it is clear that CR u[Y] � χYCR u[X]χY (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' CR u[Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' H] � χYCR u[X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' H]χY), and hence can be regarded as a subset of CR u[X] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' CR u[X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' H]) with support in Y × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Similarly, C∗ u(Y) � χYC∗ u(X)χY can be regarded as a C∗-subalgebra in C∗ u(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we will not tell the difference in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For S > 0, denote BY S := � x∈X B(ℓ2(B(x, S) ∩ Y)), whose elements will be written as b = ([bx(y, z)]y,z∈B(x,S)∩Y)x∈X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We also consider the map ψY S : C∗ u(Y) � χYC∗ u(X)χY −→ BY S by a �→ ([a(y, z)]y,z∈B(x,S)∩Y)x∈X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall the notions of completely positive map and completely bounded map: A self-adjoint closed subspace F of a unital C∗-algebra B such that 1B ∈ F is called an operator system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A linear map φ from F to a C∗-algebra C is said to be completely positive if the map φ(n) = φ ⊗ id : F ⊗ Mn(C) → C ⊗ Mn(C) is positive for every n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' A linear map θ : F → C is said to be completely bounded if the sequence {∥θ(n) : F⊗Mn(C) → C⊗Mn(C)∥} is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Denote ∥θ∥cb := supn∈N ∥θ(n)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We have the following characterisation for partial ONL, which is analogous to [37, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The proof is almost identical, hence omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 38 QIN WANG AND JIAWEN ZHANG Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space and U ⊆ βX an invariant open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then the following are equivalent: (1) X has partial ONL towards Uc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) there exists c ∈ (0, 1] such that for any R > 0 there exist S > 0 and D ⊆ X with D ⊇ Uc satisfying condition (α): for any a ∈ CR u[D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' H] there exists a non-zero ζ ∈ ℓ2(X) ⊗ H with diam(supp(ζ)) ≤ S and c∥a∥ · ∥ζ∥ ≤ ∥aζ∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (3) for any c ∈ (0, 1) and R > 0, there exist S > 0 and D ⊆ X with D ⊇ Uc satisfying condition (α);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (4) for any c ∈ (0, 1) and R > 0, there exist S > 0 and D ⊆ X with D ⊇ Uc satisfying condition (β): for any a ∈ CR u[D] there exists a non-zero ξ ∈ ℓ2(X) with diam(supp(ξ)) ≤ S and c∥a∥ · ∥ξ∥ ≤ ∥aξ∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (5) for any ε, R > 0 there exist S > R and D ⊆ X with D ⊇ Uc such that ∥(ψD S |CR u[D])−1 : ψD S (CR u[D]) −→ CR u[D]∥ < 1 + ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (6) for any ε, R > 0 there exist S > R and D ⊆ X with D ⊇ Uc such that ∥(ψD S |CR u[D])−1 : ψD S (CR u[D]) −→ CR u[D]∥cb < 1 + ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We record the following, which comes directly from Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space, U ⊆ βX an invariant open subset and L = L(U) the associated ideal in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then X has partial ONL towards Uc if and only if for any c ∈ (0, 1) and R > 0 there exist S > 0 and Y ∈ L(U) satisfying the following: for any a ∈ CR u[Yc] there exists a non-zero ξ ∈ ℓ2(X) with diam(supp(ξ)) ≤ S and c∥a∥ · ∥ξ∥ ≤ ∥aξ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Finally, we can mimic the proof of [37, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1] using Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2′, Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='10 instead, and reach the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The proof is almost identical, and hence omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space and U ⊆ βX be an invariant open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then X has partial Property A towards Uc if and only if X has partial ONL towards Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' To end this subsection, we study a permanence property of partial ONL, which will help to prove the main result in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let X be a space and L an ideal in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Assume that X can be decomposed into X = X1 ∪ X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consider (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3) Li := {Y ∩ Xi : Y ∈ L} for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then it is routine to check that Li is an ideal in Xi for i = 1, 2, and L = {Y1 ∪ Y2 : Yi ∈ Li, i = 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' With respect to the decomposition above, we now show that partial ONL is preserved under finite unions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' With the notation as above, assume that Xi has partial ONL towards βXi \\ U(Li) for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then X has partial ONL towards βX \\ U(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' One way to prove Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='13 is to follow the proof of [16, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3] with minor changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Here we choose another approach using Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 that for a subset Z ⊆ X, the closure Z in βX is homeomorphic to βZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we will regard them as the same object in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 39 Now we can easily transfer the restriction of ideals in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3) to that of invariant open subsets (the proof is straightforward, hence omitted): Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let X = X1 ∪ X2, L be an ideal in X and U = U(L) ⊆ βX be the associated invariant open subset of βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then U ∩ Xi is an invariant open subset of Xi = βXi, which corresponds to the ideal Li in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3) for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Although U ∩ Xi is invariant in βXi, generally it is not invariant in βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This coincides with the fact that χXiC∗ u(X)χXi � C∗ u(Xi) is a just subalgebra in C∗ u(X) rather than an ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Instead, we consider the spatial ideal IXi recalled in Section 4, and prove the following permanence property for partial Property A: Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let X = X1 ∪ X2, U ⊆ βX be an invariant open subset and Ui = U ∩ Xi for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' If Xi has partial Property A towards βXi \\ Ui for i = 1, 2, then X has partial Property A towards βX \\ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For any R > 0 and i = 1, 2, set Ui(R) := � Y∈L(Ui) NR(Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Clearly, Ui(R) is an invariant open subset of NR(Xi) = β(NR(Xi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, it follows from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 that NR(Xi) has partial Property A towards NR(Xi) \\ Ui(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By definition, this means that the groupoid G(NR(Xi))NR(Xi)\\Ui(R) is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence as a subgroupoid, G(NR(Xi))NR(Xi)\\U is amenable, which further implies that the groupoid � R>0 G(NR(Xi))NR(Xi)\\U is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='14 that � R NR(Xi) \\ U is invariant in βX and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='16 implies that G(X)� R NR(Xi)\\U = � R>0 G(NR(Xi))NR(Xi)\\U, which is hence amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that � R NR(X1) ∪ � R NR(X2) = βX, and hence due to the extension property we obtain that G(X)βX\\U = G(X)� R NR(X1)\\U ∪ G(X)� R NR(X2)\\U is amenable as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Combining Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='12 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='15, we conclude Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Characterisation for I(U) = ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Having established all the necessary ingre- dients above, now we present the main result of this section: Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 and U ⊆ βX be a countably generated invariant open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then the following are equivalent: (1) X has partial Property A towards βX \\ U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) ˜I(U) = I(U);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (3) the ideal IG of all ghost operators is contained in I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that U = X is countably generated, and hence Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='16 recovers [36, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3] (see Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 and Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Borrowing the language of [36], condition (3) in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='16 says that all the ghosts can be busted in the geometric ideal I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We follow the outline of the proof for [36, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Firstly, we need a modified version of [36, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2]: 40 QIN WANG AND JIAWEN ZHANG Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space, U ⊆ βX be a countably generated invariant open subset and L = L(U) the associated ideal in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Assume that X does not have partial ONL towards Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then there exist κ ∈ (0, 1), R > 0, a sequence (Tn) in Cu[X], a sequence (Bn) of finite subsets of X and a sequence (Sn) of positive real numbers such that: (a) (Sn) is an increasing sequence tending to infinity as n → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (b) each Tn is positive and has norm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (c) for n � m, then Bn ∩ Bm = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (d) each Tn is supported in Bn × Bn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (e) for each n and ξ ∈ ℓ2(X) with ∥ξ∥ = 1 and diam(suppξ) ≤ Sn, then ∥Tnξ∥ ≤ κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (f) for each Y ∈ L(U), there exists n such that Bn ∩ Y = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Fixing a basepoint x0 ∈ X, consider the decomposition X = X(1) ∪ X(2) with X(1) := � m even {x ∈ X : m2 ≤ d(x, x0) < (m + 1)2} and X(2) := � m odd {x ∈ X : m2 ≤ d(x, x0) < (m + 1)2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Set Li := {Y ∩ X(i) : Y ∈ L} for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By assumption, X does not have partial Property A towards Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence without loss of generality, we can assume that X(1) does not have partial Property A towards βX(1) \\ U(L1) thanks to Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that this implies that U(L1) � βX(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is clear that L1 is also countably generated, and hence according to Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6 there exists a countable subset {Y1, Y2, · · · , Yn, · · · } of L1 such that L1 = {Z ⊆ X(1) : ∃ n ∈ N such that Z ⊆ Yn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' In the sequel, we fix such a sequence {Y1, Y2, · · · , Yn, · · · }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Due to Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='11, we know that there exist c ∈ (0, 1) and R > 0 such that for any Y ∈ L1 and S > 0, there exists T ∈ CR u[X(1) \\ Y] with ∥T∥ = 1 satisfying: for any ξ ∈ ℓ2(X(1)) with diam(suppξ) ≤ S and ∥ξ∥ = 1, then ∥Tξ∥ < c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We call such an operator (R, c, S, Y)-localised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Replacing T by T∗T (and R by 2R and c by √c), we see that there exist c ∈ (0, 1) and R > 0 such that for any Y ∈ L1 and S > 0, there exists a positive (R, c, S, Y)-localised operator of norm one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let us fix such c and R in the rest of the proof, and set κ := 2c 1+c < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that X(1) can be decomposed into: X(1) := � m∈N Xm where each Xm is finite and d(Xm, Xn) > R for any n � m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence each T ∈ CR u[X(1)] splits as a block diagonal sum of finite rank operators T = � m T(m) where T(m) ∈ B(ℓ2(Xm)), with respect to this decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Take S1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By assumption, there exists a positive (R, c, S1, Y1)-localised op- erator T ∈ CR u[X(1) \\ Y1] with norm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that ∥T∥ = supm ∥T(m)∥, and then there exists m1 ∈ N such that ∥T(m1)∥ > 1+c 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We set T1 := T(m1)/∥T(m1)∥ and denote B1 := Xm1 ∩ (X(1) \\ Y1), which is nonempty since T has support in B1 × B1 by as- sumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then for any ξ ∈ ℓ2(X(1)) with ∥ξ∥ = 1 and diam(supp(ξ)) ≤ S1, we GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 41 have ∥T1ξ∥ ≤ 2c 1 + c = κ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we take S2 > max � diam � � k≤m1 Xk � , 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By assumption, there exists a positive (R, c, S2, Y2)-localised operator T ∈ CR u[X(1) \\ Y2] with norm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Again there exists m2 such that ∥T(m2)∥ > 1+c 2 , which forces m2 > m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We set T2 := T(m2)/∥T(m2)∥ and denote B2 := Xm2 ∩(X(1)\\Y2), which is nonempty since T has support in B2×B2 by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Similarly for any ξ ∈ ℓ2(X(1)) with ∥ξ∥ = 1 and diam(supp(ξ)) ≤ S2, we have ∥T2ξ∥ ≤ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Inductively, we can construct a sequence (Tn) in Cu[X(1)] ⊆ Cu[X], a sequence (Bn) of finite subsets of X(1) ⊆ X and a sequence (Sn) of positive real numbers satisfying condition (a)-(e) in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Furthermore for each Z ∈ L(U), there exists Yn containing Z∩X(1) for some n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' By construction, we know that Bn ∩Yn = ∅ and Bn ⊆ X(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we have: Bn ∩ Z = Bn ∩ X(1) ∩ Z ⊆ Bn ∩ Yn = ∅, which provides condition (f) and concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Comparing Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='17 with [36, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2], we note that condition (f) is the only extra condition added in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It seems hard to write condition (f) in the language of the invariant open subset U instead of the ideal L(U), which indicates the importance of using the notion of ideals in spaces as mentioned in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we are in the position to prove Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' “(1) ⇒ (2)” is contained in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2, and “(2) ⇒ (3)” holds trivially since IG = ˜I(X) ⊆ ˜I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence it suffices to show “(3) ⇒ (1)”, and we follow the outline of the proof for [36, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Assume that X does not have partial Property A towards Uc, then it follows from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='12 that X does not have partial ONL towards Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then from Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='17, there exist κ ∈ (0, 1), R > 0, a sequence (Tn) in Cu[X], a sequence (Bn) of finite subsets of X and a sequence (Sn) of positive real numbers satisfying condition (a)-(f) therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we consider the operator T := � n Tn, which is a positive operator in Cu[X] of norm one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Now we take a continuous function f : [0, 1] → [0, 1] such that supp f ⊆ [1+κ 2 , 1] and f(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consider the operator f(T) ∈ C∗ u(X), which is positive, norm one, and admits a decomposition f(T) = � n f(Tn), where each f(Tn) ∈ B(ℓ2(Bn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We will show that f(T) ∈ ˜I(X) \\ I(U), and hence conclude a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' First we show that f(T) � I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2) that I(U) = {T′ ∈ Cu[X] : supp(T′) ⊆ Y × Y for some Y ∈ L(U)}, 42 QIN WANG AND JIAWEN ZHANG Now for any T′ ∈ Cu[X] with supp(T′) ⊆ Y × Y for some Y ∈ L(U), condition (f) in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='17 implies that there exists n such that Bn ∩ Y = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we have: ∥ f(T) − T′∥ ≥ ∥χBn f(T)χBn − χBnT′χBn∥ = ∥ f(Tn) − 0∥ = 1, which implies that f(T) � I(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' On the other hand, using the same argument as for [36, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3] (since Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='17 provides all the conditions required in [36, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2]), we obtain that f(T) is a ghost operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence according to Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5, we have f(T) ∈ ˜I(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore, we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Open questions Here we collect several open questions around this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' First recall from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4 that for a space (X, d), any ideal I in the uniform Roe algebra C∗ u(X) must lie between I(U) and ˜I(U) for U = U(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' However, the structure of the lattice IU = {I is an ideal in C∗ u(X) : U(I) = U} in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1) is still unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that for any invariant open subset V ⊇ U of βX, the ideal I(V)∩ ˜I(U) belongs to the lattice IU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Unfortunately, we do not know whether these ideals can bust every element in IU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we pose the following: Question 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space and U ⊆ βX be an invariant open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Can we describe elements in the lattice IU = {I is an ideal in C∗ u(X) : U(I) = U} in details?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For I ∈ IU, can we find an invariant open subset V ⊇ U such that I = I(V) ∩ ˜I(U)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note that an answer to the above question together with Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4 will provide a full description for the ideal structure of the uniform Roe algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Our next question concerns minimal points discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall that minimal points in the Stone- ˇCech boundary correspond to maximal ideals in the uniform Roe algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' However as shown in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8, there exist a number of non-minimal points in the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence it would be interesting to explore a practical approach to distinguish minimal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Question 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Given a space (X, d), can we find a practical approach to distinguish minimal points in the Stone- ˇCech boundary ∂βX?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Note from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 that the answer might not be easy even in the elementary case that X = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Our last questions concern the assumption of countably generatedness used in Section 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall that in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8 we prove that the inclusion ιU : I(U) ֒→ ˜I(U) induces an isomorphism in K-theory when the space is coarsely embeddable and U is countably generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We ask the following: Question 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Does the inclusion ιU : I(U) ֒→ ˜I(U) induce an isomorphism in K-theory for coarsely embeddable X without the assumption that U is countably generated?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Also note that even under the assumption of countable generatedness, we are merely able to show that (ιI)∗ : K∗(I ∩ Cu[X]) −→ K∗(I) is injective in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we also pose the following: GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 43 Question 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For an ideal I in C∗ u(X) with U(I) countably generated and X coarsely embeddable, is the map (ιI)∗ : K∗(I ∩ Cu[X]) −→ K∗(I) surjective for ∗ = 0, 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Our final question is designed for Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Recall that the assumption of countably generatedness plays an important role in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='17, which is in turn crucial in the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Hence we ask the following: Question 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let (X, d) be a space and U ⊆ βX be an invariant open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' If ˜I(U) = I(U), can we deduce that X has partial Property A towards βX \\ U?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Ultrafilters Here we collect some basic knowledge on ultrafilters, which is used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The material should be fairly well-known (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', [9, Appendix A], [34, Chapter 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4] or [43, Appendix A]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' While some of the results might not be standard and we have not dug into the reference, we include the proofs for convenience to readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let X be a set and P(X) be its power set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' An ultrafilter on X is a family U ⊆ P(X) satisfying the following: (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1) ∅ � U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2) for A, B ∈ U, then A ∩ B ∈ U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3) for A ∈ U and A ⊆ B, then B ∈ U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4) for any A ⊆ X, either A ∈ U or X \\ A ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For an ultrafilter U on X, we can associate a function ω : P(X) → {0, 1} by setting ω(A) = 1 if and only if A ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It follows from (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1)-(I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4) above that ω is a finitely additive {0, 1}-valued probability measure on P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Conversely for such a function ω on P(X), we can associate a family Uω := {A ⊆ X : ω(A) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is clear that Uω is an ultrafilter on X, and these two procedures are inverse to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Therefore throughout the paper, we slide between these two notions freely without further explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a ∈ X, it is clear that the family {A ∈ P(X) : a ∈ A} is an ultrafilter on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Such an ultrafiler is called principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' An ultrafilter which is not principal is called non-principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' An argument using Zorn’s lemma shows that non-principal ultrafilters always exist whenever X is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The following is well-known (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', [43, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2]): Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let ω be an ultrafilter on a set X, and D ⊆ X with ω(D) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let f : D → Y be a function from D to a compact Hausdorff topological space Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then there exists a unique point y ∈ Y such that for any open neighbourhood U of y, we have ω(f −1(U)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The unique point in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2 is called the ultralimit of f along ω or the ω-limit of f, denoted by limω f or lima→ω f(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' We record the following localisation result: Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let U be an ultrafilter on a set X, and A ⊆ X with A ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then we have: (1) {S ∩ A : S ∈ U} = {S ⊆ A : S ∈ U} is an ultrafilter on A, denoted by UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) U = {S ⊆ X : S ∩ A ∈ UA} = {S ⊆ X : ∃ S′ ∈ UA such that S′ ⊆ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' 44 QIN WANG AND JIAWEN ZHANG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It follows from Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1 that {S ∩ A : S ∈ U} = {S ⊆ A : S ∈ U}, and hence (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='1)-(I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3) hold for UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Concerning (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4): given B ⊆ A, if B ∈ U then B ∈ UA as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' if B � U then X \\ B ∈ U, and hence A \\ B = A ∩ (X \\ B) ∈ UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This is straightforward, hence omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ We can also extend an ultrafilter on a subset to the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The proof is straightforward, hence omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let Y be a subset of a set X, and U0 an ultrafilter on Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Define U := {S ⊆ X : S ∩ Y ∈ U0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then U is an ultrafilter on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The following result provides an approach to combine a family of ultrafilters into a single one: Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let {Xi}i∈I be a family of sets, and Ui be an ultrafilter on Xi for each i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let ω0 be an ultrafilter on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consider the set X := � i∈I Xi and define: U := � � i∈I Ai ⊆ � i∈I Xi : ∃ J ⊆ I with ω0(J) = 1 such that ∀i ∈ J, Ai ∈ Ui � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then U is an ultrafilter on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Firstly, it is clear that ∅ � U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Assume that � i∈I Ai and � i∈I Bi ∈ U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', there exist JA, JB ⊆ I with ω0(JA) = ω0(JB) = 1 such that Ai ∈ Ui for any i ∈ JA and Bi ∈ Ui for any i ∈ JB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Consider (� i∈I Ai) ∩ (� i∈I Bi) = � i∈I(Ai ∩ Bi) and J = JA ∩ JB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then ω0(J) = 1 and for each i ∈ J, Ai and Bi are in Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' This implies that Ai ∩ Bi ∈ Ui, which concludes (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' It is clear that (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='3) holds for U and finally, we consider (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Given A = � i∈I Ai ⊆ X, denote J := {i ∈ I : Ai ∈ Ui}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' If ω0(J) = 1, then it follows that A ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Otherwise, assume that ω0(J) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then we consider X \\ A = � i∈I(Xi \\ Ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Then I \\ J = {i ∈ I : Xi \\ Ai ∈ Ui} and ω0(I \\ J) = 1, which implies that X \\ A ∈ U and concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' □ Recall that ultrafilters can also be characterised by the Stone- ˇCech compactifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' More precisely, we have the following (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', [34, Chapter 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4]): Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Let X be a set and βX be the Stone- ˇCech compactification of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (1) Given ω ∈ βX, the family {A ⊆ X : ω ∈ A} is an ultrafilter on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (2) Given an ultrafilter U on X, the intersection � A∈U A consists of a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The procedures above are inverse to each other, and hence βX can be characterised by ultrafilters on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Moreover, points in ∂βX correspond to non-principal ultrafilters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Thanks to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7, we will also use ultrafilters and points in the Stone- ˇCech compactification freely without further explanation throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For convenience, we also record that for D ⊆ X, its closure D in βX satisfies: D = {ω ∈ βX : ω(D) = 1} and D is clopen in βX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The topology of βX is generated by {D : D ⊆ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS 45 Finally we recall the following, which can be proved either directly using the universal property of the Stone- ˇCech compactification or deduced directly from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='4, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='5 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='7: Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' For a subset Z ⊆ X, the closure Z in βX is homeomorphic to βZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' References [1] Claire Anantharaman-Delaroche and Jean Renault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Amenable groupoids, volume 36 of Mono- graphies de L’Enseignement Math´ematique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' L’Enseignement Math´ematique, Geneva, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' With a foreword by Georges Skandalis and Appendix B by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Germain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' [2] Florent P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Baudier, Bruno M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Braga, Ilijas Farah, Ana Khukhro, Alessandro Vignati, and Rufus Willett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Uniform Roe algebras of uniformly locally finite metric spaces are rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', 230(3):1071–1100, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' [3] Paul Baum and Alain Connes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Geometric K-theory for Lie groups and foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Enseign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', 46(1/2):3–42, 2000 (firstly circulated in 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' [4] Paul Baum, Alain Connes, and Nigel Higson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Classifying space for proper actions and K- theory of group C∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Contemporary Mathematics, 167:241–241, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' [5] Bruno M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Braga, Yeong Chyuan Chung, and Kang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Coarse Baum-Connes conjecture and rigidity for Roe algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Funct.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Higher index theory for certain expanders and Gromov monster groups, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', 229(3):1380–1416, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' [47] Guoliang Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' The coarse Baum-Connes conjecture for spaces which admit a uniform embed- ding into Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=', 139(1):201–240, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Wang) Research Center for Operator Algebras, and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, School of Mathematical Sciences, East China Nor- mal University, Shanghai, 200241, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Email address: qwang@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='ecnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='cn (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Zhang) School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai, 200433, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content=' Email address: jiawenzhang@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQfJgxB/content/2301.04921v1.pdf'} diff --git a/KdE0T4oBgHgl3EQfigFl/content/tmp_files/2301.02446v1.pdf.txt b/KdE0T4oBgHgl3EQfigFl/content/tmp_files/2301.02446v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cca759d75c675d1cf30f6c753d4a6bb34e6faad4 --- /dev/null +++ b/KdE0T4oBgHgl3EQfigFl/content/tmp_files/2301.02446v1.pdf.txt @@ -0,0 +1,10946 @@ +Optimal Scaling Results for a Wide Class of Proximal MALA +Algorithms +Francesca R. Crucinio∗1, Alain Durmus2, Pablo Jim´enez3, and Gareth O. Roberts4 +1CREST, ENSAE Paris +2Centre de Math´ematiques Appliqu´ees, Ecole Polytechnique, France, Institut +Polytechnique de Paris +3Sorbonne Universit´e and Universit´e Paris Cit´e, CNRS, Laboratoire de +Probabilit´es, Statistique et Mod´elisation, F-75005 Paris, France +4Department of Statistics, University of Warwick +Abstract +We consider a recently proposed class of MCMC methods which uses proximity maps in- +stead of gradients to build proposal mechanisms which can be employed for both differentiable +and non-differentiable targets. These methods have been shown to be stable for a wide class +of targets, making them a valuable alternative to Metropolis-adjusted Langevin algorithms +(MALA); and have found wide application in imaging contexts. The wider stability properties +are obtained by building the Moreau-Yoshida envelope for the target of interest, which depends +on a parameter λ. In this work, we investigate the optimal scaling problem for this class of +algorithms, which encompasses MALA, and provide practical guidelines for the implementation +of these methods. +Contents +1 +Introduction +2 +2 +Proximal MALA Algorithms +5 +3 +Optimal scaling of Proximal MALA +7 +3.1 +Differentiable targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3.2 +Laplace target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +4 +Practical Implications and Numerical Simulations +13 +4.1 +Numerical Experiments +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +5 +Discussion +15 +∗Corresponding author: francesca.crucinio@gmail.com +1 +arXiv:2301.02446v1 [stat.CO] 6 Jan 2023 + +6 +Proof of the Result for the Laplace distribution +16 +6.1 +Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +6.2 +Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +6.3 +Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +6.4 +Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +A Proof of Theorem 1 +38 +A.1 Auxiliary Results for the Proof of Case (a) . . . . . . . . . . . . . . . . . . . . . . . . +39 +A.2 Auxiliary Results for the Proof of Case (b) +. . . . . . . . . . . . . . . . . . . . . . . +47 +A.3 Auxiliary Results for the Proof of Case (c) . . . . . . . . . . . . . . . . . . . . . . . . +51 +A.4 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +54 +B Numerical Experiments +54 +B.1 +Differentiable Targets +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +54 +B.2 +Laplace Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +55 +C Taylor Expansions for the Results on Differentiable Targets +63 +C.1 Coefficients of the Taylor Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . +63 +C.1.1 +Case (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +63 +C.1.2 +Case (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +63 +C.1.3 +Case (c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +64 +C.2 +Taylor Expansions of the Log-acceptance Ratio . . . . . . . . . . . . . . . . . . . . . +66 +C.2.1 +R1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +66 +C.2.2 +R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +67 +C.3 Derivatives of the Proximity Map for Differentiable Targets . . . . . . . . . . . . . . +71 +D Moments and Integrals for the Laplace Distribution +73 +D.1 Moments of Acceptance Ratio for the Laplace Distribution +. . . . . . . . . . . . . . +73 +D.2 Bound on Second Moment of Acceptance Ratio for the Laplace Distribution . . . . . +79 +D.3 Additional Integrals for the Laplace Distribution . . . . . . . . . . . . . . . . . . . . +83 +D.4 Integrals for Moment Computations +. . . . . . . . . . . . . . . . . . . . . . . . . . . +84 +D.4.1 +First Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +84 +D.4.2 +Second Moment +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +87 +D.4.3 +Third Moment +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +88 +1 +Introduction +Gradient-based Markov chain Monte Carlo methods have proved to be very successful at sampling +from high-dimensional target distributions [9]. +The key to their success is that in many cases +their mixing time appears to be better than their competitor algorithms which do not use gradient +information (see for example [34]), while their implementation has similar computational cost. +Indeed, gradients of target densities can often be computed with computational complexity (in +dimension d) which scales no worse than evaluation of the target density itself. +Gradient-based MCMC methods are mainly motivated from stochastic processes constructed +to have the target density as limiting distribution [25, 8, 6, 44]. Our analysis will concentrate on +2 + +Metropolis Adjusted Langevin Algorithm (MALA) and its proximal variants which are based on +the Langevin diffusion +dLt = dBt + ∇ log π(Lt) +2 +dt , +(1) +where π denotes the target density with respect to the Lebesgue measure. It is well-known that un- +der appropriate conditions, (1) defines a continuous-time Markov process associated with a Markov +semigroup which is reversible with respect to π. From this observation, it has been suggested to +use a Euler-Maruyama (EM) approximation of (1). This scheme has been popularized in statistics +by [20] and referred to as the Unadjusted Langevin Algorithm (ULA) in [36]. Due to the time- +discretization, ULA does not have π as stationary distribution. To address this problem, [39] and +independently Besag in his contribution to [20] proposed to add a Metropolis acceptance step at +each iteration of the EM scheme, leading to the Metropolis Adjusted Langevin Algorithm (MALA) +following [36] who also derive basic stability analysis. The accept/reject step in this algorithm +confers two significant advantages: it ensures that the resulting algorithm has exactly the correct +invariant distribution, while step sizes can be chosen larger than in the unadjusted case as there +is not need to make step size small to reduce discretization error. +On the other hand, MALA +algorithms are typically hard to analyze theoretically (see e.g. [7, 13, 16]). However, [34] (see also +[5, 32]) have established that MALA has better convergence properties than the Random Walk +Metropolis (RWM) algorithm with respect to the dimension d from an optimal scaling perspective +(see also [33]). +Whereas gradient-based methods have been successively applied and offer interesting features, +they are typically less robust than their vanilla alternatives (for example see [36]) while intuition +suggests, and existing underpinning theory requires, that target densities need to be sufficiently +smooth for the gradients to be aiding Markov chain convergence. Moreover, while gradient-based +MCMC have been successful for smooth densities, there is no reason to believe that they should be +effective for densities which are not differentiable at a subset D ⊆ Rd. For non-smooth densities, +[30] proposes modified gradient-based algorithms. Their proposed P-MALA algorithm is inspired +by the proximal algorithms popular in the optimization literature (e.g. [29]). The main idea is +to approximate the (possibly non differentiable but) log-concave target density π ∝ exp(−G) by +substituting the potential G with its Moreau-Yoshida envelope Gλ (see (3) below for its definition), +to obtain a distribution πλ whose level of smoothness is controlled by the proximal parameter λ > 0. +Given this smooth approximation to π one can then build proposals based on time discretizations +of the Langevin diffusion targeting πλ [30, 14]: +ξk+1 = ξk − σ2 +2 ∇Gλ(ξk) + σZk+1 , +(2) +where σ2 > 0 is a fixed stepsize and (Zk)k∈N∗ is a sequence of i.i.d. zero-mean Gaussian random +variables with identity covariance matrix. Our aims in this paper are broadly to provide theo- +retical underpinning for a slightly larger family of proximal MALA algorithms, analyze how these +methods scale with dimension, and to give insights and practical guidance into how they should be +implemented supported by the theory we establish. +Proximal optimization and MCMC methods proved to be particularly well-suited for image +estimation, where penalties involving the sparsity inducing norms are common [30, 14, 43]. Similar +targets are also common in sparse regression contexts [2, 19, 46]. In these situations, the set of +non-differentiability points for the target density π is a null set for the Lebesgue measure, and, +3 + +following [12], we shall focus on this case. However, in contrast to the conclusions of [12] for RWM, +we shall demonstrate that optimal scaling of proximal MALA may be affected by non-smoothness. +More precisely, in this work, we first extend the results of [31] and consider a wider range of +proximal MALA algorithms, as well as a wider class of finite dimensional target distributions. We +let both the steps size σ2 and the regularization parameter λ depend on the dimension d of the +target and find that the scaling properties of proximal MALA depend on the relative speed at which +λ and σ converge to 0 as d → ∞. We start by considering a class of sufficiently differentiable target +distributions π to which MALA can also be applied, to allow direct comparison between MALA +and proximal MALA and thus between a gradient-based method and one which approximates the +gradient through proximal operators. When λ goes to 0 at least as fast as σ2, we find that the +scaling properties of proximal MALA are equivalent to those of MALA (i.e. σ2 should decay as +d−1/3; see Theorem 1–(b), Theorem 1–(c) and Theorem 2); when λ converges to 0 more slowly than +σ2, proximal MALA is less efficient than MALA with σ2 decaying as d−1/2 (Theorem 1–(a)). As +in some cases the proximal operator for a given distribution π is cheaper to compute than ∇ log π +[29, 11, 30], we anticipate that proximal MALA with an appropriately tuned λ might provide a +cheaper alternative to MALA retaining similar scaling properties. +We then turn to the optimal scaling of proximal MALA applied to the Laplace distribution +π(x) ∝ e−|x|. We focus on this particular non-smooth target since it is the most widely used in +applications of proximal MALA, including image deconvolution [30, 14, 43], LASSO, and sparse +regression [2, 19, 46]. We establish that non-differentiability of the target even at one point leads +to a different optimal scaling than MALA. In particular, the step size has to scale as d−2/3 and not +as d−1/3. +This appears to be a new optimal scaling for Metropolis MCMC algorithms which is between +the one of RWM and MALA. From the conclusion for smooth target distributions, we restrict our +study to the choice of λ going to 0 at least as fast as σ2. +The proof of the result for the differentiable case extends that of [34] for MALA, while the +structure of the proof for the Laplace target is similar to that of [12] and constitutes the main +element of novelty in this paper. As a special case of the result for the Laplace distribution, we also +obtain the optimal scaling for MALA on Laplace targets. We point out that the strategy adopted +in the proof of this result is not unique to the Laplace distribution, and could be applied to other +distributions provided that the required integrals can be obtained. +To sum up, our main contributions are: +1) We extend the result of [31] beyond the Gaussian case, covering all finite dimensional (suffi- +ciently) differentiable targets, and show that, in some cases, proximal MALA affords the same +scaling properties of MALA if the proximal parameter λ is chosen appropriately. +2) Motivated by applications in imaging and sparse regression applications, we study the scaling of +proximal MALA methods for the Laplace target, and show that for values of λ decaying sufficiently +fast, the optimal scaling of proximal MALA, i.e. the choice for σ2, is different from the one for +MALA on differentiable targets and is of order d−2/3. +3) We use the insights obtained with the aforementioned results to provide practical guidelines for +the selection of the proximal parameter λ. +Organization of the paper +The paper is structured as follows. In Section 2, we rigorously +introduce the class of proximal MALA algorithms that are studied and discuss related works on +optimal scaling for MCMC algorithms. In Section 3.1 we state the main result for differentiable +targets, showing that the scaling properties of proximal MALA depend on the relative speed at +4 + +which λ goes to 0 with respect to σ. In Section 3.2 we obtain a scaling limit for proximal MALA +when π is a Laplace distribution, as a special case of our result we also obtain the scaling properties +of a sub-gradient version of MALA for this target. +We collect in Section 4 the main practical +takeaways from these results and discuss possible extensions in Section 5. Finally, in Section 6 we +prove the result for the Laplace distribution. The proof of the result for differentiable targets is +postponed to Appendix A. +2 +Proximal MALA Algorithms +We now introduce the general class of proximal MALA algorithms, first studied in [30]. This class +of algorithms aims at sampling from a density with respect to the Lebesgue measure on Rd of the +form π(x) = exp(−G(x))/ +� +Rd exp(−G(˜x))d˜x, with G satisfying the following assumption +A0. The function G : Rd → R is convex, proper and lower semi-continuous. +The main idea behind proximal MALA is to approximate the (possibly non differentiable) target +density π by approximating the potential G with its Moreau-Yoshida envelope Gλ : Rd → R defined +for λ > 0 by +Gλ(x) = min +u∈Rd[G(u) + ∥u − x∥2/(2λ)] . +(3) +Since G is supposed to be convex, by [38, Theorem 2.26], the Moreau-Yoshida envelope is well- +defined, convex and continuously differentiable with +∇Gλ(x) = λ−1(x − proxλ +G(x)) , +proxλ +G(x) = arg min +u∈Rd[G(u) + ∥u − x∥2/(2λ)] . +(4) +The proximity operator x �→ proxλ +G(x) behaves similarly to a gradient mapping and moves points +in the direction of the minimizers of G. In the limit λ → 0 the quadratic penalty dominates (4) +and the proximity operator coincides with the identity operator, i.e. proxλ +G(x) = x; in the limit +λ → ∞, the quadratic penalty term vanishes and (4) maps all points to the set of minimizers of G. +It was shown in [14, Proposition 1] that, under A0, +� +Rd exp(−Gλ(x))dx < ∞, and therefore +the probability density πλ ∝ exp(−Gλ) is well-defined. In addition, it has been shown that ∥π − +πλ∥TV → 0 as λ → 0. Based on this observation and since as we have emphasized πλ is now +continuously differentiable, it has been suggested in [30, 14] to use the discretization of the Langevin +diffusion associated with πλ given by (2), which can be rewritten using (4) as +ξk+1 = +� +1 − σ2 +2λ +� +ξk + σ2 +2λ proxλ +G(ξk) + σZk+1 . +(5) +Similarly to other MCMC methods based on discretizations of the Langevin diffusion (e.g. +[36]), one can build unadjusted schemes which target πλ, expecting draws from these schemes to +be close to draws from π for small enough λ, or add a Metropolis-Hastings step to ensure that the +resulting algorithm targets π. Unadjusted proximal MCMC methods have been analyzed in [14]; +in this paper we focus on Metropolis adjusted proximal MCMC methods and study their scaling +properties. More precisely, at each step k and given the current state of the Markov chain Xk, +a candidate Yk+1 is generated from the transition density associated to (5), (x, y) �→ q(x, y) = +ϕ(y; [1−σ2/(2λ)]x+σ2 proxλ +G(x)/2λ, σ2 Id), where ϕ(· ; u, Σ) stands for the d-dimension Gaussian +density with mean u and covariance matrix Σ. Given Xk and Yk+1, Then, the next state is set as: +Xk+1 = Yk+1bk+1 + Xk(1 − bk+1) , bk+1 = 1R+ +�π(Yk+1)q(Yk+1, Xk) +π(Xk)q(Xk, Yk+1) ∧ 1 − Uk+1 +� +, +(6) +5 + +where (Ui)i∈N∗ is a sequence of i.i.d. uniform random variables on [0, 1]. +The value of λ characterizes how close the distribution πλ is to the original target π and therefore +how good the proposal is. Small values of λ provide better approximations to π and therefore better +proposals (see [14, Proposition 1]), while larger values of λ provide higher levels of smoothing for +non-differentiable distributions (see [30, Figure 1]). In the case λ = σ2/2 we obtain the special case +of proximal MALA referred to as P-MALA in [30]. +The main contribution of this paper is to analyze the optimal scaling for proximal MALA defined +by (6). +Optimal scaling and related works +We briefly summarize here some examples of MCMC +algorithms and their optimal scaling results; a full review is out of the scope of this paper and +we only mention algorithms to which we will compare proximal MALA in the development of this +work. +Popular examples of Metropolis MCMC are RWM and MALA. RWM uses as a proposal the +transition density (x, y) �→ ϕ(y ; x, σ2 Id), where σ2 > 0. The MALA scheme uses as proposal +(x, y) �→ ϕ(y ; x + (σ2/2)∇ log π(x), σ2 Id). As we will show in Section 3.1, proximal MALA can +be considered as an extension of MALA. +A natural question to address when implementing Metropolis adjusted algorithms is how to set +the parameter σ2 (variance parameter for RWM, step size parameter for MALA) to maximize the +efficiency of the algorithm. Small values of σ2 result in higher acceptance probability and cause +sticky behaviour, while large values of σ2 result in a high number of rejections with the chain +(Xk)k≥0 moving slowly [35]. Optimal scaling studies aim to address this question by investigating +how σ2 should behave with respect to the dimension d of the support of π in the high dimensional +setting d → ∞, to obtain the best compromise. +The standard optimal scaling set-up considers the case of d-dimensional targets πd which are +product form, i.e. +πd(xd) = +d +� +i=1 +π(xd +i ) , +(7) +where xd +i stands for the i-th component of xd and π is a one-dimensional probability density with +respect to the Lebesgue measure. +Under appropriate assumptions on the regularity of π, and +assuming that the MCMC algorithm is initialized at stationarity, the optimal value of σ2 scales as +ℓ/dα with ℓ > 0, α = 1 for RWM [33] and α = 1/3 for MALA [34]. +By setting α to these values, it is then possible to show that each as d → ∞ each 1-dimensional +component of the Markov chain defined by RWM and MALA, appropriately rescaled in time, +converges to the Langevin diffusion +dLt = h(ℓ)1/2dBt − h(ℓ) +2 [log π]′(x)dt , +where (Bt)t≥0 is a standard Brownian motion and h(ℓ), referred to as speed function of the diffusion, +is a function of the parameter ℓ > 0 that we may tune. Indeed, it is well-known that (Lh(ℓ)t)t≥0 is +a solution of the Langevin diffusion (1). As a result, we may identify the values of ℓ maximizing +h(ℓ) for the algorithms at hand to approximate the fastest version of the Langevin diffusion. The +optimal values for ℓ results in an optimal average acceptance probability of 0.234 for RWM and +0.574 for MALA. +6 + +The scaling properties allow to get an intuition of the efficiency of the corresponding algorithms: +RWM requires O(d) steps to achieve convergence on a d-dimensional target, i.e. its efficiency is +O(d−1), while MALA has efficiency O(d−1/3). While these results are asymptotic in d, the insights +obtained by considering the limit case d → ∞ prove to be useful in practice [35]. +In the context of non-smooth and even discontinuous target distributions, studying the simpler +RWM algorithm applied to a class of distributions on compact intervals, [27, 28] show that the lack +of smoothness effects the optimal scaling of RWM with respect to dimension d. More precisely, +they show that for a class of discontinuous densities which includes the uniform distribution on +[0, 1], the optimal scaling of RWM is of order O(d−2). On the other hand, in the case where the set +of non-differentiability D of π is a null set with respect to the Lebesgue measure, [12] shows that +under appropriate conditions, including Lp differentiability, the optimal scaling of RWM is of order +O(d−1) still. +The scaling properties of proximal MALA have been partially investigated in [31], which shows +that P-MALA, obtained when λ = σ2/2, has the same scaling properties of MALA for the finite +dimensional Gaussian density and for a class of infinite dimensional target measures (Theorem 2.1 +and Theorem 5.1 therein, respectively). +3 +Optimal scaling of Proximal MALA +We consider the same set up of [34] and briefly recalled above. +Given a real-valued function +g : R → R satisfying A0 we consider the i.i.d. d-dimensional target specified by (7) with +π(x) ∝ exp(−g(x)) . +(8) +Since for any xd, G(xd) = �d +i=1 g(xd +i ), we have by [29, Section 2.1] +proxλ +G(xd) = (proxλ +g(xd +1), . . . , proxλ +g(xd +d))⊤ . +It follows that the distribution of the proposal (10) with target πd in (7)-(8) is also product form +qd(xd, yd) = �d +i=1 q(xd +i , yd +i ) , +q(xd +i , yd +i ) = +1 +(2πσ2)1/2 exp +� +−(yd +i −xd +i +σ2g′[proxλ +g (xd +i )]/2) +2 +2σ2 +� +, +with λ > 0. For any dimension d ∈ N∗, we denote by (Xd +k)k∈N the Markov chain defined by the +Metropolis recursion (6) with target distribution πd and proposal density qd and associated to the +sequence of candidate moves +Y d +k+1 = +� +1 − σ2 +2λ +� +Xd +k + σ2 +2λ proxλ +G(Xd +k) + σZd +k+1 . +(9) +As mentioned in the introduction, the focus of this work is on investigating the optimal depen- +dence of the proposal variance σ2 on the dimension d of the target π. In this section, we make the +dependence of the proposal variance on the dimension explicit and let σ2 +d = ℓ2/d2α and λd = c2/2d2β +for some α, β > 0 and some constants c, ℓ independent on d. Thus, we can write λd as a function of +σd, λd = σ2m +d r/2, where we defined r = c2/ℓ2m > 0 and m = β/α. By writing λd as a function of +σd we can decouple the effect of the constants c, ℓ from that of the dependence on d (i.e. α, β). The +value of m controls the relative speed at which σd and λd converge to 0 as d → ∞, when m = 1, +7 + +σd and λd decay to 0 at the same rate, for m > 1 the decay of λd is faster than that of σd and for +m < 1 the decay of λd is slower than that of σd. The parameter r allows to refine the comparison +between σd and λd as β = α. In the limit r → 0 (i.e. λd/σd → 0 if α = β), the proposal (10) +coincides with that of MALA, in the case m = 1, r = 1 we get the P-MALA algorithm studied +in [30, 31]. Note that for all other values of r, m we have a family of proposals whose behaviour +depends on r and m. +3.1 +Differentiable targets +We start with the case where π is continuously differentiable. Since MALA can be applied to this +class of targets, the results obtained in this section allow direct comparison of proximal MALA +algorithms with MALA and thus between gradient-based algorithms (MALA) and algorithms that +use proximal operator-based approximations of the gradient (proximal MALA). If G = − log π is +continuously differentiable, using [3, Corollary 17.6], proxλ +G(x) = −λ∇G(proxλ +G(x)) + x, and (5) +reduces to +ξk+1 = ξk − σ2 +2 ∇G(proxλ +G(ξk)) + σZk+1 . +(10) +Hence, the value of λ controls how close to ξk is the point at which the gradient is evaluated. +For λ → 0, the proximal MALA proposal becomes arbitrarily close to that of MALA, while, as λ +increases (10) moves away from MALA. +Our main result, Theorem 1 below, shows that the relative speed of decay (i.e. m) influences +the optimal scaling of the resulting proximal MALA algorithm, while the constant r influences the +speed function of the limiting diffusion. +We make the following assumptions on the regularity of g. +A1. g is a C8-function whose derivatives are bounded by some polynomial: there exists k0 ∈ N +such that +sup +x∈R +max +i∈{0,...,8}[g(i)(x)/(1 + |x|k0)] < ∞ . +Note that under A0 and A1, [14, Lemma A.1] implies that +� +R xk exp(−g(x))dx < ∞ for any +k ∈ N. We also assume that the sequence of proximal MALA algorithms is initialized at stationarity. +A2. For any d ∈ N∗, Xd +0 has distribution πd. +The assumptions above closely resemble those of [34] used to obtain the optimal scaling results +for MALA. In particular, A1 ensures that we can approximate the log-acceptance ratio in (6) with +a Taylor expansion, while A2 avoids technical complications due to the transient phase of the +algorithm. We discuss how the latter assumption could be relaxed in Section 5. +For technical reasons, and to allow direct comparisons with the results established in [34] for +MALA, we will also consider the following regularity assumption +A3. The function g′ is Lipschitz continuous. +We denote by Ld +t the linear interpolation of the first component of the discrete time Markov +chain (Xd +k)k≥0 obtained with the generic proximal MALA algorithm described above +Ld +t = (⌈d2αt⌉ − d2αt)Xd +⌊d2αt⌋,1 + (d2αt − ⌊d2αt⌋)Xd +⌈d2αt⌉,1 , +(11) +8 + +where ⌊·⌋ and ⌈·⌉ denote the lower and upper integer part functions, respectively, and denote by +Xd +k,1 the first component of Xd +k. The following result shows that in the limit d → ∞ the properties +of proximal MALA depend on the relative speed at which σ2 +d = ℓ2/d2α and λd = c2/2d2β converge +to 0. Recall that we set r = c2/ℓ2m > 0 and under A2, consider for any d ∈ N∗, +ad(ℓ, r) = E +� πd(Y d +1 )qd(Y d +1 , Xd +0) +πd(Xd +0)qd(Xd +0, Y d +1 ) ∧ 1 +� +. +(12) +Theorem 1. Assume A0, A1 and A2. For any d ∈ N∗, let σ2 +d = ℓ2/d2α and λd = c2/2d2β with +α, β > 0. Then, the following statements hold. +(a) If α = 1/4, β = 1/8 and r > 0, we have limd→+∞ ad(ℓ, r) = 2Φ +� +−ℓ2K1(r)/2 +� +, where Φ is the +distribution function of a standard normal and +K2 +1(r) = r2 +4 E +�� +g′′(Xd +0,1)g′(Xd +0,1) +�2� +. +If in addition, A3 holds. +(b) If α = 1/6, β = 1/6 and r > 0, we have limd→+∞ ad(ℓ, r) = 2Φ +� +−ℓ3K2(r)/2 +� +, where Φ is the +distribution function of a standard normal and +K2 +2(r) = +�r +8 + r2 +4 +� +E +� +g′′(Xd +0,1)g′(Xd +0,1) +2� ++ +� 1 +16 + r +8 +� +E +� +g′′(Xd +0,1)3� ++ 5 +48E +� +g′′′(Xd +0,1)2� +. +(c) If α = 1/6, β > 1/6 and r > 0, we have limd→+∞ ad(ℓ, r) = 2Φ +� +−ℓ3K2(0)/2 +� +, where Φ is the +distribution function of a standard normal. +In addition, in all these cases, as d → ∞ the process (Ld +t )t≥0 converges weakly to the Langevin +diffusion +dLt = h(ℓ, r)1/2dBt − h(ℓ, r) +2 +g′(x)dt , +(13) +where (Bt)t≥0 denotes standard Brownian motion and h(ℓ, r) = ℓ2a(ℓ, r) is the speed of the diffusion, +setting a(ℓ, r) = limd→∞ ad(ℓ, r). If α = 1/4, β = 1/8, for any r > 0, ℓ �→ h(ℓ, r) is maximized at +the unique value of ℓ such that a(ℓ, r) = 0.452; while if α = 1/6, β = m/6 with m ≥ 1 and r > 0, +ℓ �→ h(ℓ, r) is maximized at the unique value of ℓ such that a(ℓ, r) = 0.574. +Proof. The proof follows that of [34, Theorem 1, Theorem 2] and is postponed to Appendix A. +The theorem above shows that the relative speed at which λd converges to 0 influences the +scaling of the resulting proximal algorithm. In case (c), m > 1 and λd decays with d at a faster +rate than σ2 +d. This causes the proximity map (4) to collapse onto the identity and therefore the +proposal (10) is arbitrarily close to that of MALA. The resulting scaling limit also coincides with +that of MALA established in [34, Theorem 1, Theorem 2]. +9 + +If λd and σ2 +d decay at the same rate (case (b)), the amount of gradient information provided by +the proximity map is controlled by r. Comparing our result for case (b) with [34, Theorem 1] we +find that +K2 +2(0) = 1 +16E +� +g′′(Xd +0,1)3� ++ 5 +48E +� +g′′′(Xd +0,1)2� += K2 +MALA; +thus, we have +K2 +2(r) = K2 +2(0) + +�r +8 + r2 +4 +� +E +� +g′′(Xd +0,1)2g′(Xd +0,1)2� ++ r +8E +� +g′′(Xd +0,1)3� += K2 +MALA + +�r +8 + r2 +4 +� +E +� +g′′(Xd +0,1)2g′(Xd +0,1)2� ++ r +8E +� +g′′(Xd +0,1)3� +≥ K2 +MALA , +since the convexity of g implies that g′′ ≥ 0. In particular, K2 +2(r) is an increasing function of r +achieving its minimum when r → 0 (i.e. MALA), see Figure 1(a). +In case (a), m = 1/2 and λd decays more slowly than σ2 +d. +As a consequence, the gradient +information provided by the proximity map is smaller than in cases (b)–(c), and the resulting +scaling differs from that of MALA. The value of K2 +1(r) is increasing in r and the speed of the +corresponding diffusion also depends on r (see Figure 1(a) gray lines and Figure 1(b)). +Example 1 (Gaussian target). Take g(x) = x2/2, proxg +λ(x) = x/(1 + λ). In this case, g′ is Lipschitz +continuous and we have K2 +1(r) = r2/4, K2 +2(r) = +� +1 + 4r + 4r2� +/16 and K2 +2(0) = K2 +MALA = 1/16. +The corresponding speeds are given in Figure 1(a). Optimizing for m = 1, r = 0 (MALA) and +m = 1, r = 1 (P-MALA) we obtain +hMALA(ℓ, r) = 1.5639, +hP-MALA(ℓ, r) = 0.7519, +achieved with ℓMALA = 1.6503 and ℓP-MALA = 1.1443, respectively. For Gaussian targets, MALA +is geometrically ergodic [13], and therefore the optimal choice in terms of speed of convergence is +MALA which is obtained for r = 0. The result for r = 1 and m = 1 are also given in [31, Theorem +2.1]. +Example 2 (Target with light tails). Take g(x) = x4, which gives a normalized distribution with +normalizing constant 2Γ(5/4). The proximity map is +proxλ +g(x) = 1 +2 +� +3� +9λ2x + +√ +54λ4x2 + 3λ3 +32/3λ +− +1 +3� +27λ2x + 3 +√ +54λ4x2 + 3λ3 +� +. +In this case g′ is not Lipschitz continuous and therefore we only consider (a), for which we have +K2 +1(r) = 144r2Γ(11/4)/Γ(5/4). The corresponding speed is given in Figure 1(b). +3.2 +Laplace target +As discussed in the introduction, proximal MALA has been widely used to quantify uncertainty in +imaging applications, in which target distributions involving the ℓ1 norm are particularly common +[30, 14, 1, 46]. +Here, we consider πL +d to be the product of d i.i.d. Laplace distributions as in (7), +πL +d (xd) = +d +� +i=1 +πL(xd +i ), for xd ∈ Rd, where πL(x) = 2−1 exp(−|x|) . +(14) +10 + +r +Speed +(a) Gaussian target +r +Speed +(b) Light tail target +Figure 1: Value of K for i = 1, 2 and speed of the corresponding Langevin diffusion as a function +of r for a Gaussian target and a light tail target. We denote by h1 the speed obtained in case (a), +by h2 that obtained in (b). In case (c) both K3 and the speed h3 are constant w.r.t. r and coincide +with that of MALA. For the Gaussian target we report the results for case (a)–(c) while for the +light tail target we only report (a). +For this particular choice of one-dimensional target distribution, the corresponding potential G is +x �→ |x| and satisfies A0. Then, the proximity map is given by the soft thresholding operator [29, +Section 6.1.3] +proxλ +G(x) = (x − sgn(x)λ)1{|x| > λ} , +(15) +where sgn : R → {−1, 1} is the sign function, given by sgn(x) = −1 if x ≤ 0, and sgn(x) = +1 otherwise. This operator is a continuous but not continuously differentiable map whose non- +differentiability points are the extrema of the interval [−λ, λ] and are controlled by the value of the +proximity parameter λ. +Plugging (15) in (9), the proximal MALA algorithm applied to πL +d proposes component-wise +for i = 1, . . . , d +Y d +k+1,i = Xd +k,i − σ2 +d +2 sgn(Xd +k,i)1{|Xd +k,i| > λd} − σ2 +d +2λd +Xd +k,i1{|Xd +k,i| ≤ λd} + σdZd +k+1,i . +(16) +For Xd +k,i close to 0 (i.e. the point of non-differentiability) the proximal MALA proposal is a biased +random walk around Xd +k,i, while outside the region [−λd, λd] the proposal coincides with that of +MALA. As λd → 0 the region in which the proximal MALA proposal coincide with that of MALA +increases and when λd ≈ 0 the region [−λd, λd] in which the proposal corresponds to a biased +random walk is negligible, as confirmed by the asymptotic acceptance rate in Theorem 2. +We also consider the case λd = 0 for any d. Then, the proposal (16) becomes the proposal +for the subgradient version of MALA: Y d +k+1,i = Xd +k,i − (σ2 +d/2) sgn(Xd +k,i) + σdZd +k+1,i, referred to as +sG-MALA. +The proof of the optimal scaling for the Laplace distribution follows the structure of that of +[12] for Lp-mean differentiable distributions. We start by characterizing the asymptotic acceptance +11 + +ratio of a generic proximal MALA algorithm; contrary to Theorem 1 for differentiable targets, in +the limit d → ∞ the properties of proximal MALA do not depend on the relative speed at which +σ2 +d = ℓ2/d2α and λd = c2/2d2β converge to 0, as long as λd decays at least at the same rate as σ2 +d. +In this regime, the region in which the proposal (16) corresponds to a biased random walk proposal +is negligible, and therefore we obtain the same scaling obtained with λd = 0 and corresponding to +sG-MALA. +Theorem 2. Assume A2 and consider the sequence of target distributions {πL +d }d∈N∗ given in (14). +For any d ∈ N∗, let σ2 +d = ℓ2/d2α and λd = c2/2d2β with α = 1/3 and β = m/3 for m ≥ 1. Then, we +have limd→∞ ad(ℓ, r) = aL(ℓ) = 2Φ(−ℓ3/2/(72π)1/4), where (ad(ℓ, r))d∈N∗ is defined in (12), with +r = c2/ℓ2m, and Φ is the distribution function of a standard normal. +Proof. The proof is postponed to Section 6.1. +Note that the asymptotic average acceptance rate aL(ℓ) does not depend on r and as a result +on c. +Having identified the possible scaling for proximal MALA with Laplace target, we are now ready +to show weak convergence to the appropriate Langevin diffusion. To this end, we adapt the proof +strategy followed in [22] and [12]. +As for the differentiable case, consider the linear interpolation (Ld +t )t≥0 of the first component +of the Markov chain (Xd +k)k≥0 given in (11). For any d ∈ N∗, denote by νd the law of the process +(Ld +t )t≥0 on the space of continuous functions from R+ to R, C(R+, R), endowed with the topology +of uniform convergence over compact sets and its corresponding σ-field. We first show that the +sequence (νd)d∈N∗, admits a weak limit point as d → ∞. +Proposition 1. Assume A2 and consider the sequence of target distributions {πL +d }d∈N∗ given +in (14). For any d ∈ N∗, let σ2 +d = ℓ2/d2α and λd = c2/2d2β with α = 1/3 and β = m/3. The +sequence (νd)d∈N∗ is tight in M1 (C(R+, R)), the set of probability measures acting on C(R+, R). +Proof. See Section 6.2. +By Prokhorov’s theorem, the tightness of (νd)d∈N∗ implies existence of a weak limit point ν. In +our next result, we give a sufficient condition to show that any limit point of (νd)d∈N∗ coincides +with the law of a solution of: +dLt = [hL(ℓ)]1/2dBt − hL(ℓ) +2 +sgn(Lt)dt . +(17) +To this end, we consider the martingale problem (see [42]) associated with (17), that we now +present. Let us denote by C∞ +c (R, R) the subset of functions of C(R, R) which are infinitely many +times differentiable and with compact support, and define the generator of (17) for V ∈ C∞ +c (R, R) +by +LV (x) = hL(ℓ) +2 +[V ′′(x) − sgn(x)V ′(x)] . +(18) +Denote by (Wt)t≥0 the canonical process on C(R+, R), Wt : {ws}s≥0 �→ wt and the corresponding +filtration by (Ft)t≥0. A probability measure ν is said to solve the martingale problem associated +12 + +with (17) with initial distribution πL, if the pushforward of ν by W0 is πL and if for all V ∈ +C∞ +c (R, R), the process +� +V (Wt) − V (W0) − +� t +0 +LV (Wu)du +� +t≥0 +is a martingale with respect to ν and the filtration (Ft)t≥0. +The following proposition gives a +sufficient condition to prove that ν is a solution of the martingale problem: +Proposition 2. Suppose that for any V ∈ C∞ +c (R, R), m ∈ N, ρ : Rm → R bounded and continuous, +and for any 0 ≤ t1 ≤ ... ≤ tm ≤ s ≤ t: +lim +d→+∞ Eνd +�� +V (Wt) − V (Ws) − +� t +s +LV (Wu)du +� +ρ(Wt1, ..., Wtm) +� += 0 . +Then any limit point of (νd)d∈N∗ on M1 (C(R+, R)) is a solution to the martingale problem associated +with (17). +Proof. See Section 6.3. +Finally, we use this sufficient condition to establish that any limit point of (νd)d∈N∗ is a solution +of the martingale problem for (17). Uniqueness in law of solutions of (17) allows to conclude that +(Ld +t )t≥0 converges weakly to the Langevin diffusion (17), which establishes our main result. +Theorem 3. The sequence of processes {(Ld +t )t≥0 : d ∈ N∗} converges in distribution towards +(Lt)t≥0, solution of (17) as d → ∞, with hL(ℓ) = ℓ2aL(ℓ) and aL defined in Theorem 2. +In +addition, hL is maximized at the unique value of ℓ such that aL(ℓ) = 0.360. +Proof. See Section 6.4. +4 +Practical Implications and Numerical Simulations +The optimal scaling results in Sections 3.1 and 3.2 provide some guidance on the choice of the +parameters σ and λ of proximal MALA algorithms, suggesting that smaller values of λ provide +better efficiency in terms of number of steps necessary to convergence (Theorem 1). +However, a number of other factors must be taken into account. First, as shown in [26, 37, 36, 21] +the convergence properties of Metropolis adjusted algorithms are influenced by the shape of the +target distribution and, in particular, by its tail behavior. Secondly, when comparing proximal +MALA algorithms with gradient-based methods (e.g. MALA) one must take into account the cost +of obtaining the gradients, whether this comes from automatic differentiation algorithms or from +evaluating a potentially complicated gradient function. On the other hand, proximity mappings can +be quickly found or approximated solving convex optimization problems which have been widely +studied in the convex optimization literature (e.g. [29, Chapter 6], [11] and [30, Section 3.2.3]). +In terms of convergence properties, we are usually interested in the family of distributions for +which the discrete time Markov chain produced by our algorithm is geometrically ergodic, together +with the optimal scaling results briefly recalled in Section 2. Normally, the ergodicity results are +given by considering the one-dimensional class of distributions E(β, γ) introduced in [36] and defined +for γ > 0 and 0 < β < ∞ by +E(β, γ) : +� +π : R → [0, +∞) : π(x) ∝ exp +� +−γ|x|β� +, |x| > x0 for some x0 > 0 +� +. +13 + +As observed by [24], there usually is a trade-off between ergodicity and optimal scaling results, +algorithms providing better optimal scaling results tend to be geometrically ergodic for a smaller +set of targets (e.g. MALA w.r.t. RWM). +As suggested by Theorem 1, the scaling properties of proximal MALA on differentiable targets +are close to those of MALA. This leads to a natural comparison between the two algorithms. First, +we observe that A0 rules out targets for which G is not convex and therefore restricts the families +E(β, γ) to β ≥ 1. To compare MALA with proximal MALA we therefore focus on distributions +with β ≥ 1. +It is shown in [36] that MALA is geometrically ergodic for targets in E(β, γ) with 1 ≤ β ≤ 2 +(with some caveat for β = 2). Theorem 1–(b) and (c) show that in this case proximal MALA +has the same scaling properties of MALA but in case (b) the asymptotic speed of convergence +decays as the constant r increases (Figure 1(a)), with the maximum achieved for r → 0, for which +proximal MALA collapses onto MALA. Since MALA is geometrically ergodic, and achieves better +(or equivalent) scaling properties than proximal MALA, it would be natural to prefer MALA to +proximal MALA for this set of targets. However, if the gradient is costly to obtain, one might +instead consider to use proximal MALA with a small λ, to retain scaling properties as close as +possible to that of MALA but to reduce the computational cost of evaluating the gradient. +In the case of differentiable targets with light-tails (i.e. β > 2), MALA is known not to be +geometrically ergodic [36, Section 4.2] while the ergodicity properties of proximal MALA have only +been partially studied in [30, Section 3.2.2] for the case λ = σ2/2 (P-MALA). As shown in [30, +Section 2.1], given a distribution π ∈ E(β, γ) with β ≥ 1, the distribution πλ obtained using the +potential (3) belongs to E(β′, γ′), where β′ = min(β, 2) and γ′ depending on λ. This suggests that +proximal MALA is likely to be ergodic for appropriate choices of λ; a first result in this direction +is given in [30, Corollary 3.2] for the P-MALA case λ = σ2/2. Theorem 1–(c) restricts the sets of +available λs showing that for light-tail distributions (for which A3 does not hold) λ should decay +at half the speed of σ2. Studying the ergodicity properties of proximal MALA in function of the +parameter λ is, of course, an interesting problem that we leave for future work. +For the Laplace distribution, Theorem 2 shows that the value of λ does not influence the +asymptotic acceptance ratio of proximal MALA, as long as λ decays with d at least as fast as σ2. +The scaling properties and the asymptotic speed h(ℓ) in Theorem 3 do not depend on λ and coincide +with that of the sG-MALA (obtained for λ = 0). Hence, in terms of optimal scaling, there does not +seem to be a difference between proximal MALA and sG-MALA for the Laplace distribution. +4.1 +Numerical Experiments +To illustrate the results established in Section 3.1 and 3.2 we consider here a small collection +of simulation studies. +The aim of these studies is to empirically confirm the optimal scalings +identified in Theorem 1 and 2, investigate the dimension d at which the asymptotic acceptance +ratio limd→∞ ad(ℓ, r) well approximates the empirical average acceptance ratio and, consequently, +for which dimensions d we can expect the optimal asymptotic acceptances in Theorem 1 and 2 to +guarantee maximal speed h(ℓ, r) (approximated by the expected squared jumping distance, see, e.g. +[18]) for the corresponding diffusion. We summarize here our findings, a more detailed discussion +can be found in Appendix B. +For the differentiable case, we consider the Gaussian distribution in Example 1 and four algorith- +mic settings which correspond to the three cases identified in Theorem 1 and MALA. The different +values of r and m influence the dimension required to observe convergence to the theoretical limit +14 + +in Theorem 1: for r → 0 and m = 1 (MALA) and m = 1/2, r = 1 (corresponding to Theorem 1–(a)) +the theoretical limit is already achieved for d of order 102, while in the cases m = 3, r = 2 and +m = r = 1 (corresponding to Theorem 1–(c) and (b), respectively) our simulation result match the +theoretical limit only for d of order 105 or higher. +The results for the Laplace case are similar, with the case m > 1 requiring a higher d to observe +convergence to the theoretical limit. +In general, we find that the optimal average acceptance ratios in Theorem 1 guarantee maximal +speed h(ℓ, r) for d sufficiently large (for small d the optimal acceptance ratio often differs from the +optimal asymptotic one, see, e.g. [40, Section 2.1]). +5 +Discussion +In this work we analyze the scaling properties of a wide class of proximal MALA algorithms intro- +duced in [30, 14] for smooth targets and for the Laplace distribution. We show that the scaling +properties of proximal MALA are influenced by the relative speed at which the proximal parameter +λd and the proposal variance σd decay to 0 as d → ∞ and suggest practical ways to choose λd as a +function of σd to guarantee good results. +In the case of smooth targets, we provide a detailed comparison between proximal MALA and +MALA, showing that proximal MALA scales no better than MALA (Theorem 1). In particular, +Theorem 1–(a) shows that if λd is too large w.r.t. σd then the efficiency of proximal MALA is +of order O(d−1/2) and therefore worse than the O(d−1/3) of MALA, suggesting that λd should +be chosen to decay approximately as σd, if possible. If λd decays sufficiently fast, then MALA +and proximal MALA have similar scaling properties and, in the case in which the proximity map +is cheaper to compute that the gradient, one can build proximal MALA algorithms which are as +efficient as MALA in terms of scaling but more computationally efficient. +In the case of the Laplace distribution, we show that the scaling of proximal MALA is O(d−2/3) +for any λd decaying sufficiently fast w.r.t. σd and, in the limit λd ≈ 0, we obtain a novel optimal +scaling result for MALA on Laplace targets. +As discussed in Section 4, our analysis provides some guidance on the choice of the parameters +that need to be specified to implement proximal MALA, but this analysis should be complemented +by an exploration of the ergodicity properties of proximal MALA to obtain a comprehensive descrip- +tion of the algorithms. We conjecture that for sufficiently large values of λ, proximal MALA applied +to light tail distributions will be exponentially ergodic; establishing exactly how large should λ be +to guarantee fast convergence is an interesting question that we leave for future work. Obtaining +these results would open the doors to adaptive tuning strategies for proximal MALA, which are +likely to produce better results than those given by the strategies currently used. +The set up under which we carried out our analysis closely resembles that of [34]; we anticipate +that A2 could be relaxed following similar ideas as those in [10, 22] and that our analysis could be +extended to d-dimensional targets πd possessing some dependence structure following the approach +of [40, 4, 45]. Finally, the analysis carried out for the Laplace distribution could be extended to +other piecewise smooth distributions provided that the moments necessary for the proof in Section 6 +can be computed. +15 + +6 +Proof of the Result for the Laplace distribution +In this section we prove the results in Section 3.2 which give the scaling properties of proximal +MALA (and sG-MALA) for the Laplace distribution. We collect technical results (e.g. moment +computations, bounds, etc.) in Appendix D. +We recall that σ2 +d = ℓ2/d2α and λd = c2/2d2β for some α, β > 0 and some constants c, ℓ +independent on d. +Thus, we can write λd as a function of σd, λd = σ2m +d r/2, where we define +r = c2/ℓ2m > 0 and m = β/α. +In order to study the scaling limit of proximal MALA with Laplace target, consider the mapping +bd : R2 → R given by +bd : (x, z) �→ z − σd +2 sgn(x)1 +� +|x| > σ2m +d r/2 +� +− +1 +σ2m−1 +d +rx1 +� +|x| ≤ σ2m +d r/2 +� +, +(19) +which allows us to write the proposal as Y d +1,i = Xd +0,i + σdbd(Xd +0,i, Zd +1,i) , for any i ∈ {1, . . . , d}. +We consider also the function φd : R2 → R, given by +φd : (x, z) �→ log π(x + σdbd(x, z))q(x + σdbd(x, z), x) +π(x)q(x, x + σdbd(x, z)) +(20) += |x| − |x + σdbd(x, z)| + z2 +2 +− +1 +2σ2 +d +�σ2 +d +2 sgn [x + σdbd(x, z)] 1 +� +|x + σdbd(x, z)| > σ2m +d r +2 +� +− σdbd(x, z) ++ +1 +σ2(m−1) +d +r +[x + σdbd(x, z)] 1 +� +|x + σdbd(x, z)| ≤ σ2m +d r +2 +��2 +. +We introduce, for i ∈ {1, . . . , d}, φd,i = φd(Xd +0,i, Zd +1,i) for the sake of conciseness. This allows us to +rewrite ad(ℓ, r), defined in (12), in the following way, +ad(ℓ, r) = E +� +exp +� d +� +i=1 +φd,i +� +∧ 1 +� +. +(21) +Remark 1. Under A2, the families of random variables (bd(Xd +0,i, Zd +1,i))i∈{1,...,d} and (φd,i)i∈{1,...,d} +are i.i.d. +6.1 +Proof of Theorem 2 +The proof of Theorem 2 uses the first three moments of φd,1, whose computation is postponed to +Appendix D.1, and is an application of Lindeberg’s central limit theorem. +To identify the optimal scaling for the Laplace distribution, we look for those values of α such +that �d +i=1 E[φd,i] and Var(�d +i=1 φd,i) converge to a finite value. Using Remark 1, we have that, +d +� +i=1 +E [φd,i] = d E [φd,1] +and +Var +� d +� +i=1 +φd,i +� += d Var (φd,1) . +(22) +16 + +Then, using the integrals in Appendix D.1, we find that the only value of α for which (22) converge to +a finite value with the variance strictly positive is α = 1/3 as confirmed empirically in Appendix B.2. +Having identified α = 1/3, we can then proceed applying Lindeberg’s CLT. +Proof of Theorem 2. We start by showing that the acceptance ratio converges to a Gaussian dis- +tribution. Define µd = E[φd,1] and Fd,i = σ((Xd +0,j, Zd +1,j), 1 ≤ j ≤ i), the natural filtration for +(Xd +0,i, Zd +1,i)d∈N,1≤i≤d. The square-integrable martingale sequence +� +�Sd,i = +i +� +j=1 +Wd,i, Fd,i +� +� +d∈N∗,1≤i≤d +where Wd,i = φd,i − µd, forms a triangular array, to which we can apply the corresponding CLT +(e.g. [41, Theorem 4, page 543]). In particular, we have that, +lim +d→∞ +d +� +i=1 +E +� +W 2 +d,i | Fd,i−1 +� += lim +d→∞ d Var (φd,1) = +2ℓ3 +3 +√ +2π , +as shown in Proposition 17 in Appendix D.1. It remains to verify Lindeberg’s condition: for ε > 0, +lim +d→∞ dE +� +W 2 +d,11 {|Wd,1| > ε} +� += 0 . +In order to verify Lindeberg’s condition we verify the stronger Lyapunov condition: there exists +ϵ > 0 such that +lim +d→∞ dE +� +W 2+ϵ +d,1 +� += 0 . +Pick ϵ = 1 and expand the cube using µd = E[φd,i], +E +� +W 3 +d,1 +� += E +� +φ3 +d,i +� +− 3µdE +� +φ2 +d,i +� ++ 2µ3 +d . +(23) +By Proposition 16 in Appendix D.1, we have limd→∞ dµ3 +d = 0, limd→∞ µd = 0, and, by Proposi- +tion 17 in Appendix D.1, +lim +d→∞ dE +� +φ2 +d,i +� += +2ℓ3 +3 +√ +2π . +Finally, for the remaining term in (23) we use Proposition 18 in Appendix D.1 to show that +limd→∞ dE[φ3 +d,i] = 0. The above and the fact that, by Proposition 16 in Appendix D.1, +lim +d→∞ dµd = − +ℓ3 +3 +√ +2π , +show, by Lindeberg’s CLT, that the acceptance ratio converges in law to a normal random variable +�Z with mean −ℓ3/(3 +√ +2π) and variance 2ℓ3/(3 +√ +2π). +To conclude the proof, we plug this convergence into (21), since x �→ ex ∧ 1 is a continuous and +bounded mapping, we have that +lim +d→∞ exp +� d +� +i=1 +φd,i +� +∧ 1 +d= e +� +Z ∧ 1 +and +lim +d→∞ ad(ℓ, r) = E +� +e +� +Z ∧ 1 +� +, +where the limit does not depend on r. Defining aL(ℓ) = limd→∞ ad(ℓ, r) and using [33, Proposition +2.4], we have the result. +17 + +6.2 +Proof of Proposition 1 +We are interested in the law νd of the linear interpolant (Ld +t )t≥0, defined in (11), of the first +component of the chain (Xd +k)k∈N. Let us recall the definition of the chain: assumption A2 gives +the initial distribution πd, then, for any k ∈ N, the proposal Y d +k+1 = (Y d +k+1,i)1≤i≤d is defined in (16) +with σ2 +d = ℓ2/d2α, λd = σ2m +d r/2 with α = 1/3 and m ≥ 1. With the notations introduced in +Section 6, we can rewrite (16), for any i ∈ {1, . . . , d}, +Y d +k+1,i = Xd +k,i + σdbd(Xd +k,i, Zd +k+1,i) , +(24) +where bd is defined in (19) with r = c2/ℓ2m. From there, we apply the acceptance-rejection step +described in (6), we additionally define the acceptance event Ad +k+1 = +� +bd +k+1 = 1 +� +. +We can now expand the expression of the linear interpolant Ld +t , for t ≥ 0, using (6), (11) and +the definition of Ad +k+1, +Ld +t = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Xd +⌊d2αt⌋,1 + (d2αt − ⌊d2αt⌋) +� +σdZd +⌈d2αt⌉,1 − σ2 +d +2 sgn(Xd +⌊d2αt⌋,1) +� +1Ad +⌈d2αt⌉ +if |Xd +⌊d2αt⌋,1| > σ2m +d +r +2 +Xd +⌊d2αt⌋,1 + (d2αt − ⌊d2αt⌋) +� +σdZd +⌈d2αt⌉,1 − +1 +σ2(m−1) +d +rXd +⌊d2αt⌋,1 +� +1Ad +⌈d2αt⌉ +otherwise +, +(25) +or, equivalently, +Ld +t = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Xd +⌈d2αt⌉,1 − (⌈d2αt⌉ − d2αt) +� +σdZd +⌈d2αt⌉,1 − σ2 +d +2 sgn(Xd +⌊d2αt⌋,1) +� +1Ad +⌈d2αt⌉ +if |Xd +⌊d2αt⌋,1| > σ2m +d +r +2 +Xd +⌈d2αt⌉,1 − (⌈d2αt⌉ − d2αt) +� +σdZd +⌈d2αt⌉,1 − +1 +σ2(m−1) +d +rXd +⌊d2αt⌋,1 +� +1Ad +⌈d2αt⌉ +otherwise +. +In order to prove Proposition 1, we consider Kolmogorov’s criterion for tightness (see [23, The- +orem 23.7]): the sequence (νd)d≥1 is tight if +E +� +(Ld +t − Ld +s)4� +≤ γ(t)(t − s)2 , +for some non-decreasing positive function γ, all 0 ≤ s ≤ t and all d ∈ N∗ and the sequence (Ld +0)d∈N∗ +is tight. The latter condition is straightforward to check, since by A2 the distribution of Ld +0 = Xd +0,1 +is πL for all d ∈ N∗. +Proof of Proposition 1. Consider E +� +(Ld +t − Ld +s)4� +, if ⌊d2αs⌋ = ⌊d2αt⌋, the inequality follows straight- +forwardly recalling that the moments of normal distributions are bounded: in the case |Xd +⌊d2αt⌋,1| = +|Xd +⌊d2αs⌋,1| > σ2m +d r/2 it follows directly from the boundedness of the sgn function, while in the case +|Xd +⌊d2αt⌋,1| = |Xd +⌊d2αs⌋,1| ≤ σ2m +d r/2 we exploit the boundedness of Xd +⌊d2αt⌋,1 itself. +For all 0 ≤ s ≤ t such that ⌈d2αs⌉ ≤ ⌊d2αt⌋, we can distinguish three cases. +18 + +Case 1 +If |Xd +⌊d2αt⌋,1| > σ2m +d r/2 and |Xd +⌊d2αs⌋,1| > σ2m +d r/2, then +Ld +t − Ld +s = Xd +⌊d2αt⌋,1 − Xd +⌈d2αs⌉,1 ++ (d2αt − ⌊d2αt⌋) +� +σdZd +⌈d2αt⌉,1 − σ2 +d +2 sgn(Xd +⌊d2αt⌋,1) +� +1Ad +⌈d2αt⌉ ++ (⌈d2αs⌉ − d2αs) +� +σdZd +⌈d2αs⌉,1 − σ2 +d +2 sgn(Xd +⌊d2αs⌋,1) +� +1Ad +⌈d2αs⌉ . +Using H¨older’s inequality and the fact that 0 ≤ d2αt − ⌊d2αt⌋ ≤ 1 (and similarly for s) we have +E +� +(Ld +t − Ld +s)4� +≤ CE +�� +Xd +⌊d2αt⌋,1 − Xd +⌈d2αs⌉,1 +�4� ++ C (d2αt − ⌊d2αt⌋)2 +d4α +E +�� +ℓZd +⌈d2αt⌉,1 +�4 ++ +ℓ4 +24d4α +� ++ C (⌈d2αs⌉ − d2αs)2 +d4α +E +�� +ℓZd +⌈d2αs⌉,1 +�4 ++ +ℓ4 +24d4α +� +. +Recalling that the moments of Zd are bounded and that d2αs ≤ ⌈d2αs⌉ ≤ ⌊d2αt⌋ ≤ d2αt, it follows +E +� +(Ld +t − Ld +s)4� +≤ C +� +(t − s)2 + E +�� +Xd +⌊d2αt⌋,1 − Xd +⌈d2αs⌉,1 +�4�� +. +(26) +Case 2 +If |Xd +⌊d2αt⌋,1| > σ2m +d r/2 and |Xd +⌊d2αs⌋,1| ≤ σ2m +d r/2 or |Xd +⌊d2αt⌋,1| ≤ σ2m +d r/2 and |Xd +⌊d2αs⌋,1| > +σ2m +d r/2. We only describe the argument for the first case, the second case follows from analogous +steps. Take +Ld +t − Ld +s = Xd +⌊d2αt⌋,1 + (d2αt − ⌊d2αt⌋) +� +σdZd +⌈d2αt⌉,1 − σ2 +d +2 sgn(Xd +⌊d2αt⌋,1) +� +1Ad +⌈d2αt⌉ +− Xd +⌈d2αs⌉,1 − (⌈d2αs⌉ − d2αs) +� +σdZd +⌈d2αs⌉,1 − +1 +σ2(m−1) +d +r +Xd +⌊d2αs⌋,1 +� +1Ad +⌈d2αs⌉ . +Proceeding as above, we find that +E +� +(Ld +t − Ld +s)4� +≤ C +� +(t − s)2 + E +�� +Xd +⌊d2αt⌋,1 − Xd +⌈d2αs⌉,1 +�4� ++(⌈d2αs⌉ − d2αs)4E +� +� +� +1 +σ2(m−1) +d +r +Xd +⌊d2αs⌋ +�4� +� +� +� , +and recalling that |Xd +⌊d2αs⌋,1| ≤ σ2m +d r/2 we have that |Xd +⌊d2αs⌋,1|/(rσ2(m−1) +d +) ≤ σ2 +d/2. Using this +and the same arguments as above, we have +E +� +(Ld +t − Ld +s)4� +≤ C +� +(t − s)2 + E +�� +Xd +⌊d2αt⌋,1 − Xd +⌈d2αs⌉,1 +�4�� +. +(27) +19 + +Case 3 +If |Xd +⌊d2αt⌋,1| ≤ σ2m +d r/2 and |Xd +⌊d2αs⌋,1| ≤ σ2m +d r/2, then +Ld +t − Ld +s = Xd +⌊d2αt⌋,1 + (d2αt − ⌊d2αt⌋) +� +σdZd +⌈d2αt⌉,1 − +1 +σ2(m−1) +d +r +Xd +⌊d2αt⌋,1 +� +1Ad +⌈d2αt⌉ +− Xd +⌈d2αs⌉,1 + (⌈d2αs⌉ − d2αs) +� +σdZd +⌈d2αs⌉,1 − +1 +σ2(m−1) +d +r +Xd +⌊d2αs⌋,1 +� +1Ad +⌈d2αs⌉ . +Using the boundedness of moments of Gaussian distributions and of Xd +⌊d2αt⌋,1, Xd +⌊d2αs⌋,1, we have +E +� +(Ld +t − Ld +s)4� +≤ C +� +(t − s)2 + E +�� +Xd +⌊d2αt⌋,1 − Xd +⌈d2αs⌉,1 +�4�� +. +(28) +Putting (26), (27) and (28) together and using Lemma 1 below we obtain +E +�� +Ld +t − Ld +s +�4� +≤ C +� +(t − s)2 + +4 +� +p=2 +� +⌊d2αt⌋ − ⌈d2αs⌉ +�p +d2αp +� +≤ C(t − s)2 + C +4 +� +p=2 +d2αp (t − s)p +d2αp +≤ C +� +2 + t + t2� +(t − s)2 , +which concludes the proof. +We are now ready to state and prove Lemma 1: +Lemma 1. There exists C > 0 such that for any k1, k2 ∈ N with 0 ≤ k1 < k2, +E +�� +Xd +k2,1 − Xd +k1,1 +�4� +≤ C +4 +� +p=2 +(k2 − k1)p +d2αp +. +Proof. Recalling the definition of the proposal in (24) and the notations of (19) we can write +E +�� +Xd +k2,1 − Xd +k1,1 +�4� += E +� +� +� +k2 +� +k=k1+1 +σdbd +� +Xd +k−1,1, Zd +k,1 +� +1Ad +k +�4� +� . +Then, we expand all acceptance or rejection terms between k1 and k2 and use H¨older’s inequality +to obtain +E +�� +Xd +k2,1 − Xd +k1,1 +�4� +≤ σ4 +dE +� +� +� +k2 +� +k=k1+1 +bd +� +Xd +k−1,1, Zd +k,1 +� +�4� +� ++ σ4 +dE +� +� +� +k2 +� +k=k1+1 +bd +� +Xd +k−1,1, Zd +k,1 +� +1(Ad +k)c +�4� +� , +20 + +where bd is defined in (19). Using again H¨older’s inequality, for the first term we have +E +� +� +� +k2 +� +k=k1+1 +bd +� +Xd +k−1,1, Zd +k,1 +� +�4� +� ≤ C +� +� +�E +� +� +� +k2 +� +k=k1+1 +Zd +k,1 +�4� +� ++ σ4 +d +24 E +� +� +� +k2 +� +k=k1+1 +sgn +� +Xd +k−1,1 +� +1 +� +|Xd +k−1,1| > σ2m +d r/2 +� +�4� +� ++σ4 +d +24 E +� +� +� +k2 +� +k=k1+1 +1 +σ2m−1 +d +rXd +k−1,11 +� +|Xd +k−1,1| ≤ σ2m +d r/2 +� +�4� +� +� +� +� +≤ C +� +3(k2 − k1)2 + 2σ4 +d +24 (k2 − k1)4 +� +, +(29) +where the last line follows using the moments of Zd +k,1 and the boundedness of Xd +k−1,1 in the set +{|Xd +k−1,1| ≤ σ2m +d r/2}. +Using a Binomial expansion of the rejection term, we obtain +E +� +� +� +k2 +� +k=k1+1 +bd +� +Xd +k−1,1, Zd +k,1 +� +1(Ad +k) +c +�4� +� = +� +E +� 4 +� +i=1 +bd +� +Xd +mi−1,1, Zd +mi,1 +� +1(Admi) +c +� +, +(30) +where the sum is over the quadruplets (mi)1≤i≤4 with mi ∈ {k1 + 1, . . . , k2}. +We separate the terms in the sum according to their cardinality, let us denote, for j ∈ {1, . . . , 4}, +Ij = +� +(m1, . . . , m4) ∈ {k1 + 1, . . . , k2}4 : # {m1, . . . , m4} = j +� +; +and define, for any (m1, . . . , m4) ∈ {k1 + 1, . . . , k2}4, � +Xd +0 = Xd +0 and for any i ∈ {1, . . . , d}, +� +Xd +k+1,i = � +Xd +k,i + 1{m1−1,...,m4−1}c(k)1�Ad +k+1σdbd +� +� +Xd +k,i, Zd +k+1,i +� +, +where +�Ad +k+1 = +� +Uk+1 ≤ exp +� d +� +i=1 +φd +� +� +Xd +k,i, Zd +k+1,i +��� +, +(31) +and φd in (20). Denote by F the σ-algebra generated by the process � +Xd and observe that on the +event +4� +j=1 +� +Ad +mj +�c +, Xd is equal to � +Xd. +We consider now the terms in the sum (30). +(i) If (m1, . . . , m4) ∈ I4, then the mis are all distinct and +E +� +� +4 +� +j=1 +bd +� +Xd +mj−1,1, Zd +mj,1 +� +1� +Admj +�c +������ +F +� +� = E +� +� +4 +� +j=1 +bd +� +� +Xd +mj−1,1, Zd +mj,1 +� +1� +�Admj +�c +������ +F +� +� . +21 + +However, {bd( � +Xd +mj−1,1, Zd +mj,1)1(�Ad +mj )c}j=1,...,4 are independent conditionally on F. Thus, +E +� +� +4 +� +j=1 +bd +� +� +Xd +mj−1,1, Zd +mj,1 +� +1� +�Admj +�c +������ +F +� +� = +4 +� +j=1 +E +� +bd +� +� +Xd +mj−1,1, Zd +mj,1 +� +1� +�Admj +�c +����F +� += +4 +� +j=1 +E +� +bd +� +� +Xd +mj−1,1, Zd +mj,1 +� +× +� +1 − exp +� d +� +i=1 +φd +� +� +Xd +mj−1,i, Zd +mj,i +��� ++ +�����F +� +, +by integrating the uniform variables Umj in (31). +Recalling the definition of bd in (19), we can bound the expectation above with +�����E +� +bd +� +� +Xd +mj−1,1, Zd +mj,1 +� � +1 − exp +� d +� +i=1 +φd +� +� +Xd +mj−1,i, Zd +mj,i +��� ++ +�����F +������ +(32) +≤ +���E +��σd +2 sgn +� +� +Xd +mj−1,1 +� +1 +� +| � +Xd +mj−1,1| > σ2m +d r/2 +� +− +1 +σ2m−1 +d +r +� +Xd +mj−1,11 +� +| � +Xd +mj−1,1| ≤ σ2m +d r/2 +�� +× +� +1 − exp +� d +� +i=1 +φd +� +� +Xd +mj−1,i, Zd +mj,i +��� ++ +�����F +������ ++ +�����E +� +Zd +mj,1 +� +1 − exp +� d +� +i=1 +φd +� +� +Xd +mj−1,i, Zd +mj,i +��� ++ +�����F +������ . +For the first one, we use the boundedness of the sgn function and of � +Xd +mj−1,1 in the set +{| � +Xd +mj−1,1| ≤ σ2m +d r/2} to obtain +���E +��σd +2 sgn +� +� +Xd +mj−1,1 +� +1 +� +| � +Xd +mj−1,1| > σ2m +d r/2 +� +− +1 +σ2m−1 +d +r +� +Xd +mj−1,11 +� +| � +Xd +mj−1,1| ≤ σ2m +d r/2 +�� +× +� +1 − exp +� d +� +i=1 +φd +� +� +Xd +mj−1,i, Zd +mj,i +��� ++ +�����F +������ +≤ σd +2 E +������ +� +1 − exp +� d +� +i=1 +φd +� +� +Xd +mj−1,i, Zd +mj,i +��� ++ +����� +�����F +� +≤ σd +2 . +(33) +We can write the second term as +E +� +Zd +mj,1 +� +1 − exp +� d +� +i=1 +φd +� +� +Xd +mj−1,i, Zd +mj,i +��� ++ +�����F +� += E +� +G +� +� +Xd +mj−1,1, +d +� +i=2 +φd +� +� +Xd +mj−1,i, Zd +mj,i +�������F +� +, +22 + +where we define G(a, b) = E +� +Z (1 − exp (φd (a, Z) + b))+ +� +with Z a standard Gaussian. Be- +cause the function x �→ (1 − exp(x))+ is 1-Lipschitz, we have, using Cauchy-Schwarz and +Lemma 3 in Appendix D.2, +��E +� +Z (1 − exp (φd (a, Z) + b))+ +� +− E +� +Z (1 − exp (b))+ +��� ≤ E [|Z| |φd (a, Z)|] +≤ E +� +Z2�1/2 E +� +φd (a, Z)2�1/2 +≤ E +� +φd (a, Z)2�1/2 +≤ Cd−α . +However, E +� +Z (1 − exp (b))+ +� += E [Z] (1 − exp (b))+ = 0, and therefore +�����E +� +G +� +� +Xd +mj−1,1, +d +� +i=2 +φd +� +� +Xd +mj−1,i, Zd +mj,i +�������F +������ ≤ Cd−α . +(34) +Combining equations (32), (33) and (34) and recalling that σd = ℓd−α, we have +�����E +� +bd +� +� +Xd +mj−1,1, Zd +mj,1 +� � +1 − exp +� d +� +i=1 +φd +� +� +Xd +mj−1,i, Zd +mj,i +��� ++ +�����F +������ ≤ Cd−α , +(35) +from which follows that +� +(m1,...,m4)∈I4 +�����E +� 4 +� +i=1 +bd +� +Xd +mi−1,1, Zd +mi,1 +� +1(Admi) +c +������ ≤ +� +(m1,...,m4)∈I4 +E +� +� +4 +� +j=1 +C +dα +� +� +≤ +�k2 − k1 +4 +� C +d4α ≤ C (k2 − k1)4 +d4α +, +(36) +using that |I4| = +�k2−k1 +4 +� +. +(ii) If (m1, .., m4) ∈ I3, only three of the mis take distinct values; proceeding as in case (i), we +have +������ +E +� +� +3 +� +j=1 +bd +� +Xd +mj−1,1, Zd +mj,1 +�1+δ1,j +1� +Admj +�c +������ +F +� +� +������ += +3 +� +j=1 +�����E +� +bd +� +� +Xd +mj−1,1, Zd +mj,1 +�1+δ1,j +� +1 − exp +� d +� +i=1 +φd +� +� +Xd +mj−1,i, Zd +mj,i +��� ++ +�����F +������ , +where δ1,j denotes a Dirac’s delta. For the terms j ̸= 1, we use (35), while for the term j = 1 +23 + +we bound the indicator function by 1 to obtain +������ +E +� +� +3 +� +j=1 +bd +� +Xd +mj−1,1, Zd +mj,1 +�1+δ1,j +1� +Admj +�c +������ +F +� +� +������ +≤ +����E +� +bd +� +� +Xd +m1−1,1, Zd +m1,1 +�2����F +����� +3 +� +j=2 +C +dα +≤ +� +3 + 2σ2 +d +22d2α +� C2 +d2α ≤ C 1 +d2α , +where the second-to-last inequality follows using the same approach taken for (29) and recall- +ing that σd = ℓd−α. Hence, +� +(m1,...,m4)∈I3 +�����E +� 4 +� +i=1 +bd +� +Xd +mi−1,1, Zd +mi,1 +� +1(Admi) +c +������ +(37) +≤ C +�k2 − k1 +3 +� 1 +d2α ≤ C (k2 − k1)3 +d2α +. +(iii) If (m1, .., m4) ∈ I2, we have two different cases: the mis take the two values twice or three +mis have the same value. For the first one, we have, bounding the indicator function with 1, +E +� +�E +� +� +2 +� +j=1 +bd +� +Xd +mj−1,1, Zd +mj,1 +�2 +1� +Admj +�c +������ +F +� +� +� +� +≤ E +� +� +2 +� +j=1 +E +� +bd +� +� +Xd +mj−1,1, Zd +mj,1 +�2����F +�� +� . +Since, conditionally on F, the random variables inside the expectation are Gaussians with +bounded mean and variance 1, we have, using the same approach taken for (29), +E +� +� +2 +� +j=1 +E +� +bd +� +� +Xd +mj−1,1, Zd +mj,1 +�2����F +�� +� ≤ +� +1 + 2σ2 +d +22 +�2 +≤ C . +The second case follows similarly +������ +E +� +�E +� +� +2 +� +j=1 +bd +� +Xd +mj−1,1, Zd +mj,1 +�1+2δ1,j +1� +Admj +�c +������ +F +� +� +� +� +������ +≤ E +� +�E +� +� +2 +� +j=1 +���bd +� +� +Xd +mj−1,1, Zd +mj,1 +���� +1+2δ1,j +������ +F +� +� +� +� ≤ C , +24 + +where δ1,j denotes a Dirac’s delta. Therefore, +� +(m1,...,m4)∈I2 +�����E +� 4 +� +i=1 +� +bd(Xd +mi−1,1, Zd +mi,1 +� +1(Admi) +c +������ +(38) +≤ C +��4 +2 +� ++ +�4 +3 +�� �k2 − k1 +2 +� +≤ C(k2 − k1)2 . +(iv) If (m1, .., m4) ∈ I1 (i.e. all mis take the same value), we bound the indicator function by 1 +and, using the same approach taken for (29), we find +E +� +bd +� +Xd +m1−1,1, Zd +m1,1 +�4 1(Adm1) +c +� +≤ C +� +3 + 2σ4 +d +24 +� +≤ C , +since σd = ℓd−α and d ∈ N. Hence, +� +(m1,...,m4)∈I1 +�����E +� 4 +� +i=1 +bd +� +Xd +m1−1,1, Zd +m1,1 +� +1(Ad +mi) +c +������ ≤ C +�k2 − k1 +1 +� += C(k2 − k1) . +(39) +The result follows combining (36), (37), (38) and (39) in (30). +6.3 +Proof of Proposition 2 +We start by proving the following lemma. +Lemma 2. Let ν be a limit point of the sequence of laws (νd)d≥1 of {(Ld +t )t≥0 | d ∈ N}. Then for +any t ≥ 0, the pushforward measure of ν by Wt is πL(dx) = exp(−|x|)dx/2. +Proof. Using (25), we have +E +����Ld +t − Xd +⌊d2αt⌋,1 +��� +� +≤ E +�����(d2αt − ⌊d2αt⌋) +� +σdZd +⌈d2αt⌉,1 − σ2 +d +2 sgn(Xd +⌊d2αt⌋,1) +� +1|Xd +⌊d2αt⌋,1|>σ2m +d +r/21Ad +⌈d2αt⌉ +���� +� ++ E +������(d2αt − ⌊d2αt⌋) +� +σdZd +⌈d2αt⌉,1 − +1 +σ2(m−1) +d +r +Xd +⌊d2αt⌋,1 +� +1|Xd +⌊d2αt⌋,1|≤σ2m +d +r/21Ad +⌈d2αt⌉ +����� +� +≤ (d2αt − ⌊d2αt⌋) +� +σdE +����Zd +⌈d2αt⌉,1) +��� +� ++ σ2 +d +2 E +����sgn(Xd +⌊d2αt⌋,1) +��� +� ++ +1 +σ2(m−1) +d +r +E +����Xd +⌊d2αt⌋,1 +��� 1|Xd +⌊d2αt⌋,1|≤σ2m +d +r/2 +�� +≤ (d2αt − ⌊d2αt⌋) +� +ℓ +dα E +� +(Zd +⌈d2αt⌉,1)2�1/2 ++ +ℓ2 +2d2α + +1 +σ2(m−1) +d +r +E +�σ2m +d r +2 +�� +≤ (d2αt − ⌊d2αt⌋) +� ℓ +dα + +ℓ2 +2d2α + +ℓ2 +2d2α +� +≤ C +dα , +25 + +where we used Cauchy-Schwarz inequality and the fact that the moments of Zd +⌈d2αt⌉,1 are bounded. +The above guarantees that, +lim +d→∞ E +����Ld +t − Xd +⌊d2αt⌋,1 +��� +� += 0 . +As (νd)d≥1 converges weakly towards ν, for any Lipschitz bounded function ψ : R → R, +lim +d→∞ E +� +ψ +� +Xd +⌊d2αt⌋,1 +�� += lim +d→∞ E +� +ψ +� +Ld +t +�� += Eν [ψ(Wt)] . +The result follows since Xd +⌊d2αt⌋,1 is distributed according to πL(dx) = exp(−|x|)dx/2 for any t ≥ 0 +and d ∈ N. +We are now ready to prove Proposition 2: +Proof of Proposition 2. Let ν be a limit point of (νd)d≥1. +We start by showing that if for any +V ∈ C∞ +c (R, R), m ∈ N, any bounded and continuous mapping ρ : Rm → R and any 0 ≤ t1 ≤ · · · ≤ +tm ≤ s ≤ t, ν satisfies +Eν +�� +V (Wt) − V (Ws) − +� t +s +LV (Wu)du +� +ρ(Wt1, . . . , Wtm) +� += 0 , +(40) +then ν is a solution to the martingale problem associated with L. +Let Fs denote the σ-algebra generated by +{ρ(Wt1, . . . , Wtm) : m ∈ N, ρ : Rm → R bounded and continuous, and 0 ≤ t1 ≤ · · · ≤ tm ≤ s} . +Then, +Eν +� +V (Wt) − V (Ws) − +� t +s +LV (Wu)du +����Fs +� += 0 , +showing that the process +� +V (Wt) − V (W0) − +� t +0 +LV (Wu)du +� +t≥0 +is a martingale w.r.t. ν and the filtration (Ft)t≥0. +To prove (40), it is enough to show that for any V ∈ C∞ +c (R, R), m ∈ N and any bounded and +continuous mapping ρ : Rm → R and any 0 ≤ t1 ≤ · · · ≤ tm ≤ s ≤ t, the mapping +Ψs,t : w �−→ +� +V (wt) − V (ws) − +� t +s +LV (wu)du +� +ρ (wt1, . . . , wtm) , +is continuous on a ν-almost sure subset of C(R+, R). Let +W = {w ∈ C(R+, R) : wu ̸= 0 for almost any u ∈ [s, t]} . +Since w ∈ Wc if and only if +� t +s 1{0}(wu)du > 0, using Lemma 2 and the Fubini–Tonelli’s theorem, +Eν +�� t +s +1{0}(Wu)du +� += +� t +s +Eν � +1{0}(Wu) +� +du = +� t +s +πL({0})du = 0 , +26 + +and we have that ν(Wc) = 0. +Since w �→ wu is continuous for any u ≥ 0, so are w �→ V (wu) and w �→ ρ(wt1, . . . , wtm). +Thus, it is enough to prove that the mapping w �→ +� t +s LV (wu)du is continuous. +Let (wn)n≥0 +be a sequence in C(R+, R) that converges to w ∈ W in the uniform topology on compact sets. +Let u be such that wu ̸= 0, therefore, since the sgn function is continuous in a neighbourhood +of wu, limn→∞ LV (wn +u) = LV (wu), thus limn→∞ LV (wn +u) = LV (wu) for almost any u ∈ [s, t]. +Finally, using the boundedness of the sequence (LV (wn +u))n≥0 and Lebesgue’s dominated convergence +theorem, +lim +n→∞ +� t +s +LV (wn +u)du = +� t +s +LV (wu)du , +which proves that the mappings Ψs,t are continuous on W. +6.4 +Proof of Theorem 3 +Let us introduce, for any n ∈ N, Fd +n,1 = σ({Xd +k,1, 0 ≤ k ≤ n}), the σ-algebra generated by the first +components of {Xd +k | 0 ≤ k ≤ n}. We also introduce for any V ∈ C∞ +c (R, R) +M d +n(V ) = ℓ +dα +n−1 +� +k=0 +V ′(Xd +k,1) +× +� +bd +� +Xd +k,1, Zd +k+1,1 +� +1Ad +k+1 − E +� +bd +� +Xd +k,1, Zd +k+1,1 +� +1Ad +k+1 +���Fd +k,1 +�� +(41) ++ +ℓ2 +2d2α +n−1 +� +k=0 +V ′′(Xd +k,1) +× +� +bd +� +Xd +k,1, Zd +k+1,1 +�2 1Ad +k+1 − E +� +bd +� +Xd +k,1, Zd +k+1,1 +�2 1Ad +k+1 +���Fd +k,1 +�� +. +where bd is defined in (19). +The proof of Theorem 3 follows using the sufficient condition in Proposition 2, the tightness of +the sequence (νd)d≥1 established in Proposition 1 and Proposition 3 below. +Proof. Using Proposition 1, Proposition 2 and Proposition 3 below, it is enough to show that for +any V ∈ C∞ +c (R, R), m ≥ 1, any 0 ≤ t1 ≤ · · · ≤ tm ≤ s ≤ t and any bounded and continuous +mapping ρ : Rm → R, +lim +d→∞ E +�� +M d +⌈d2αt⌉(V ) − M d +⌈d2αs⌉(V ) +� +ρ(Ld +t1, ..., Ld +tm) +� += 0 , +where, for any n ≥ 1, M d +n(V ) is given by (41). However, this is straightforwardly obtained by taking +successively the conditional expectations with respect to Fd +k,1 for k = ⌈d2αt⌉, . . . , ⌈d2αs⌉. +Proposition 3. For any 0 ≤ s ≤ t, V ∈ Cc(R, R) we have +lim +d→∞ E +�����V +� +Ld +t +� +− V +� +Ld +s +� +− +� t +s +LV +� +Ld +u +� +du − +� +M d +⌈d2αt⌉ (V ) − M d +⌈d2αs⌉ (V ) +����� +� += 0 , +(42) +where (Ld +t )t≥0 is defined in (11). +27 + +Proof. The process (Ld +t )t≥0 is piecewise linear, thus it has finite variation. For any τ ≥ 0, we define +dLd +τ = d2ασdbd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1) +� +1Ad +⌈d2ατ⌉dτ . +Thus, recalling that σd = ℓd−α and using the fundamental theorem of integral calculus for piecewise +C1 maps +V +� +Ld +t +� +− V +� +Ld +s +� += ℓdα +� t +s +V ′ � +Ld +τ +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +� +1Ad +⌈d2ατ⌉dτ , +(43) +where bd is defined in (19). A Taylor expansion of V ′ with Lagrange remainder about Xd +⌊d2ατ⌋,1 +gives +V ′ � +Ld +τ +� += V ′ � +Xd +⌊d2ατ⌋,1 +� ++ ℓ +dα +� +d2ατ − ⌊d2ατ⌋ +� +V ′′ � +Xd +⌊d2ατ⌋,1 +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +� +1Ad +⌈d2ατ⌉ ++ +ℓ2 +2d2α +� +d2ατ − ⌊d2ατ⌋ +�2 V (3) (χτ) bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +� +1Ad +⌈d2ατ⌉ , +where for any point τ ∈ [s, t], there exists χτ ∈ [Xd +⌊d2ατ⌋,1, Y d +τ,1]. Substituting the above into (43) +we obtain +V +� +Ld +t +� +− V +� +Ld +s +� += ℓdα +� t +s +V ′ � +Xd +⌊d2ατ⌋,1 +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +� +1Ad +⌈d2ατ⌉dτ ++ ℓ2 +� t +s +� +d2ατ − ⌊d2ατ⌋ +� +V ′′ � +Xd +⌊d2ατ⌋,1 +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +�2 +1Ad +⌈d2ατ⌉dτ +(44) ++ ℓ3 +2dα +� t +s +� +d2ατ − ⌊d2ατ⌋ +�2 V (3) (χτ) bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +�3 +1Ad +⌈d2ατ⌉dτ . +Since V (3) is bounded, using Fubini-Tonelli’s theorem and recalling the definition of bd in (19), we +have that +ℓ3 +2dα E +����� +� t +s +� +d2ατ − ⌊d2ατ⌋ +�2 V (3) (χτ) bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +�3 +1Ad +⌈d2ατ⌉dτ +���� +� +≤ C ℓ3 +2dα +� t +s +E +�����Zd +⌈d2ατ⌉,1 +��� + +ℓ +2dα +�3� +dτ −→ +d→∞ 0 , +since the moments of Zd +⌈d2ατ⌉,1 are bounded. +For the second term in (44), we observe that most of the integrand is piecewise constant since +the process Xd +⌊d2ατ⌋,1 evolves in discrete time. Then, for any integer d2αs ≤ k ≤ d2αt − 1, +� (k+1)/d2α +k/d2α +� +d2ατ − ⌊d2ατ⌋ +� +V ′′ � +Xd +⌊d2ατ⌋,1 +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +�2 +1Ad +⌈d2ατ⌉dτ += +1 +2d2α V ′′ � +Xd +k,1 +� +bd +� +Xd +k,1, Zd +k+1,1 +�2 1Ad +k+1 += 1 +2 +� (k+1)/d2α +k/d2α +V ′′ � +Xd +⌊d2ατ⌋,1 +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +�2 +1Ad +⌈d2ατ⌉dτ . +28 + +Thus, we can write +I = +� t +s +� +d2ατ − ⌊d2ατ⌋ +� +V ′′ � +Xd +⌊d2ατ⌋,1 +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +�2 +1Ad +⌈d2ατ⌉dτ += I1 + I2 , +where we define +I2 = 1 +2 +� t +s +V ′′ � +Xd +⌊d2ατ⌋,1 +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +�2 +1Ad +⌈d2ατ⌉dτ , +and +I1 = +�� ⌈d2αs⌉/d2α +s ++ +� t +⌊d2αt⌋/d2α +� � +d2ατ − ⌊d2ατ⌋ − 1 +2 +� +V ′′ � +Xd +⌊d2ατ⌋,1 +� +× bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +�2 +1Ad +⌈d2ατ⌉dτ . +In addition, we have +I1 = +1 +2d2α +� +d2αs − ⌊d2αs⌋ +� � +⌈d2αs⌉ − d2αs +� +V ′′ � +Xd +⌊d2αs⌋,1 +� +bd +� +Xd +⌊d2αs⌋,1, Zd +⌈d2αs⌉,1 +�2 +1Ad +⌈d2αs⌉ ++ +1 +2d2α +� +d2αt − ⌊d2αt⌋ +� � +⌈d2αt⌉ − d2αt +� +V ′′ � +Xd +⌊d2αt⌋,1 +� +bd +� +Xd +⌊d2αt⌋,1, Zd +⌈d2αt⌉,1 +�2 +1Ad +⌈d2αt⌉ , +and, since V ′′ and the moments of Zd +⌈d2αt⌉,1 are bounded, limd→∞ E [|I1|] = 0. Thus, +lim +d→∞ E +���V +� +Ld +t +� +− V +� +Ld +s +� +− Is,t +��� += 0 , +where +Is,t = +� t +s +� +ℓdαV ′ � +Xd +⌊d2ατ⌋,1 +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +� +(45) ++ ℓ2 +2 V ′′ � +Xd +⌊d2ατ⌋,1 +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +�2 +1Ad +⌈d2ατ⌉ +� +dτ . +Next, we use (18) and write +� t +s +LV +� +Ld +τ +� +dτ = +� t +s +hL(ℓ) +2 +� +V ′′ � +Xd +⌊d2ατ⌋1 +� +− sgn +� +Xd +⌊d2ατ⌋,1 +� +V ′ � +Xd +⌊d2ατ⌋,1 +�� +dτ − T d +3 , +(46) +where we define +T d +3 = +� t +s +� +LV +� +Xd +⌊d2ατ⌋,1 +� +− LV +� +Ld +τ +�� +dτ . +Finally, we write the difference M d +⌈d2αt⌉(V ) − M d +⌈d2αs⌉(V ) as the integral of a piecewise constant +29 + +function +M d +⌈d2αt⌉(V ) − M d +⌈d2αs⌉(V ) = Is,t +(47) +− +� t +s +� +ℓdαV ′ � +Xd +⌊d2ατ⌋,1 +� +E +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +� +1Ad +⌈d2ατ⌉ +���Fd +⌊d2ατ⌋,1 +� ++ℓ2 +2 V ′′ � +Xd +⌊d2ατ⌋,1 +� +E +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +�2 +1Ad +⌈d2ατ⌉ +����Fd +⌊d2ατ⌋,1 +�� +dτ +− T d +4 − T d +5 , +where T d +4 and T d +5 account for the difference between the sum in (41) and the integral, and are +defined as +T d +4 = − ℓ +dα +� +⌈d2αt⌉ − d2αt +� +V ′ � +Xd +⌊d2αt⌋,1 +� � +bd +� +Xd +⌊d2αt⌋,1, Zd +⌈d2αt⌉,1 +� +1Ad +⌈d2αt⌉ +−E +� +bd +� +Xd +⌊d2αt⌋,1, Zd +⌈d2αt⌉,1 +� +1Ad +⌈d2αt⌉ +���Fd +⌊d2αt⌋,1 +�� +− +ℓ2 +2d2α +� +⌈d2αt⌉ − d2αt +� +V ′′ � +Xd +⌊d2αt⌋,1 +� � +bd +� +Xd +⌊d2αt⌋,1, Zd +⌈d2αt⌉,1 +�2 +1Ad +⌈d2αt⌉ +−E +� +bd +� +Xd +⌊d2αt⌋,1, Zd +⌈d2αt⌉,1 +�2 +1Ad +⌈d2αt⌉ +����Fd +⌊d2αt⌋,1 +�� +, +and T d +5 = −T d +4 with t substituted by s. +Putting (45), (46) and (47) together we obtain +Is,t − +� t +s +LV +� +Ld +τ +� +dτ − +� +M d +⌈d2αt⌉(V ) − M d +⌈d2αs⌉(V ) +� += T d +1 + T d +2 + T d +3 + T d +4 + T d +5 , +where T d +1 takes into account all the terms involving V ′(Xd +⌊d2ατ⌋,1), and T d +2 the terms involving +V ′′(Xd +⌊d2ατ⌋,1): +T d +1 = +� t +s +V ′ � +Xd +⌊d2ατ⌋,1 +� +× +� +ℓdαE +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +� +1Ad +⌈d2ατ⌉ +���Fd +⌊d2ατ⌋,1 +� ++ hL(ℓ) +2 +sgn +� +Xd +⌊d2ατ⌋,1 +�� +dτ , +T d +2 = +� t +s +V ′′ � +Xd +⌊d2ατ⌋,1 +� +× +�ℓ2 +2 E +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +�2 +1Ad +⌈d2ατ⌉ +����Fd +⌊d2ατ⌋,1 +� +− hL(ℓ) +2 +� +dτ . +To obtain (42) it is then sufficient to prove that for any 1 ≤ i ≤ 5, limd→∞ E +���T d +i +��� += 0. +Since V ′, V ′′ are bounded and bd is bounded in expectation because the moments of Zd +⌈d2ατ⌉,1 +are bounded, it is easy to show that limd→∞ E +���T d +i +��� += 0 for i = 4, 5. For T d +3 , we write T d +3 = +30 + +hL(ℓ)(T d +3,1 − T d +3,2)/2, where +T d +3,1 = +� t +s +� +V ′′ � +Xd +⌊d2ατ⌋,1 +� +− V ′′ � +Ld +τ +�� +dτ , +T d +3,2 = +� t +s +� +sgn +� +Xd +⌊d2ατ⌋,1 +� +V ′ � +Xd +⌊d2ατ⌋,1 +� +− sgn +� +Ld +τ +� +V ′ � +Ld +τ +�� +dτ . +Using Fubini-Tonelli’s theorem, the convergence of Xd +⌊d2ατ⌋,1 to Ld +τ in Lemma 2 and Lebesgue’s +dominated convergence theorem we obtain +E +���T d +3,1 +��� +≤ +� t +s +E +����V ′′ � +Xd +⌊d2ατ⌋,1 +� +− V ′′ � +Ld +τ +���� +� +dτ −→ +d→∞ 0 . +We can further decompose T d +3,2 as +T d +3,2 = +� t +s +� +sgn +� +Xd +⌊d2ατ⌋,1 +� +− sgn +� +Ld +τ +�� +V ′ � +Xd +⌊d2ατ⌋,1 +� +dτ ++ +� t +s +sgn +� +Ld +τ +� � +V ′ � +Xd +⌊d2ατ⌋,1 +� +− V ′ � +Ld +τ +�� +dτ . +Proceeding as for T d +3,1, it is easy to show that the second integral converges to 0 as d → ∞. We +then bound the first integral by +E +����� +� t +s +� +sgn +� +Xd +⌊d2ατ⌋,1 +� +− sgn +� +Ld +τ +�� +V ′ � +Xd +⌊d2ατ⌋,1 +� +dτ +���� +� +≤ C +� t +s +E +����sgn +� +Xd +⌊d2ατ⌋,1 +� +− sgn +� +Ld +τ +���� +� +dτ . +However, since {sgn(Xd +⌊d2ατ⌋,1) ̸= sgn(Ld +τ)} ⊂ {sgn(Xd +⌊d2ατ⌋,1) ̸= sgn(Xd +⌈d2ατ⌉,1)}, using Lemma 4 in +Appendix D.3 we have that +E +����sgn +� +Xd +⌊d2ατ⌋,1 +� +− sgn +� +Ld +τ +���� +� += 2E +� +1� +sgn +� +Xd +⌊d2ατ⌋,1 +� +̸=sgn(Ldτ ) +� +� += 2E +� +1� +sgn +� +Xd +⌊d2ατ⌋,1 +� +̸=sgn +� +Xd +⌈d2ατ⌉,1 +�� +� +−→ +d→∞ 0 . +The above and the dominated converge theorem show that +E +����� +� t +s +� +sgn +� +Xd +⌊d2ατ⌋,1 +� +− sgn +� +Ld +τ +�� +V ′ � +Xd +⌊d2ατ⌋,1 +� +dτ +���� +� +−→ +d→∞ 0 . +Consider then T d +1 , recalling that the derivatives of V are bounded, we have +E +���T d +1 +��� +≤ +� t +s +CE +����ℓdαE +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +� +1Ad +⌈d2ατ⌉ +���Fd +⌊d2ατ⌋,1 +� ++ hL(ℓ) +2 +sgn +� +Xd +⌊d2ατ⌋,1 +����� +� +dτ +≤ +� t +s +C +� +E +����D(1) +1,τ +��� +� ++ E +����D(1) +2,τ +��� +�� +dτ , +31 + +where we define +D(1) +1,τ = ℓdαE +� +Zd +⌈d2ατ⌉,11Ad +⌈d2ατ⌉ +���Fd +⌊d2ατ⌋,1 +� +, +D(1) +2,τ = hL(ℓ) +2 +sgn +� +Xd +⌊d2ατ⌋,1 +� +− ℓdα +�σd +2 sgn(Xd +⌊d2ατ⌋,1)1|Xd +⌊d2ατ⌋,1|>σ2m +d +r/2 + +1 +σ2m−1 +d +rXd +⌊d2ατ⌋,11|Xd +⌊d2ατ⌋,1|≤σ2m +d +r/2 +� +× E +� +1Ad +⌈d2ατ⌉ +���Fd +⌊d2ατ⌋,1 +� +. +Let us start with D(1) +1,τ: +D(1) +1,τ = ℓdαE +� +Zd +⌈d2ατ⌉,1 +� +1 ∧ exp +� d +� +i=1 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +��������Fd +⌊d2ατ⌋,1 +� +, +where φd is given in (20). Then, by independence of the components of Zd +⌈d2ατ⌉, we have +E +� +Zd +⌈d2ατ⌉,1 +� +1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +��������Fd +⌊d2ατ⌋,1 +� += E +� +Zd +⌈d2ατ⌉,1 +� +E +� +1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +�������Fd +⌊d2ατ⌋,1 +� += 0 . +This allows us to write +E +� +|D(1) +1,τ| +� +≤ ℓdαE +� +|Zd +⌈d2ατ⌉,1| +�����1 ∧ exp +� d +� +i=1 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +�� +− 1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +������� +� +. +However, x �→ 1 ∧ exp(x) is a 1-Lipschitz function, thus +E +� +|D(1) +1,τ| +� +≤ ℓdαE +� +|Zd +⌈d2ατ⌉,1| +���φd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +���� +� +, +and D(1) +1,τ → 0 as d → ∞ by Lemma 5 in Appendix D.3. +For D(1) +2,τ, we observe that +− σd +2 1|Xd +⌊d2ατ⌋,1|≤σ2m +d +r/2 ≤ +1 +σ2m−1rXd +⌊d2ατ⌋,11|Xd +⌊d2ατ⌋,1|≤σ2m +d +r/2 ≤ σd +2 1|Xd +⌊d2ατ⌋,1|≤σ2m +d +r/2 . +(48) +Distinguishing between Xd +⌊d2ατ⌋,1 < 0 and Xd +⌊d2ατ⌋,1 ≥ 0, it follows that +|D(1) +2,τ| ≤ +���sgn +� +Xd +⌊d2ατ⌋,1 +���� +× +���� +hL(ℓ) +2 +− ℓdα �σd +2 1|Xd +⌊d2ατ⌋,1|>σ2m +d +r/2 + σd +2 1|Xd +⌊d2ατ⌋,1|≤σ2m +d +r/2 +� +E +� +1Ad +⌈d2ατ⌉ +���Fd +⌊d2ατ⌋,1 +����� +≤ 1 +2 +���hL(ℓ) − ℓ2E +� +1Ad +⌈d2ατ⌉ +���Fd +⌊d2αr⌋,1⌋ +���� , +32 + +where we recall that σd = ℓd−α with α = 1/3. Using the triangle inequality we obtain +2E +� +|D(1) +2,τ| +� +≤ E +������hL(ℓ) − ℓ2E +� +1 ∧ exp +� d +� +i=1 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +�������Fd +⌊d2ατ⌋,1 +������ +� +≤ E +������hL(ℓ) − ℓ2E +� +1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +�������Fd +⌊d2ατ⌋,1 +������ +� ++ ℓ2E +������1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +�� +− 1 ∧ exp +� d +� +i=1 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +������� +� +, +where we used Jensen’s inequality to remove the conditional expectation in the last term. Recalling +that x �→ 1 ∧ exp(x) is 1-Lipschitz, we can then bound the second term +ℓ2E +������1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +�� +− 1 ∧ exp +� d +� +i=1 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +������� +� +≤ ℓ2E +����φd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +���� +� +, +(49) +≤ ℓ2E +� +φd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +�2�1/2 +, +where the final expectation converges to zero as d → ∞ by Proposition 17. For the remaining term +in D(1) +2,τ, since (Xd +⌊d2ατ⌋,i, Zd +⌊d2ατ⌋,i)2≤i≤n is independent of Fd +⌊d2ατ⌋,1, we have +ℓ2E +� +1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +�������Fd +⌊d2ατ⌋,1 +� += ℓ2E +� +1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +��� +, +and, using again the fact that x �→ 1 ∧ exp(x) is 1-Lipschitz, we have +�����hL(ℓ) − ℓ2E +� +1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +�������� +≤ +�����hL(ℓ) − ℓ2E +� +1 ∧ exp +� d +� +i=1 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +�������� ++ ℓ2E +����φd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +���� +� +. +The last term goes to 0 as shown in (49), and, as hL(ℓ) = ℓ2aL(ℓ), with +aL(ℓ) = lim +d→∞ E +� +1 ∧ exp +� d +� +i=1 +φd,i +�� +, +33 + +by Theorem 2, we obtain +lim +d→∞ +�����hL(ℓ) − ℓ2E +� +1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +������� +� += 0 , +showing that D(1) +2,τ → 0 as d → ∞. To obtain convergence of T d +1 , we observe that for any τ ∈ +[s, t], D(1) +1,τ and D(1) +2,τ follow the same distributions as D(1) +1,s and D(1) +2,s, since for any k ∈ N, Xd +k has +distribution πL +d . Therefore, the convergence towards zero of E[|D(1) +1,τ|] and E[|D(1) +2,τ|] is uniform for +τ ∈ [s, t], which gives us T d +1 → 0 as d → ∞. +Finally, consider T d +2 . Using analogous arguments to those used for T d +1 , we obtain +E +� +|T d +2 | +� +≤ C +� t +s +ℓ2 +2 E +�����E +� +bd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +�2 +1Ad +⌈d2ατ⌉ +����Fd +⌊d2ατ⌋,1 +� +− aL(ℓ, r) +���� +� +dτ +≤ C +� t +s +ℓ2 +2 +� +E +� +|D(2) +1,τ| +� ++ E +� +|D(2) +2,τ| +� +E +� +|D(2) +3,τ| +�� +dτ , +where we define +D(2) +1,τ = E +�� +Zd +⌈d2ατ⌉,1 +�2 +1Ad +⌈d2ατ⌉ +����Fd +⌊d2ατ⌋,1 +� +− aL(ℓ, r) , +D(2) +2,τ = +�σd +2 sgn(Xd +⌊d2ατ⌋,1)1|Xd +⌊d2ατ⌋,1|>σ2m +d +r/2 + +1 +σ2m−1 +d +rXd +⌊d2ατ⌋,11|Xd +⌊d2ατ⌋,1|≤σ2m +d +r/2 +�2 +× E +� +1Ad +⌈d2ατ⌉ +���Fd +⌊d2ατ⌋,1 +� +, +D(2) +3,τ = 2 +�σd +2 sgn(Xd +⌊d2ατ⌋,1)1|Xd +⌊d2ατ⌋,1|>σ2m +d +r/2 + +1 +σ2m−1 +d +rXd +⌊d2ατ⌋,11|Xd +⌊d2ατ⌋,1|≤σ2m +d +r/2 +� +× E +� +Zd +⌈d2ατ⌉,11Ad +⌈d2ατ⌉ +���Fd +⌊d2ατ⌋,1 +� +. +Using (48), Cauchy-Schwarz’s inequality and the fact that the moments of Zd +⌈d2ατ⌉,1 are bounded +we have +E +� +|D(2) +2,τ| +� +≤ σ2 +d +4 +−→ +d→∞ 0 , +E +� +|D(2) +3,τ| +� +≤ Cσd −→ +d→∞ 0 , +since σd = ℓd−α with α = 1/3. The remaining term is bounded similarly to D(1) +2,τ, using the fact +that x �→ 1 ∧ exp(x) is 1-Lipschitz, we have +E +� +|D(2) +3,τ| +� +≤ E +������E +�� +Zd +⌈d2ατ⌉,1 +�2 +� +1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +��������Fd +⌊d2ατ⌋,1 +� +− aL(ℓ, r) +����� +� ++ E +�� +Zd +⌈d2ατ⌉,1 +�2 ���φd +� +Xd +⌊d2ατ⌋,1, Zd +⌈d2ατ⌉,1 +���� +� +. +34 + +The second expectation is bounded as (49) using Cauchy-Schwarz’s inequality and Proposition 17. +For the first expectation, we use the conditional independence of the components of Zd +⌈d2ατ⌉ and +write +E +�� +Zd +⌈d2ατ⌉,1 +�2 +� +1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +��������Fd +⌊d2ατ⌋,1 +� += E +�� +Zd +⌈d2ατ⌉,1 +�2� +E +�� +1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +���� += E +�� +1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +���� +. +It follows that E[|D(2) +3,τ|] → 0 as d → ∞ since, by Theorem 2, +�����E +�� +1 ∧ exp +� d +� +i=2 +φd +� +Xd +⌊d2ατ⌋,i, Zd +⌈d2ατ⌉,i +���� +− aL(ℓ, r) +����� → 0 . +Combining the results for T d +i , i = 1, . . . , 5 we obtain the result. +Acknowledgments +F.R.C. and G.O.R. acknowledge support from the EPSRC (grant # EP/R034710/1). +G.O.R. +acknowledges further support from the EPSRC (grant # EP/R018561/1) and the Alan Turing +Institute. A.D. acknowledges support from the Lagrange Mathematics and Computing Research +Center. The authors would like to thank ´Eric Moulines for helpful discussions. +For the purpose of open access, the author has applied a Creative Commons Attribution (CC +BY) licence to any Author Accepted Manuscript version arising from this submission. +References +[1] S. Agrawal, D. Vats, K. �Latuszy´nski, and G. O. Roberts. Optimal scaling of MCMC beyond +Metropolis. arXiv preprint arXiv:2104.02020, 2021. +[2] Y. F. Atchad´e. A Moreau-Yosida approximation scheme for a class of high-dimensional poste- +rior distributions. arXiv preprint arXiv:1505.07072, 2015. +[3] H. H. Bauschke and P. L. Combettes. Convex analysis and monotone operator theory in Hilbert +spaces, volume 408. Springer, 2011. +[4] M. B´edard and J. S. Rosenthal. Optimal scaling of Metropolis algorithms: Heading toward +general target distributions. Canadian Journal of Statistics, 36(4):483–503, 2008. +[5] A. Beskos, G. Roberts, and A. 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Proximal MCMC for Bayesian inference of constrained and +regularized estimation. arXiv preprint arXiv:2205.07378, 2022. +A +Proof of Theorem 1 +The proof of Theorem 1 follows that of [34, Theorem 1, Theorem 2] and consists of four propositions +showing convergence of the log-acceptance probability to a normal random variable and (weak) +convergence of the process (11) to a Langevin diffusion. +We start by recalling and defining a number of quantities that we will use in the following proofs. +Recall that σd = ℓ/dα, that λd = σ2m +d r/2 where m ≥ 1/2 and r > 0 are to be chosen according to +the different cases in Theorem 1. Recalling the expression of the proposal given in (9) and using +the simplification given in (10), we define the proposal with starting point xd ∈ Rd, +yd(xd, zd) = xd − σ2 +d +2 ∇G +� +proxσ2m +d +r/2 +G +(xd) +� ++ σdzd , +where zd ∈ Rd. Since G is the d-times tensor product of g, the i-th component of the proposal only +depends on the i-th components of xd and zd. Thus, we introduce the notation for any x, z ∈ R, +yd(x, z) = x − σ2 +d +2 g′ � +proxσ2m +d +r/2 +g +(x) +� ++ σdz . +With these notations, we obtain the proposal for the chain (Xd +k)k≥0 using Y d +k = yd(Xd +k, Zd +k+1) = +(yd(Xd +k,i, Zd +k+1,i))i∈{1,...,d}. +Let us define the generator of the discrete process (Xd +k)k≥0 for all +38 + +V ∈ C∞ +c (Rd, R), i.e. infinitely differentiable R-valued multivariate functions with compact support, +and any xd ∈ Rd, +LdV (xd) = d2αE +�� +V (yd(xd, Zd +1)) − V (xd) +� πd(yd(xd, Zd +1))qd(yd(xd, Zd +1), xd) +πd(xd)qd(xd, yd(xd, Zd +1)) +∧ 1 +� += d2αE +� +� +V (yd(xd, Zd +1)) − V (xd) +� +d +� +i=1 +exp +� +φd(xd +i , Zd +1,i) +� +∧ 1 +� +, +where the expectation is w.r.t. Zd +1 = (Zd +1,i)i∈{1,...,d}, a d-dimensional standard normal random +variable, and where we defined +φd(x, z) = log π(yd(x, z))q(yd(x, z), x) +π(x)q(x, yd(x, z)) +(50) += g(yd(x, z)) − g(x) + log q(yd(x, z), x) − log q(x, yd(x, z)) . +In the remainder we will work with one-dimensional functions V ∈ C∞ +c (R, R) applied to the first +component of xd so that +LdV (xd) = d2αE +�� +V (yd(xd +1, Zd +1,1)) − V (xd +1) +� πd(yd(xd, Zd +1))qd(yd(xd, Zd +1), xd) +πd(xd)qd(xd, yd(xd, Zd +1)) +∧ 1 +� += d2αE +� +� +V (yd(xd +1, Zd +1,1)) − V (xd +1) +� +d +� +i=1 +exp +� +φd +� +xd +i , Zd +1,i +�� +∧ 1 +� +. +(51) +We also define �Ld to be a variant of Ld in which the first component of the acceptance ratio is +omitted: +�LdV (xd) = d2αE +� +� +V (yd(xd +1, Zd +1,1)) − V (xd +1) +� +d +� +i=2 +exp +� +φd +� +xd +i , Zd +1,i +�� +∧ 1 +� +. +(52) +We further define the generator of the Langevin diffusion (13) +LV (x) = h(ℓ, r) +2 +[V ′′(x) − g′(x)V ′(x)] , +(53) +where h(ℓ, r) = ℓ2a(ℓ, r) is the speed of the diffusion and a(ℓ, r) = limd→∞ ad(ℓ, r) is given in +Theorem 1. +We will make use of the derivatives of g in (8) up to order 8, which we denote by g′, g′′, g′′′ and +g(k) for all k = 4, . . . , 8. +We recall that (gλ)′ is Lipschitz continuous with Lipschitz constant λ−1 [38, Proposition 12.19] +and that (gλ)′(x) = λ−1(proxλ +g(x)−x), hence proxλ +g is Lipschitz continuous with Lipschitz constant +1. +A.1 +Auxiliary Results for the Proof of Case (a) +First, we characterize the limit behaviour of the acceptance ratio (12). +Proposition 4. Under A0, A1 and A2, if α = 1/4, β = 1/8 and r > 0, then +39 + +(i) the log-acceptance ratio (50) satisfies +φd(x, z) = d−1/2C2(x, z) + d−3/4C3(x, z) + d−1C4(x, z) + C5(x, z, σd) , +where C2(x, z) is given in (57), C3 and C4 are polynomials in z and the derivatives of g, such +that E[C3(Xd +0,1, Zd +1,1)] = 0 and E[C2(Xd +0,1, Zd +1,1)2] = −2E[C4(Xd +0,1, Zd +1,1)]; +(ii) there exists sets Fd ⊆ Rd with d2απd(F c +d) → 0 such that +lim +d→∞ sup +xd∈Fd +E +������ +d +� +i=2 +φd(xd +i , Zd +1,i) − d−1/2 +d +� +i=2 +C2(xd +i , Zd +1,i) + ℓ4K1(r)2 +8 +����� +� += 0 , +(54) +where K1(r) is given in Theorem 1–(a). +Proof. Take one component of the log-acceptance ratio +φd(x, z) = g(yd(x, z)) − g(x) + log q(yd(x, z), x) − log q(x, yd(x, z)) , +with yd(x, z) = x−σ2 +dg′(proxσ2m +d +r/2 +g +(x))/2+σdz. We have that φd(x, z) = R1(x, z, σd)+R2(x, z, σd), +where +R1(x, z, σ) = −g +� +x − σ2 +2 g′ � +proxσ2mr/2 +g +(x) +� ++ σz +� ++ g(x) , +R2(x, z, σ) = 1 +2z2 − 1 +2 +� +z − σ +2 g′ +� +proxσ2mr/2 +g +� +x + σz − σ2 +2 g′ � +proxσ2mr/2 +g +(x) +��� +−σ +2 g′ � +proxσ2mr/2 +g +(x) +��2 +. +(55) +Following the approach of [34] we approximate φd(x, z) with a Taylor expansion about σd → 0. +(i) Using a Taylor expansion of order 5, we obtain +φd(x, z) = d−1/2C2(x, z) + d−3/4C3(x, z) + d−1C4(x, z) + C5(x, z, σd) , +(56) +where +C2(x, z) = ℓ2 +2 (−rzg′′(x)g′(x)) , +(57) +C3(x, z) and C4(x, z) are given in Appendix C.1.1 and we use the integral form for the re- +mainder +C5(x, z, σd) = +� σd +0 +∂5 +∂σ5 R(x, z, σ) +���� +σ=u +(σd − u)4 +4! +du , +with u between 0 and σd and the derivatives of R1 and R2 given in Appendix C.2. +In +addition, integrating by parts and using the moments of Zd +1,1 we find that E[C2(Xd +0,1, Zd +1,1)] = +E[C3(Xd +0,1, Zd +1,1)] = 0 and +2E +� +C4(Xd +0,1, Zd +1,1) +� ++ E +� +C2(Xd +0,1, Zd +1,1)2� += 0 . +40 + +(ii) To construct the sets Fd, consider, for j = 3, 4, Fd,j = F 1 +d,j ∩ F 2 +d,j where we define +F 1 +d,j = +� +xd ∈ Rd : +����� +d +� +i=2 +E +� +Cj(xd +i , Zd +1,i) − Cj(Xd +0,i, Zd +1,i) +� +����� ≤ d5/8 +� +, +and +F 2 +d,j = +� +xd ∈ Rd : +����� +d +� +i=2 +Vj(xd +i ) − E +� +Vj(Xd +0,i) +� +����� ≤ d6/5 +� +, +where Vj(x) := Var(Cj(x, Zd +1,1)). +Using Markov’s inequality and the fact that Cj, Vj are +bounded by polynomials since g and its derivatives are bounded by polynomials, it is easy to +show that d1/2πd((F 1 +d,j)c) → 0 and d1/2πd((F 2 +d,j)c) → 0, from which follows d1/2πd(F c +d,j) → 0 +as d → ∞. To prove L1 convergence of Cj for j = 3, 4, observe that +E +� +� +� d +� +i=2 +Cj(xd +i , Zd +1,i) − E +� +Cj(Xd +0,1, Zd +1,1) +� +�2� +� += +d +� +i=2 +Vj(xd +i ) + +� d +� +i=2 +E +� +Cj(xd +i , Zd +1,i) − Cj(Xd +0,1, Zd +1,1) +� +�2 +, +and that, for xd ∈ Fd,j, we have +E +� +� +� d +� +i=2 +Cj(xd +i , Zd +1,i) − E +� +Cj(Xd +0,1, Zd +1,1) +� +�2� +� ≤ E +� +Vj(xd +1) +� +(d − 1) + d6/5 + d5/4 . +Thus, the third and fourth term in the Taylor expansion (56) converge in L1 to 0 and +−ℓ4K2 +1(r)/8 respectively. Now, consider C5(xd +i , Zd +1,i, σd). We can bound +∂5R +∂σ5 (x, z, σ) with +the derivatives of g evaluated at +x + σ2 +2 proxσ2mr/2 +g +(x) + σz +and +proxσ2mr/2 +g +(x) . +Under our assumptions, the derivatives of g are bounded by polynomials M0, it follows that +there exist polynomials p of the form +A +� +1 + +� +proxσ2mr/2 +g +(x) +�N� � +1 + zN� � +1 + xN� � +1 + σN� +, +for sufficiently large A and sufficiently large even integer N, such that +����g(k) +� +x + σ2 +2 proxσ2mr/2 +g +(x) + σdz +����� ∨ +���g(k) � +proxσ2mr/2 +g +(x) +���� +≤ p(proxσ2mr/2 +g +(x), x, z, σd) . +41 + +In addition, | proxσ2mr/2 +g +(x)| ≤ C(1 + |x|) for some C ≥ 1, and we can bound +p(proxσ2mr/2 +g +(x), x, z, σ) ≤ A +� +1 + zN� � +1 + x2N� � +1 + σN� +. +Therefore, we have +E +���C5(xd +i , Zd +1,i, σd) +��� +≤ AE +� +1 + (Zd +1,i)N� � +1 + (xd +i )2N� � σd +0 +(1 + uN)(σd − u)4 +4! +du +≤ AE +� +1 + (Zd +1,i)N� � +1 + (xd +i )2N� +d−5/2 +≤ A +� +1 + (xd +i )2N� +d−5/2 , +where the last inequality follows since all the moments of Zd +1 are bounded. Let us denote +p(x) = A +� +1 + x2N� +and +Fd,5 = +� +xd ∈ Rd : +�����d−1 +d +� +i=1 +p(xd +i ) − E +� +p(Xd +0,i) +� +����� < 1 +� +. +By Chebychev’s inequality we have πd(F c +d,5) ≤ Var(p(Xd +0,1))d−1. Additionally, for all xd ∈ +Fd,5, +d +� +i=2 +E +���C5(xd +i , Zd +1,i, σd) +��� +≤ +d +� +i=2 +d−5/2 � +E +� +p(Xd +0,1) +� ++ d−1� +≤ d−3/2 � +E +� +p(Xd +0,1) +� ++ 1 +� +. +Finally, set Fd = ∩5 +j=3Fd,j. On Fd the last three terms of (56) converge uniformly in L1, +and (54) follows using the triangle inequality. +Next, we compare the generator Ld and �Ld in (51) and (52) respectively. +Proposition 5. Under A0, A1 and A2, if α = 1/4, β = 1/8 and r > 0, there exists sets Sd ⊆ Rd +with d2απd(Sc +d) → 0 such that for any V ∈ C∞ +c (R, R) +lim +d→∞ sup +xd∈Sd +���LdV (xd) − �LdV (xd) +��� = 0 , +and +lim +d→∞ sup +xd∈Sd +E +������ +� +exp +� d +� +i=1 +φd(xd +i , Zd +1,i) +� +∧ 1 +� +− +� +exp +� d +� +i=2 +φd(xd +i , Zd +1,i) +� +∧ 1 +������ +� += 0 . +Proof. The function x �→ exp(x) ∧ 1 is Lipschitz continuous with Lipschitz constant 1, hence +���LdV (xd) − �LdV (xd) +��� ≤ d2αE +���V +� +yd(xd +1, Zd +1,1) +� +− V (xd +1) +�� |R(xd +1, Zd +1,1, σd)| +� +, +42 + +where R(x, z, σ) = R1(x, z, σ) + R2(x, z, σ) as in (55). Using a Taylor expansion of order 1 about +σ = 0 with integral remainder: +R(x, z, σ) = R(x, z, 0) + ∂R +∂σ (x, z, σ) +���� +σ=0 +σ + +� σ +0 +∂2R +∂σ2 (x, z, σ) +���� +σ=u +(σ − u)du , +we obtain +R(x, z, σ) = +� σ +0 +∂2R +∂σ2 (x, z, σ) +���� +σ=u +(σ − u)du , +where ∂2R +∂σ2 (x, z, σ) is bounded by the derivatives of g evaluated at +x + σ2 +2 proxσ2mr/2 +g +(x) + σz +and +proxσ2mr/2 +g +(x) . +Under our assumptions, the derivatives of g are bounded by polynomials M0, it follows that there +exist polynomials p of the form +A +� +1 + +� +proxσ2mr/2 +g +(x) +�N� � +1 + zN� � +1 + xN� � +1 + σN� +, +for sufficiently large A and sufficiently large even integer N, such that +����g(k) +� +x + σ2 +2 proxσ2mr/2 +g +(x) + σz +����� ∨ +���g(k) � +proxσ2mr/2 +g +(x) +���� ≤ p(proxσ2mr/2 +g +(x), x, z, σ) . +Proceeding as in Proposition 4, we can bound +p(proxσ2mr/2 +g +(x), x, z, σ) ≤ A +� +1 + zN� � +1 + x2N� � +1 + σN� +. +Therefore, we have +��R +� +xd +1, Zd +1,1, σd +��� ≤ A +� +1 + (Zd +1,1)N� � +1 + (xd +1)2N� +× +� σd +0 +(1 + uN)(σd − u)du ≤ A +� +1 + (Zd +1,1)N� � +1 + (xd +1)2N� σ2 +d +2 . +(58) +Since V ∈ C∞ +c (R, R), there exists a constant C such that +��V +� +yd(xd +1, Zd +1,1) +� +− V (xd +1) +�� ≤ C +��yd(xd +1, Zd +1,1) − xd +1 +�� +≤ C +� +σd|Zd +1,1| + σ2 +d +2 +���g′ � +proxσ2m +d +r/2 +g +(xd +1) +���� +� +. +Recalling that g′(proxλ +g(x)) = (gλ)′(x) with (gλ)′ Lipschitz continuous with Lipschitz constant λ−1, +we have +���g′ � +proxσ2m +d +r/2 +g +(xd +1) +���� ≤ +2 +σ2m +d r(1 + |xd +1|) , +43 + +and +��V +� +yd(xd +1, Zd +1,1) +� +− V (xd +1) +�� ≤ C +� +σd|Zd +1,1| + σ2−2m +d +r +� +1 + |xd +1| +�� +(59) +≤ Cσd +� +|Zd +1,1| + 1 +r +� +1 + |xd +1| +�� +, +since m = 1/2. Combining (58) and (59) we obtain +d2α ��V +� +yd(xd +1, Zd +1,1) +� +− V (xd +1) +�� ��R(xd +1, Zd +1,1, σd) +�� +≤ Cσd +� +1 + (Zd +1,1)N� � +1 + (xd +i )2N� � +|Zd +1,1| + 1 +r (1 + |xd +1|) +� +, +for some C > 0. +Set Sd to be the set in which 1 + (xd +1)2N+1 ≤ dα/2, applying Markov’s inequality we obtain +d2απd(Sc +d) = d2απd +�� +1 + (xd +1)2N+1�5 ≥ d5α/2� +≤ d−α/2E +�� +1 + (xd +1)2N+1�5� +−→ +d→∞ 0 . +Recalling that |Zd +1,1| and 1 + (Zd +1,1)N are bounded, we have that +sup +xd∈Sd +���LdV (xd) − �LdV (xd) +��� ≤ Cdα/2 ℓ +dα −→ +d→∞ 0 . +The second results follows from (58) using the same argument. +The following result considers the convergence to the generator of the Langevin diffusion (53). +Proposition 6. Under A0, A1 and A2, if α = 1/4, β = 1/8 and r > 0, there exists sets Td ⊆ Rd +with d2απd(T c +d) → 0 as d → ∞, such that for any V ∈ C∞ +c (R, R) +lim +d→∞ sup +xd∈Td +����d2αE +� +V +� +yd(xd +1, Zd +1,1) +� +− V (xd +1) +� +− ℓ2 +2 (V ′′(xd +1) + g′(xd +1)V ′(xd +1)) +���� = 0 . +Proof. Take +yd(xd +1, Zd +1,1) = xd +1 + σ2 +d +2 g′ � +proxσ2m +d +r/2 +g +(xd +1) +� ++ σdZd +1,1 , +and use a Taylor expansion of order 2 of +W(x, z, σ) = V +� +x + σ2 +2 g′ � +proxσ2mr/2 +g +(x) +� ++ σz +� +, +about σ = 0 with integral remainder: +W(x, z, σ) = W(x, z, 0) + ∂W +∂σ (x, z, σ) +���� +σ=0 +σ + 1 +2 +∂2W +∂σ2 (x, z, σ) +���� +σ=0 +σ2 ++ +� σ +0 +∂3W +∂σ3 (x, z, σ) +���� +σ=u +(σ − u)2 +2 +du . +44 + +Using the derivatives +W(x, z, 0) = V (x) , +∂W +∂σ (x, z, σ) +���� +σ=0 += V ′(x)z , +∂2W +∂σ2 (x, z, σ) = V ′′(x)z2 + V ′(x)g′(x) , +and recalling that E +� +Zd +1,1 +� += 0, E +� +(Zd +1,1)2� += 1, we have +E +� +V +� +yd(xd +1, Zd +1,1) +� +− V (xd +1) +� += σ2 +d +2 +� +V ′′(xd +1) + V ′(xd +1)g′(xd +1) +� ++ E +�� σd +0 +∂3W +∂σ3 (xd +1, Zd +1,1, σ) +���� +σ=u +(σd − u)2 +2 +du +� +. +Proceeding as in the previous proposition, we can bound +���� +� σd +0 +∂3W +∂σ3 (xd +1, Zd +1,1, σ) +�� +σ=u +(σd − u)2 +2 +du +���� ≤ A +� +1 + (Zd +1,1)N� � +1 + (xd +i )2N� +d−3α . +Setting Td to be the set in which (1 + (xd +1)2N) ≤ dα/2, the result follows by applying Markov’s +inequality as in Proposition 5. +Before proceeding to stating and proving the last auxiliary result, let us denote by ψ1 : R → +[0, +∞) the characteristic function of the distribution N(0, K2 +1(r)), where K2 +1(r) is given in Theo- +rem 1–(a), +ψ1(t) = exp(−t2ℓ2K1(r)2/2) , +and by ψd +1(xd; t) = +� +exp(itw)Qd +1(xd; dw) the characteristic functions associated with the law +Qd +1(xd; ·) = L +� +d−1/2 +d +� +i=2 +C2(xd +i , Zd +1,i) +� +, +Proposition 7. Under A0, A1 and A2, if α = 1/4, β = 1/8 and r > 0, there exists a sequence of +sets Hd ⊆ Rd such that +(i) +lim +d→∞ d2απd(Hc +d) = 0 , +(ii) for all t ∈ R +lim +d→∞ sup +xd∈Hd +|ψd +1(xd; t) − ψ1(t)| = 0 , +(iii) +d−1/2 +d +� +i=2 +C2(xd +i , Zd +1,i) +L +−→ N +� +0, ℓ4K2 +1(r) +2 +� +, +where +L +−→ denotes convergence in law, +45 + +(iv) +lim +d→∞ sup +xd∈Hd +�����E +� +1 ∧ exp +� +d−1/2 +d +� +i=2 +C2(xd +i , Zd +1,i) − ℓ4K2 +1(r) +2 +�� +− 2Φ +� +−ℓ2K1(r) +2 +������ , +where Φ is the distribution function of the standard normal random variable. +Proof. +(i) Define the functions hj(x) = [−g′′(x)g′(x)]j with j = 1, . . . , 4 and let Hd = Hd,1 ∩Hd,2 +where +Hd,1 = +� +xd ∈ Rd : +����� +1 +d +d +� +i=2 +hj(xd +i ) − +� +R +hj(u)π(u)du +����� ≤ d1/3 for j = 1, . . . , 4 +� +, +Hd,2 = +� +xd ∈ Rd : |hj(xd +i )| ≤ d2/3 for i = 1, . . . , d and j = 1, . . . , 4 +� +. +Using Chebychev’s inequality, the fact that the derivatives of g are bounded by polynomials +and that π has finite moments, we have d1/2πd((Hd,1)c) → 0 as d → ∞. Similarly, by Markov’s +inequality we have d1/2πd((Hd,2)c) → 0 as d → ∞. +(ii) We follow [34, Lemma 3(b)] and decompose +|ψd +1(xd; t) − ψ1(t)| ≤ +�����ψd +1(xd; t) − +d +� +i=2 +� +1 − t2 +2dv(xd +i ) +������ +(60) ++ +����� +d +� +i=2 +� +1 − t2 +2dv(xd +i ) +� +− +d +� +i=2 +exp +� +−t2 v(xd +i ) +2d +������ ++ +����� +d +� +i=2 +exp +� +−t2 v(xd +i ) +2d +� +− exp +� +−t2 ℓ2K1(r)2 +2 +������ +where v(xd +i ) = Var(C2(xd +i , Zd +1,i)) = ℓ4E[C2(xd +i , Zd +1,i)2]. +For the first term, decompose the +characteristic function ψd +1(xd; t) = �d +i=2 θd +i (xd +i ; t) as the product of the characteristic functions +of d−1/2Wi where we define Wi = C2(xd +i , Zd +1,i), using [15, equation (3.3.3)] as in the proof of +[15, Theorem 3.4.10] we obtain +����θd +i (xd +i ; t) − +� +1 − t2 +2dv(xd +i ) +����� ≤ EZ +� |t|3 +d3/2 +|Wi|3 +3! +∧ 2|t|2 +d +|Wi|2 +2! +� +≤ EZ +� |t|3 +d3/2 +|Wi|3 +3! +; |Wi| ≤ d1/2ε +� ++ t2 +d EZ +� +|Wi|2; |Wi| > d1/2ε +� +≤ ε|t|3 +6d EZ +� +|Wi|2� ++ +t2 +ε2d2 EZ +� +|Wi|4� +for any ε > 0. For sufficiently large d, we have that t2v(xd +i )/(2d) ≤ 1 for x ∈ Hd,2, and we +can use [15, Lemma 3.4.3] +�����ψd +j (xd; t) − +d +� +i=2 +� +1 − t2 +2dv(xd +i ) +������ ≤ +d +� +i=2 +�ε|t|3 +6d EZ +� +|Wi|2� ++ +t2 +ε2d2 EZ +� +|Wi|4�� +≤ εℓ4|t|3 +6 +(K2 +1(r) + D1d−1/3) + t2 +ε2d(E +� +|Wi|4� ++ D2ℓ8d−1/4) +46 + +where the last inequality follows from the fact that xd ∈ Hd,2 and D1, D2 are positive con- +stants. For any δ > 0 we can chose ε small enough so that the first term in the above is less +than δ/2 and we can chose d sufficiently large to make the second term less than δ/2. Thus, +for any δ > 0 we can find ε > 0 and d ∈ N such that +�����ψd +1(xd; t) − +d +� +i=2 +� +1 − t2 +2dv(xd +i ) +������ < δ; +the uniform convergence then follows. The second term in (60) converges to 0 uniformly for +all xd ∈ Hd,1; while for the third term in (60) we use again [15, Lemma 3.4.3] +����� +d +� +i=2 +exp +� +−t2 v(xd +i ) +2d +� +− exp +� +−t2 ℓ2K1(r)2 +2 +������ ≤ +d +� +i=2 +t4v(xd +i )2 +4d2 +→ 0 +for all x ∈ Hd,2. The result then follows. +(iii) This is a straightforward consequence of (ii) and the L´evy’s continuity Theorem (e.g. [41, +Theorem 1, page 322]). +(iv) This statement follows directly from (iii) and [33, Proposition 2.4]. +A.2 +Auxiliary Results for the Proof of Case (b) +First, we characterize the limit behaviour of the acceptance ratio (12). The following result is an +extension of [34, Lemma 1]. +Proposition 8. Under A0, A1, A2 and A3, if α = 1/6, β = 1/6 and r > 0, then +(i) the the log-acceptance ratio (50) satisfies +φd(xd +i , Yi) = d−1/2C3(xi, Zi) + d−2/3C4(xi, Zi) ++ d−5/6C5(xi, Zi) + d−1C6(xi, Zi) + C7(xi, Zi, σd), +where C3(xi, Zi) is given in (61), C4, C5, C6 are polynomials in Zi and the derivatives of g, +such that EX [EZ [Cj(X, Z)]] = 0 for j = 3, 4, 5 and +EX +� +EZ +� +C3(X, Z)2�� += −2EX [EZ [C6(X, Z)]] ; +(ii) there exists sets Fd ⊆ Rd with d2απd(F c +d) → 0 such that +lim +d→∞ sup +xd∈Fd +E +������ +d +� +i=2 +φd(xd +i , Yi) − d−1/2 +d +� +i=2 +C3(xd +i , Zi) + ℓ6K2(r)2 +2 +����� +� += 0, +where K2(r) is given in Theorem 1–(b). +47 + +Proof. Take one component of the log-acceptance ratio +φd(xd +i , Yi) = g(Yi) − g(xd +i ) + log q(Yi, xd +i ) − log q(xd +i , Yi) +with Yi = xd +i − σ2 +d +2 g′ � +proxσ2m +d +r/2 +g +(xd +i ) +� ++ σdZi and Zi ∼ N(0, 1). Proceeding as in the proof for +case (a), we have that φd(xd +i , Yi) = R1(xd +i , Z, σd) + R2(xd +i , Z, σd) where R1, R2 are given in (55). +Following the approach of [34] we approximate φd(xd +i , Yi) with a Taylor expansion about σd = 0. +(i) Using a Taylor expansion of order 7, we find that +φd(xd +i , Yi) = d−1/2C3(xd +i , Zi) + d−2/3C4(xd +i , Zi) + d−5/6C5(xd +i , Zi) ++ d−1C6(xd +i , Zi) + C7(xd +i , Zi, σd), +where +C3(x, z) = ℓ3 +6 +�1 +2g′′′(x)z3 − 3 +2g′′(x)g′(x)z (1 + 2r) +� +, +(61) +C4(x, z), C5(x, z) and C6(x, z) are given in Appendix C.1.2 and integral form of the remainder +C7(xd +i , Zi, σd) = +� σd +0 +∂7 +∂σ7 R(xd +i , Zi, σ)|σ=ϵ +(σd − ϵ)6 +6! +dϵ, +with ϵ between 0 and σd and the derivatives of R1 and R2 are given in Appendix C.2. In +addition, integrating by parts and using the moments of Z we find that EX [EZ [C3(X, Z)]] = +EX [EZ [C4(X, Z)]] = EX [EZ [C5(X, Z)]] = 0 and +EX [EZ [C6(X, Z)]] += ℓ6 +� +− 1 +16 +� +r + 2r2� +[g′′(X)g′(X)]2 − 1 +16 +�1 +2 + r +� +g′′(X)3 − 5 +96g′′′(X)2 +� +, +which shows that EX +� +EZ +� +C3(X, Z)2 + 2C6(X, Z) +�� += 0. +(ii) The proof of this result follows using the same steps as case (a) and is analogous to that of +[34, Lemma 1]. +Next, we compare the generator Ld and �Ld in (51) and (52) respectively, extending [34, Theorem +3]. +Proposition 9. Under A0, A1, A2 and A3, if α = 1/6, β = 1/6 and r > 0, there exists sets +Sd ⊆ Rd with d2απd(Sc +d) → 0 such that for any V ∈ C∞ +c (R, R) +lim +d→∞ sup +xd∈Sd +���LdV (xd) − �LdV (xd) +��� = 0 +and +lim +d→∞ sup +xd∈Sd +E +������ +πd(Y )qd(Y , xd) +πd(xd)qd(xd, Y ) ∧ 1 − +d +� +i=2 +φd(xd +i , yi) ∧ 1 +����� +� += 0. +48 + +Proof. The function x �→ exp(x) ∧ 1 is Lipschitz continuous with Lipschitz constant 1, hence +���LdV (xd) − �LdV (xd) +��� ≤ d2αE +� +|V (Y1) − V (xd +1)||R(xd +1, Z1, σd)| +� +where R(x, z, σ) = R1(x, z, σ) + R2(x, z, σ) as in (55). Using a Taylor expansion of order 1 about +σ = 0 with integral remainder: +R(x, z, σ) = R(x, z, 0) + ∂R +∂σ (x, z, σ)|σ=0σ + +� σ +0 +∂2R +∂σ2 (x, z, σ)|σ=ϵ(σ − ϵ)dϵ, +we obtain +R(xd +1, Z, σd) = +� σd +0 +∂2R +∂σ2 (x, z, σ)|σ=ϵ(σd − ϵ)dϵ, +where ∂2R +∂σ2 (x, z, σ) is bounded by the derivatives of g evaluated at +x + σ2 +d +2 proxσ2mr/2 +g +(x) + σz +and +proxσ2mr/2 +g +(x). +Under our assumptions, the derivatives of g are bounded by polynomials M0, it follows that there +exist polynomials p of the form A +� +1 + +� +proxσ2mr/2 +g +(x) +�N� � +1 + zN� � +1 + xN� � +1 + σN� +, for suffi- +ciently large A and sufficiently large even integer N, such that +����g(k) +� +x + σ2 +2 proxσ2mr/2 +g +(x) + σz +����� ∨ +���g(k) � +proxσ2mr/2 +g +(x) +���� ≤ p(proxσ2mr/2 +g +(x), x, z, σ). +Proceeding as in Proposition 8, we can bound +p(proxσ2mr/2 +g +(x), x, z, σ) ≤ A +� +1 + zN� � +1 + x2N� � +1 + σN� +. +Therefore, we have +|R(xd +1, Z1, σd)| ≤ A +� +1 + ZN +1 +� � +1 + (xd +i )2N� +× +� σd +0 +(1 + ϵN)(σd − ϵ)dϵ ≤ A +� +1 + ZN +1 +� � +1 + (xd +1)2N� σ2 +d +2 . +Since V ∈ C∞ +c (R, R), there exists a constant C such that +|V (Y1) − V (xd +1)| ≤ C|Y1 − xd +1| ≤ C +� +σd|Z1| + σ2 +d +2 |g′ � +proxσ2m +d +r/2 +g +(xd +1) +� +| +� +. +Under A3, g′ is Lipschitz continuous and we have, for some C ≥ 1, +|g′ � +proxσ2m +d +r/2 +g +(xd +1) +� +| ≤ C(1 + | proxσ2m +d +r/2 +g +(xd +1)|) ≤ C(1 + |xd +1|), +where we used the fact that proxλ +g is 1-Lipschitz continuous for all λ > 0. The result then follows +exactly as in [34, Theorem 3]. +49 + +The following result considers the convergence to the generator of the Langevin diffusion (53) +and is a generalization of [34, Lemma 2]. +Proposition 10. Under A0, A1, A2 and A3, if α = 1/6, β = 1/6 and r > 0, there exists sets +Td ⊆ Rd with d2απd(T c +d) → 0 such that for any V ∈ C∞ +c (R, R) +lim +d→∞ sup +xd∈Td +����d2αE +� +V (Y1) − V (xd +1) +� +− ℓ2 +2 (V ′′(xd +1) + g′(xd +1)V ′(xd +1)) +���� = 0. +Proof. The proof is identical to that of Proposition 6. +Before proceeding to stating and proving the last auxiliary result, let us denote by ψ2 : R → +[0, +∞) the characteristic function of the distribution N(0, K2 +2(r)), where K2 +2(r) is given in Theo- +rem 1, +ψ2(t) = exp(−t3ℓ2K2(r)2/2), +and by ψd +2(xd; t) = +� +exp(itw)Qd +2(xd; dw) the characteristic functions associated with the law +Qd +2(xd; ·) = L +� +d−1/2 +d +� +i=2 +C3(xd +i , Zi) +� +. +The following result extends [34, Lemma 3]. +Proposition 11. Under A0, A1, A2 and A3, if α = 1/6, β = 1/6 and r > 0, there exists a +sequence of sets Hd ⊆ Rd such that +(i) +lim +d→∞ d2απd(Hc +d) = 0, +(ii) for all t ∈ R +lim +d→∞ sup +xd∈Hd +|ψd +2(xd; t) − ψ2(t)| = 0, +(iii) +d−1/2 +d +� +i=2 +C3(xd +i , Zi) +L +−→ N +� +0, ℓ6K2 +2(r) +2 +� +, +where +L +−→ denotes convergence in law, +(iv) +lim +d→∞ sup +xd∈Hd +�����EZ +� +1 ∧ exp +� +d−1/2 +d +� +i=2 +C3(xd +i , Zi) − ℓ6K2 +2(r) +2 +�� +− 2Φ +� +−ℓ3K2(r) +2 +������ . +50 + +Proof. +(i) The proof is analogous to that of Proposition 7 and follows the same steps of that of +[34, Lemma 3(a)]. +(ii) The proof is analogous to that of Proposition 7 and follows the same steps of that of [34, +Lemma 3(b)]. +(iii) This is a straightforward consequence of (ii) and the L´evy’s continuity Theorem (e.g. [41, +Theorem 1, page 322]). +(iv) This statement follows directly from (iii) and [33, Proposition 2.4]. +A.3 +Auxiliary Results for the Proof of Case (c) +First, we characterize the limit behaviour of the acceptance ratio (12). +Proposition 12. Under A0, A1, A2 and A3 and, if α = 1/6, β = m/6 for m > 1 and r > 0, +then +(i) the log-acceptance ratio (50) satisfies +φd(xd +i , Yi) = d−1/2C3(xi, Zi) + d−2/3C4(xi, Zi) ++ d−5/6C5(xi, Zi) + d−1C6(xi, Zi) + C7(xi, Zi, σd), +where C3(xi, Zi) is given in (62), C4, C5, C6 are polynomials in Zi and the derivatives of g, +such that EX [EZ [Cj(X, Z)]] = 0 for j = 3, 4, 5 and +EX +� +EZ +� +C3(X, Z)2�� += −2EX [EZ [C6(X, Z)]] ; +(ii) there exists sets Fd ⊆ Rd with d2απd(F c +d) → 0 such that +lim +d→∞ sup +xd∈Fd +E +������ +d +� +i=2 +φd(xd +i , Yi) − d−1/2 +d +� +i=2 +C3(xd +i , Zi) + ℓ6K2 +3 +2 +����� +� += 0, +where K3 = K2(0). +Proof. Take one component of the log-acceptance ratio +φd(xd +i , Yi) = g(Yi) − g(xd +i ) + log q(Yi, xd +i ) − log q(xd +i , Yi) +with Yi = xd +i − σ2 +d +2 g′ � +proxσ2m +d +r/2 +g +(xd +i ) +� ++ σdZi and Zi ∼ N(0, 1). Proceeding as in the proof for +case (a), we have that φd(xd +i , Yi) = R1(xd +i , Z, σd) + R2(xd +i , Z, σd) where R1, R2 are given in (55). +Following the approach of [34] we approximate φd(xd +i , Yi) with a Taylor expansion about σd = 0. +(i) Using a Taylor expansion of order 7, we find that +φd(xd +i , Yi) = d−1/2C3(xd +i , Zi) + d−2/3C4(xd +i , Zi) + d−5/6C5(xd +i , Zi) ++ d−1C6(xd +i , Zi) + C7(xd +i , Zi, σd), +51 + +where +C3(x, z) = ℓ3 +6 +�1 +2g′′′(x)z3 − 3 +2g′′(x)g′(x)z +� +, +(62) +C4(x, z), C5(x, z) and C6(x, z) are given in Appendix C.1.3 and integral form of the remainder +C7(xd +i , Zi, σd) = +� σd +0 +∂7 +∂σ7 R(xd +i , Zi, σ)|σ=ϵ +(σd − ϵ)6 +6! +dϵ, +with ϵ between 0 and σd and the derivatives of R1 and R2 are given in Appendix C.2. In +addition, integrating by parts and using the moments of Z we find that EX [EZ [C3(X, Z)]] = +EX [EZ [C4(X, Z)]] = EX [EZ [C5(X, Z)]] = 0 and +EX [EZ [C6(X, Z)]] = ℓ6 +� +− 1 +32g′′(X)3 − 5 +96g′′′(X)2 +� +, +which shows that EX +� +EZ +� +C3(X, Z)2 + 2C6(X, Z) +�� += 0. +(ii) The proof of this result follows using the same steps as case (a) and is analogous to that of +[34, Lemma 1]. +Next, we compare the generator Ld and �Ld in (51) and (52) respectively. +Proposition 13. Under A0, A1, A2 and A3, if α = 1/6, β = m/6 for m > 1 and r > 0, there +exists sets Sd ⊆ Rd with d2απd(Sc +d) → 0 such that for any V ∈ C∞ +c (R, R) +lim +d→∞ sup +xd∈Sd +���LdV (xd) − �LdV (xd) +��� = 0 +and +lim +d→∞ sup +xd∈Sd +E +������ +πd(Y )qd(Y , xd) +πd(xd)qd(xd, Y ) ∧ 1 − +d +� +i=2 +φd(xd +i , yi) ∧ 1 +����� +� += 0. +Proof. The proof is identical to that of Proposition 9. +The following result considers the convergence to the generator of the Langevin diffusion (53). +Proposition 14. Under A0, A1, A2 and A3, if α = 1/6, β = m/6 for m > 1 and r > 0, there +exists sets Td ⊆ Rd with d2απd(T c +d) → 0 such that for any V ∈ C∞ +c (R, R) +lim +d→∞ sup +xd∈Td +����d2αE +� +V (Y1) − V (xd +1) +� +− ℓ2 +2 (V ′′(xd +1) + g′(xd +1)V ′(xd +1)) +���� = 0. +Proof. The proof is identical to that of Proposition 6. +52 + +Before proceeding to stating and proving the last auxiliary result, let us denote by ψ3 : R → +[0, +∞) the characteristic function of the distribution N(0, K2 +3), where K2 +3 = K2 +2(0), +ψ3(t) = exp(−t2ℓ3K2 +3/2), +and by ψd +3(xd; t) = +� +exp(itw)Qd +3(xd; dw) the characteristic functions associated with the law +Qd +3(xd; ·) = L +� +d−1/2 +d +� +i=2 +C3(xd +i , Zi) +� +. +Proposition 15. Under A0, A1, A2 and A3, if α = 1/6, β = m/6 for m > 1 and r > 0, there +exists a sequence of sets Hd ⊆ Rd such that +(i) +lim +d→∞ d2απd(Hc +d) = 0, +(ii) for all t ∈ R +lim +d→∞ sup +xd∈Hd +|ψd +3(xd; t) − ψ3(t)| = 0, +(iii) +d−1/2 +d +� +i=2 +C3(xd +i , Zi) +L +−→ N +� +0, ℓ6K2 +3 +2 +� +, +where +L +−→ denotes convergence in law, +(iv) +lim +d→∞ sup +xd∈Hd +�����EZ +� +1 ∧ exp +� +d−1/2 +d +� +i=2 +C3(xd +i , Zi) − ℓ6K2 +3 +2 +�� +− 2Φ +� +−ℓ3K3 +2 +������ . +Proof. +(i) The proof is analogous to that of Proposition 7 and follows the same steps of that of +[34, Lemma 3(a)]. +(ii) The proof is analogous to that of Proposition 7 and follows the same steps of that of [34, +Lemma 3(b)]. +(iii) This is a straightforward consequence of (ii) and the L´evy’s continuity Theorem (e.g. [41, +Theorem 1, page 322]). +(iv) This statement follows directly from (iii) and [33, Proposition 2.4]. +53 + +A.4 +Proof of Theorem 1 +Proof of Theorem 1. +(a) The asymptotic acceptance rate follows by combining Propositions 4– +6 with part (iv) of Proposition 7 as in the proof of [34, Theorem 1]. +To prove the weak +convergence of the process it suffices to show that there exists events F ⋆ +d ∈ Rd such that for +all t > 0 +P +� +Ld +t ∈ F ⋆ +d for all 0 ≤ s ≤ t +� +→ 1 +and +lim +d→∞ sup +xd∈F ⋆ +d +��LdV (xd) − LV (xd) +�� +for all V ∈ C∞ +c (R, R) [17, Chapter 4, Corollary 8.7]. We take F ⋆ +d = Fd ∩ Sd ∩ Td ∩ Hd. Then, +d2απd ((F ⋆ +d )c) → 0 and +P +� +Ld +t ∈ F ⋆ +d for all 0 ≤ s ≤ t +� +→ 1 +for all fixed t. Combining Propositions 4–7 we obtain convergence of the generators. +To obtain the value of a(ℓ, r) maximizing the speed, we observe that K2 +1(r) is a function of +the ratio r = c2/ℓ2m = c2/ℓ only, we can take c ∝ ℓ1/2 so that K2 +1(r) is constant for given +c. Using the same substitution as in [34, Theorem 2] we find that h(ℓ, r) is maximized at the +unique value of ℓ such that a(ℓ, r) = 0.452. +(b) The proof is analogous to that of case (a) replacing Propositions 4, 5, 6 and 7 with Proposi- +tions 8, 9, 10 and 11. To obtain the value of a(ℓ, r) maximizing the speed, we observe that +K2 +2(r) is a function of the ratio r = c2/ℓ2m = c2/ℓ2 only, we can take c ∝ ℓ so that K2 +2(r) is +constant for given c. Using the same substitution as in [34, Theorem 2] we find that h(ℓ, r) is +maximized at the unique value of ℓ such that a(ℓ, r) = 0.574. +(c) The proof is analogous to that of case (a) replacing Propositions 4, 5, 6 and 7 with Proposi- +tions 12, 13, 14 and 15. To obtain the value of a(ℓ, r) maximizing the speed, we observe that +K2 +3 is constant w.r.t. r, we can use the same substitution as in [34, Theorem 2] we find that +h(ℓ, r) is maximized at the unique value of ℓ such that a(ℓ, r) = 0.574. +B +Numerical Experiments +B.1 +Differentiable Targets +We collect here a number of numerical experiments confirming the results in Section 3.2. To do so, +we consider the Gaussian distribution in Example 1 and four algorithmic settings summarized in +Table 1 which correspond to the three cases identified in Theorem 1 and MALA. +The first plot in Figure 2–5 show that for values of α different from those identified in Theorem 1 +the acceptance ratio ad(ℓ, r) becomes degenerate as d increases. For the values of α identified in +Theorem 1 we analyze the influence of ℓ on the acceptance ad(ℓ, r) (second plot), obtaining for +d → +∞ the expression given by Theorem 1–(a) for Figure 2, the expression in Theorem 1–(b) for +Figures 3 and 5 and that in Theorem 1–(c) for Figure 4. +54 + +Case +Figure +Algorithm +α +β +m +r +(a) +2 +Proximal MALA +1/4 +1/8 +1/2 +1 +(b) +3 +P-MALA +1/6 +1/6 +1 +1 +(c) +4 +Proximal MALA +1/6 +1/2 +3 +2 +— +5 +MALA +1/6 +1/6 +1 +≈ 0 +Table 1: Algorithm setting for the simulation study on the Gaussian target. +Finally, we consider the relationship between acceptance ratio ad(ℓ, r) and the speed of the +diffusion h(ℓ, r) approximated by the expected squared jumping distance (see, e.g. [18]) +ESJDd := d2αE +� +(Xd +0 − Xd +1)2� +. +Looking at the last plot in Figure 2–5 we find that, even for relatively small values of d, the shape of +the plot of ESJDd as a function of the acceptance ad(ℓ, r) is similar to that of the theoretical limit. +This suggests that tuning the acceptance ratio to be approximately 0.452 when α = 1/4, β = 1/8 +and approximately 0.574 when α = 1/6, β = m/6 with m ≥ 1 should generally guarantee high +efficiency. +B.2 +Laplace Target +We collect here a number of numerical experiments confirming the results for the Laplace distribu- +tion in Section 3.2. Similarly to Appendix B.1 we consider three algorithmic settings, summarized +in Table 2. +Figure +Algorithm +α +β +m +r +6 +MALA +1/3 +1/3 +1 +≈ 0 +7 +P-MALA +1/3 +1/3 +1 +1 +8 +Proximal MALA +1/3 +1 +3 +2 +Table 2: Algorithm setting for the simulation study on the Laplace target. +The first plot in Figures 6–8 shows that for values of α ̸= 1/3 the acceptance ration ad(ℓ, r) +becomes degenerate as d increases; while the second plot shows that, for sufficiently large d, the +average acceptance ratio and the ESJDd converge to aL(ℓ) and hL(ℓ) given in Theorems 2 and 3, +respectively. In the case m = 3, r = 2 in Figure 8, we find that the behaviour for low values of +d significantly differs from the limiting one. For values of d lower than 130 the ESJDd achieves +its maximum at a value of the acceptance ad(ℓ, r) different from that predicted by Theorem 3. In +practice, this might mean that for low dimensional settings the recommended choice of a(ℓ, c) = +0.360 is far from optimal. Similar behaviours for small d have also been observed in the case of +RWM and MALA (e.g., [40, Section 2.1]). +55 + +0 +10 +20 +30 +40 +50 +60 +70 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 += 1 +8 += 1 +4 += 1 +2 +d +ad(ℓ, r) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +d = 15 +d = 50 +d = 300 +d = 10000 +theoretical limit +ℓ +ad(ℓ, r) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +d=15 +d=50 +d=300 +d=10000 +theory +ad(ℓ, r) +ESJDd +Figure 2: Case (a): Proximal MALA with Gaussian target and m = 1/2, r = 1. Average ac- +ceptance rate for different choices of α (first); acceptance rate as a function of ℓ for increasing +dimension d (second); ESJDd as a function of the acceptance rate ad(ℓ, r) (third). +56 + +10 +20 +30 +40 +50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 += 1 +3 += 1 +6 += 1 +12 +d +ad(ℓ, r) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +d = 3 +d = 41 +d = 300 +d = 10000 +theoretical limit +ℓ +ad(ℓ, r) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +d=41 +d=300 +d=30000 +theory +ad(ℓ, r) +ESJDd +Figure 3: Case (b): Proximal MALA with Gaussian target and m = 1, r = 1 (P-MALA). Average +acceptance rate for different choices of α (first); acceptance rate as a function of ℓ for increasing +dimension d (second); ESJDd as a function of the acceptance rate ad(ℓ, r) (third). +57 + +10 +20 +30 +40 +50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 += 1 +3 += 1 +6 += 1 +12 +d +ad(ℓ, r) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +d = 100 +d = 1000 +d = 60000 +theoretical limit +ℓ +ad(ℓ, r) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +d=100 +d=1000 +d=60000 +theory +ad(ℓ, r) +ESJDd +Figure 4: Case (c): Proximal MALA with Gaussian target and m = 3, r = 2. Average acceptance +rate for different choices of α (first); acceptance rate as a function of ℓ for increasing dimension d +(second); ESJDd as a function of the acceptance rate ad(ℓ, r) (third). +58 + +10 +20 +30 +40 +50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 += 1 +3 += 1 +6 += 1 +12 +d +ad(ℓ, r) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +d = 3 +d = 22 +d = 41 +theoretical limit +ℓ +ad(ℓ, r) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +d=22 +d=100 +d=5000 +theory +ad(ℓ, r) +ESJDd +Figure 5: Proximal MALA with Gaussian target and m = 1, r → 0 (MALA). Average acceptance +rate for different choices of α (first); acceptance rate as a function of ℓ for increasing dimension d +(second); ESJDd as a function of the acceptance rate ad(ℓ, r) (third). +59 + +10 +20 +30 +40 +50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 += 2 +3 += 1 +3 += 1 +6 +d +ad(ℓ, r) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.2 +0.4 +0.6 +0.8 +1.0 +d = 5 +d = 26 +d = 100 +theoretical limit +ℓ +ad(ℓ, r) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +d = 15 +d = 80 +d = 1000 +theoretical limit +ad(ℓ, r) +ESJDd +Figure 6: Proximal MALA with Laplace target and m = 1, r = 0 (sG-MALA). Average acceptance +rate for different choices of α (first); acceptance rate as a function of ℓ for increasing dimension d +(second); ESJDd as a function of the acceptance rate ad(ℓ, r) (third). +60 + +0 +10 +20 +30 +40 +50 +60 +70 +80 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 += 2 +3 += 1 +3 += 1 +6 +d +ad(ℓ, r) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.2 +0.4 +0.6 +0.8 +1.0 +d = 5 +d = 50 +d = 300 +d = 10000 +theoretical limit +ℓ +ad(ℓ, r) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +d = 300 +d = 1000 +d = 15000 +theoretical limit +ad(ℓ, r) +ESJDd +Figure 7: Proximal MALA with Laplace target and m = 1, r = 1 (P-MALA). Average acceptance +rate for different choices of α (first); acceptance rate as a function of ℓ for increasing dimension d +(second); ESJDd as a function of the acceptance rate ad(ℓ, r) (third). +61 + +0 +10 +20 +30 +40 +50 +60 +70 +80 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 += 2 +3 += 1 +3 += 1 +6 +d +ad(ℓ, r) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.2 +0.4 +0.6 +0.8 +1.0 +d = 15 +d = 32 +d = 58 +d = 300 +theoretical limit +ℓ +ad(ℓ, r) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +d = 15 +d = 50 +d = 150 +d = 1000 +theoretical limit +ad(ℓ, r) +ESJDd +Figure 8: Proximal MALA with Laplace target and m = 3, r = 2. Average acceptance rate for +different choices of α (first); acceptance rate as a function of ℓ for increasing dimension d (second); +ESJDd as a function of the acceptance rate ad(ℓ, r) (third). +62 + +C +Taylor Expansions for the Results on Differentiable Tar- +gets +C.1 +Coefficients of the Taylor Expansion +We collect here the coefficients of the Taylor expansions in Propositions 4, 8 and 12. +C.1.1 +Case (a) +If α = 1/4, β = 1/8 and r > 0, then the log-acceptance ratio (50) satisfies +φd(x, z) = d−1/2C2(x, z) + d−3/4C3(x, z) + d−1C4(x, z) + C5(x, z, σd) , +where +C2(x, z) = ℓ2 +2 (−rzg′′(x)g′(x)) , +and +C3(x, z) = ℓ3 +6 +�z3 +2 g′′′(x) − 3 +2z2rg′(x)g′′′(x) − 3 +2rz2 [g′′(x)]2 + 3 +4zr2g′′′(x) [g′(x)]2 +−3 +2zg′(x)g′′(x) + 3 +2r [g′(x)]2 g′′(x) +� +, +C4(x, z) = ℓ4 +24 +� +g(4)(x) +� +z4 − zr3 +2 [g′(x)]3 − 3z3rg′(x) + 3 +2z2r2 [g′(x)]2 +� ++ g′′′(x) +� +−6z2g′(x) − 9rz3g′′(x) + 9z2r2g′(x)g′′(x) + 9rz [g′(x)]2 +−9 +2zr3 [g′(x)]2 g′′(x) − 3 +2r2 [g′(x)]3 +� ++ 12rzg′(x) [g′′(x)]2 + 3g′′(x) [g′(x)]2 − 3z2 [g′′(x)]2 + 3z2r2 [g′′(x)]3 +−6r2 [g′(x)g′′(x)]2 − 3zr3g′(x) [g′′(x)]3� +, +and we use the integral form for the remainder +C5(x, z, σd) = +� σd +0 +∂5 +∂σ5 R(x, z, σ) +���� +σ=u +(σd − u)4 +4! +du , +with u between 0 and σd and the derivatives of R1 and R2 given in Appendix C.2. +C.1.2 +Case (b) +If α = 1/6, β = 1/6 and r > 0, the the log-acceptance ratio (50) satisfies +φd(xd +i , Yi) = d−1/2C3(xi, Zi) + d−2/3C4(xi, Zi) ++ d−5/6C5(xi, Zi) + d−1C6(xi, Zi) + C7(xi, Zi, σd), +63 + +where +C3(x, z) = ℓ3 +6 +�1 +2g′′′(x)z3 − 3 +2g′′(x)g′(x)z (1 + 2r) +� +, +and +C4(x, z) = ℓ4 +24 +� +z4g(4)(x) − 6z2g′′′(x)g′(x)(1 + r) +−3z2 [g′′(x)]2 (1 + 2r) + 3g′′(x) [g′(x)]2 (1 + 2r) +� +, +C5(x, z) = ℓ5 +120 +�3 +2z5g(5)(x) − 15z3g(4)(x)g′(x)(1 + r) + 15z [g′(x)]2 g′′′(x) +�3 +2 + 3r + r2 +� ++15z(1 + 4r + 2r2)g′(x) [g′′(x)]2 − 15z3g′′(x)g′′′(x)(1 + 3r) +� +, +C6(x, z) = ℓ6 +720 +� +2z6g(6)(x) − 30 (1 + r) z4g′(x)g(5)(x) + 45 +� +2 + 4r + r2� +z2 [g′(x)]2 g(4)(x) ++ 90 +� +r + r2� +z2 [g′′(x)]3 − 15 +� +2 + 6r + 3r2� +g′′′(x) [g′(x)]3 +− 30 (1 + 4r) z4g′′(x)g(4)(x) + 45 +� +3 + 16r + 6r2� +z2g′(x)g′′(x)g′′′(x) +−45 +2 (1 + 4r) z4 [g′′′(x)]2 − 45 +2 +� +1 + 8r + 8r2� +[g′(x)g′′(x)]2 +� +, +and integral form of the remainder +C7(xd +i , Zi, σd) = +� σd +0 +∂7 +∂σ7 R(xd +i , Zi, σ)|σ=ϵ +(σd − ϵ)6 +6! +dϵ, +with ϵ between 0 and σd and the derivatives of R1 and R2 are given in Appendix C.2. +C.1.3 +Case (c) +If α = 1/6, β = m/6 for m > 1 and r > 0, then the log-acceptance ratio (50) satisfies +φd(xd +i , Yi) = d−1/2C3(xi, Zi) + d−2/3C4(xi, Zi) ++ d−5/6C5(xi, Zi) + d−1C6(xi, Zi) + C7(xi, Zi, σd), +where +C3(x, z) = ℓ3 +6 +�1 +2g′′′(x)z3 − 3 +2g′′(x)g′(x)z +� +64 + +and +C4(x, z) = ℓ4 +24 +� +� +� +� +� +� +� +� +� +z4g(4)(x) − 6z2g′′′(x)g′(x) − 3z2 [g′′(x)]2 + 3g′′(x) [g′(x)]2 +−12zrg′(x)g′′(x) +if m = 3/2 +z4g(4)(x) − 6z2g′′′(x)g′(x) − 3z2 [g′′(x)]2 + 3g′′(x) [g′(x)]2 +otherwise +, +C5(x, z) = ℓ5 +120 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +3 +2z5g(5)(x) − 15z3g(4)(x)g′(x) + 45 +2 z [g′(x)]2 g′′′(x) + 15zg′(x) [g′′(x)]2 +−15z3g′′(x)g′′′(x) − 30z2r [g′′(x)]2 + 30r [g′(x)]2 g′′(x) +−30rz2g′(x)g′′′(x) +if m = 3/2 +3 +2z5g(5)(x) − 15z3g(4)(x)g′(x) + 45 +2 z [g′(x)]2 g′′′(x) + 15zg′(x) [g′′(x)]2 +−15z3g′′(x)g′′′(x) − 60zrg′(x)g′′(x) +if m = 2 +3 +2z5g(5)(x) − 15z3g(4)(x)g′(x) + 45 +2 z [g′(x)]2 g′′′(x) ++15zg′(x) [g′′(x)]2 − 15z3g′′(x)g′′′(x) +otherwise +, +C6(x, z) = ℓ6 +720 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +2z6g(6)(x) − 30z4g′(x)g(5)(x) + 90z2 [g′(x)]2 g(4)(x) − 30g′′′(x) [g′(x)]3 +− 45 +2 [g′(x)g′′(x)]2 − 30z4g′′(x)g(4)(x) + 135z2g′(x)g′′(x)g′′′(x) +− 45 +2 z4 [g′′′(x)]2 − 90z3rg′(x)g(4)(x) + 540rzg′(x) [g′′(x)]2 ++180rzg′(x)g′′(x) + 270rzg′′′(x) [g′(x)]2 − 45zrg′′(x)g′′′(x) ++90rz3g′′′(x) +if m = 3/2 +2z6g(6)(x) − 30z4g′(x)g(5)(x) + 90z2 [g′(x)]2 g(4)(x) − 30g′′′(x) [g′(x)]3 +− 45 +2 [g′(x)g′′(x)]2 − 30z4g′′(x)g(4)(x) + 135z2g′(x)g′′(x)g′′′(x) +− 45 +2 z4 [g′′′(x)]2 − 180z2rg′(x)g′′′(x) − 180z2r [g′′(x)]2 ++180rg′′(x) [g′(x)]2 +if m = 2 +2z6g(6)(x) − 30z4g′(x)g(5)(x) + 90z2 [g′(x)]2 g(4)(x) − 30g′′′(x) [g′(x)]3 +− 45 +2 [g′(x)g′′(x)]2 − 30z4g′′(x)g(4)(x) + 135z2g′(x)g′′(x)g′′′(x) +− 45 +2 z4 [g′′′(x)]2 − 360zrg′(x)g′′(x) +if m = 5/2 +2z6g(6)(x) − 30z4g′(x)g(5)(x) + 90z2 [g′(x)]2 g(4)(x) − 30g′′′(x) [g′(x)]3 +− 45 +2 [g′(x)g′′(x)]2 − 30z4g′′(x)g(4)(x) + 135z2g′(x)g′′(x)g′′′(x) +− 45 +2 z4 [g′′′(x)]2 +otherwise +, +and integral form of the remainder +C7(xd +i , Zi, σd) = +� σd +0 +∂7 +∂σ7 R(xd +i , Zi, σ)|σ=ϵ +(σd − ϵ)6 +6! +dϵ, +with ϵ between 0 and σd and the derivatives of R1 and R2 are given in Appendix C.2. +65 + +C.2 +Taylor Expansions of the Log-acceptance Ratio +C.2.1 +R1 +Recall that R1(x, z, σ) = −g +� +x − σ2 +2 g′ � +proxσ2mr/2 +g +(x) +� ++ σz +� ++ g(x). We compute the derivatives +of R1 w.r.t. σ evaluated at 0: +R1(x, z, 0) = 0, +∂R1 +∂σ (x, z, σ)|σ=0 = −g′(x)z, +∂2R1 +∂σ2 (x, z, σ)|σ=0 = −z2g′′(x) + [g′(x)]2 , +∂3R1 +∂σ3 (x, z, σ)|σ=0 = −z3g′′′(x) + 3g′(x)g′′(x) +� +z + ∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 +� +, +∂4R1 +∂σ4 (x, z, σ)|σ=0 = −z4g(4)(x) + 6z2g′′′(x)g′(x) − 3g′′(x) [g′(x)]2 ++ 12z [g′′(x)]2 ∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 ++ 6g′(x) +× +� +g′′(x) ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 + +� ∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 +�2 +g′′′(x) +� +. +In addition, for m > 1/2 we will also use +∂5R1 +∂σ5 (x, z, σ)|σ=0 = −z5g(5)(x) + 10z3g(4)(x)g′(x) − 15zg′′′(x) [g′(x)]2 ++ 30z [g′′(x)]2 ∂2 +∂2σ proxσ2mr/2 +g +(x)|σ=0 ++ 10g′(x)g′′(x) ∂3 +∂3σ proxσ2mr/2 +g +(x)|σ=0, +∂6R1 +∂σ6 (x, z, σ)|σ=0 = −z6g(6)(x) + 15z4g(5)(x)g′(x) − 45z2g(4)(x) [g′(x)]2 + 15g′′′(x) [g′(x)]3 +− 90g′(x) [g′′(x)]2 ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +− 60zg′′(x) ∂3 +∂σ3 proxσ2mr/2 +g +(x)|σ=0 ++ 90g′′(x)g′′′(x) ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 ++ 15g′(x) +× +� +g′′(x) ∂4 +∂σ4 proxσ2mr/2 +g +(x)|σ=0 + 3 +� ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +�2 +g′′′(x) +� +. +66 + +C.2.2 +R2 +Recall that +R2(x, z, σ) = 1 +2z2 +− 1 +2 +� +z − σ +2 g′ +� +proxσ2mr/2 +g +� +x + σz − σ2 +2 g′ � +proxσ2mr/2 +g +(x) +��� +− σ +2 g′ � +proxσ2mr/2 +g +(x) +��2 +. +67 + +We compute the derivatives of R2 w.r.t. σ evaluated at 0: +R2(x, z, 0) = 0, +∂R2 +∂σ (x, z, σ)|σ=0 = zg′(x), +∂2R2 +∂σ2 (x, z, σ)|σ=0 = − [g′(x)]2 + zg′′(x) +� +z + 2 ∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 +� +, +∂3R2 +∂σ3 (x, z, σ)|σ=0 = −3g′(x)g′′(x) +� +z + 2 ∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 +� ++ 3 +2z +�� +z + ∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 +�2 +g′′′(x) ++ +� +−g′(x) + 2z +∂2 +∂σ∂x proxσ2mr/2 +g +(x)|σ=0 + ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +� +g′′(x) ++ +� ∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 +�2 +g′′′(x) + ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0g′′(x) +� +, +∂4R2 +∂σ4 (x, z, σ)|σ=0 = −3 [g′′(x)]2 +� +z + 2 ∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 +�2 +− 6g′(x) +�� +z + ∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 +�2 +g′′′(x) ++ +� +−g′(x) + 2z +∂2 +∂σ∂x proxσ2mr/2 +g +(x)|σ=0 + ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +� +g′′(x) ++ +� ∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 +�2 +g′′′(x) + ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0g′′(x) +� ++ 2zg(4)(x) +� +z + ∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 +�3 ++ 6zg′′′(x) +� +z + ∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 +� +× +� +−g′(x) + 2z +∂2 +∂σ∂x proxσ2mr/2 +g +(x)|σ=0 + ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +� ++ 2zg′′(x) +� +−3g′′(x) ∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 − 3g′(x) ∂2 +∂σ∂x proxσ2mr/2 +g +(x)|σ=0 ++3z2 +∂3 +∂σ∂x2 proxσ2mr/2 +g +(x)|σ=0 + 3z +∂3 +∂σ2∂x proxσ2mr/2 +g +(x)|σ=0 ++ ∂3 +∂σ3 proxσ2mr/2 +g +(x)|σ=0 +� ++ 2zg(4)(x) +� ∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 +�3 ++ 2zg′′(x) ∂3 +∂σ3 proxσ2mr/2 +g +(x)|σ=0 ++ 6z ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +∂ +∂σ proxσ2mr/2 +g +(x)|σ=0g′′′(x). +68 + +We then proceed to get the derivatives needed for m > 1/2: +∂5R2 +∂σ5 (x, z, σ)|σ=0 = −15zg′′(x) +� +z2g′′′(x) + +� +−g′(x) + 2 ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +� +g′′(x) +� +− 5g′(x) +� +2z3g(4)(x) + 6zg′′′(x) +� +−g′(x) + ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +� ++2g′′(x) +� +3z +∂3 +∂σ2∂x proxσ2mr/2 +g +(x)|σ=0 + ∂3 +∂σ3 proxσ2mr/2 +g +(x)|σ=0 +� ++2g′′(x) ∂3 +∂σ3 proxσ2mr/2 +g +(x)|σ=0 +� ++ 5 +2g(5)(x)z5 + 5 +2zg′′(x) ∂4 +∂σ4 proxσ2mr/2 +g +(x)|σ=0 ++ 15g(4)(x)z3 +� +−g′(x) + ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +� ++ 15 +2 zg′′′(x) +� +−g′(x) + ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +�2 ++ 15 +2 zg′′′(x) +� ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +�2 ++ 10z2g′′′(x) +� +3z +∂3 +∂σ2∂x proxσ2mr/2 +g +(x)|σ=0 + ∂3 +∂σ3 proxσ2mr/2 +g +(x)|σ=0 +� ++ 5 +2zg′′(x) +� ∂4 +∂σ4 proxσ2mr/2 +g +(x)|σ=0 − 6g′′(x) ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +−6g′(x) +∂3 +∂σ2∂x proxσ2mr/2 +g +(x)|σ=0 + 6z2 +∂4 +∂σ2∂x2 proxσ2mr/2 +g +(x)|σ=0 ++4z +∂4 +∂σ3∂x proxσ2mr/2 +g +(x)|σ=0 +� +and +∂6R2 +∂σ6 (x, z, σ)|σ=0 = −45 +2 +� +z2g′′′(x) + +� +−g′(x) + 2 ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +� +g′′(x) +�2 +− 15zg′′(x) +� +2z3g(4)(x) + 6zg′′′(x) +� +−g′(x) + ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +� ++ 2g′′(x) +� +3z +∂3 +∂σ2∂x proxσ2mr/2 +g +(x)|σ=0 + ∂3 +∂σ3 proxσ2mr/2 +g +(x)|σ=0 +� ++2g′′(x) ∂3 +∂σ3 proxσ2mr/2 +g +(x)|σ=0 +� ++ 6g′(x)A(5) − zA(6), +69 + +with +A(5) = −5 +2g(5)(x)z4 − 5 +2g′′(x) ∂4 +∂σ4 proxσ2mr/2 +g +(x)|σ=0 +− 15g(4)(x)z2 +� +−g′(x) + ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +� +− 15 +2 g′′′(x) +� +−g′(x) + ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +�2 +− 15 +2 g′′′(x) +� ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +�2 +− 10zg′′′(x) +� +3z +∂3 +∂σ2∂x proxσ2mr/2 +g +(x)|σ=0 + ∂3 +∂σ3 proxσ2mr/2 +g +(x)|σ=0 +� +− 5 +2g′′(x) +� ∂4 +∂σ4 proxσ2mr/2 +g +(x)|σ=0 − 6g′′(x) ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +−6g′(x) +∂3 +∂σ2∂x proxσ2mr/2 +g +(x)|σ=0 ++6z2 +∂4 +∂σ2∂x2 proxσ2mr/2 +g +(x)|σ=0 + 4z +∂4 +∂σ3∂x proxσ2mr/2 +g +(x)|σ=0 +� +, +A(6) = −3 +� +10g′′′(x) ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +∂3 +∂σ3 proxσ2mr/2 +g +(x)|σ=0 ++g′′(x) ∂5 +∂σ5 proxσ2mr/2 +g +(x)|σ=0 + g(6)(x)z5 ++ 10g(5)(x)z3 +� +−g′(x) + ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +� ++ 15zg(4)(x) +� +−g′(x) + ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +�2 ++ 10g(4)(x)z2 +� ∂3 +∂σ3 proxσ2mr/2 +g +(x)|σ=0 + 3z +∂3 +∂σ2∂x proxσ2mr/2 +g +(x)|σ=0 +� ++ 10g′′′(x) +� +−g′(x) + ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 +� +× +� ∂3 +∂σ3 proxσ2mr/2 +g +(x)|σ=0 + 3z +∂3 +∂σ2∂x proxσ2mr/2 +g +(x)|σ=0 +� ++ 5g′′′(x)z +� +−6g′′(x) ∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 − 6g′(x) +∂3 +∂σ2∂x proxσ2mr/2 +g +(x)|σ=0 ++6z2 +∂4 +∂σ2∂x2 proxσ2mr/2 +g +(x)|σ=0 + 3z +∂4 +∂σ3∂x proxσ2mr/2 +g +(x)|σ=0 ++ ∂4 +∂σ4 proxσ2mr/2 +g +(x)|σ=0 +� ++ g′′(x) +� +−10g′′(x) ∂3 +∂σ3 proxσ2mr/2 +g +(x)|σ=0 − 10g′(x) +∂4 +∂σ3∂x proxσ2mr/2 +g +(x)|σ=0 ++ 5z +∂5 +∂σ4∂x proxσ2mr/2 +g +(x)|σ=0 + 10z2 +∂5 +∂σ3∂x2 proxσ2mr/2 +g +(x)|σ=0 ++ 10z3 +∂5 +∂σ2∂x3 proxσ2mr/2 +g +(x)|σ=0 +−30g′(x)z +∂4 +∂σ2∂x2 proxσ2mr/2 +g +(x)|σ=0 + ∂5 +∂σ5 proxσ2mr/2 +g +(x)|σ=0 +�� +. +70 + +C.3 +Derivatives of the Proximity Map for Differentiable Targets +Recall that, in the differentiable case, proxσ2mr/2 +g +(x) = − σ2mr +2 +g′(proxσ2mr/2 +g +(x)) + x then +∂ +∂σ proxσ2mr/2 +g +(x)|σ=0 = +� +� +� +� +� +− r +2g′(x) +if m = 1/2 +0 +if m > 1/2 +∞ +otherwise +∂2 +∂σ2 proxσ2mr/2 +g +(x)|σ=0 = +� +� +� +� +� +� +� +� +� +r2 +2 g′(x)g′′(x) +if m = 1/2 +−rg′(x) +if m = 1 +0 +if m > 1 +∞ +otherwise +∂3 +∂σ3 proxσ2mr/2 +g +(x)|σ=0 = +� +� +� +� +� +� +� +� +� +− 3r3 +8 g′′′(x) [g′(x)]2 − 3r3 +4 g′(x) [g′′(x)]2 +if m = 1/2 +−3rg′(x) +if m = 3/2 +0 +if m = 1, m > 3/2 +∞ +otherwise +∂4 +∂σ4 proxσ2mr/2 +g +(x)|σ=0 = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +< ∞ +if m = 1/2 +6r2g′(x)g′′(x) +if m = 1 +−12rg′(x) +if m = 2 +0 +if m = 3/2, m > 2 +∞ +otherwise +∂5 +∂σ5 proxσ2mr/2 +g +(x)|σ=0 = +� +� +� +� +� +� +� +� +� +< ∞ +if m = 1/2 +−60rg′(x) +if m = 5/2 +0 +if m = 1, m = 3/2, m = 2, m > 5/2 +∞ +otherwise +and +∂ +∂x proxσ2mr/2 +g +(x)|σ=0 = 1, +∂(k) +∂x(k) proxσ2mr/2 +g +(x)|σ=0 = 0, +71 + +for all integers k > 1. For the mixed derivatives we have +∂2 +∂σ∂x proxσ2mr/2 +g +(x)|σ=0 = +� +� +� +� +� +− r +2g′′(x) +if m = 1/2 +0 +if m > 1/2 +∞ +otherwise +∂3 +∂σ2∂x proxσ2mr/2 +g +(x)|σ=0 = +� +� +� +� +� +� +� +� +� +r2 +2 [g′′(x)]2 + r2 +2 g′(x)g′′′(x) +if m = 1/2 +−rg′′(x) +if m = 1 +0 +if m > 1/2 +∞ +otherwise +∂4 +∂σ3∂x proxσ2mr/2 +g +(x)|σ=0 = +� +� +� +� +� +� +� +� +� +< ∞ +if m = 1/2 +−3rg′′(x) +if m = 3/2 +0 +if m = 1, m > 3/2 +∞ +otherwise +∂5 +∂σ4∂x proxσ2mr/2 +g +(x)|σ=0 = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +< ∞ +if m = 1/2 +6r2 � +g′(x)g′′′(x) + [g′′(x)]2� +if m = 1 +−12rg′′(x) +if m = 2 +0 +if m = 3/2, m > 2 +∞ +otherwise +72 + +and +∂3 +∂σ∂x2 proxσ2mr/2 +g +(x)|σ=0 = +� +� +� +� +� +− r +2g′′′(x) +if m = 1/2 +0 +if m > 1/2 +∞ +otherwise +∂4 +∂σ∂x3 proxσ2mr/2 +g +(x)|σ=0 = +� +� +� +� +� +< ∞ +if m = 1/2 +0 +if m > 1/2 +∞ +otherwise +∂4 +∂σ2∂x2 proxσ2mr/2 +g +(x)|σ=0 = +� +� +� +� +� +� +� +� +� +< ∞ +if m = 1/2 +−rg′′′(x) +if m = 1 +0 +if m > 1/2 +∞ +otherwise +∂5 +∂σ3∂x2 proxσ2mr/2 +g +(x)|σ=0 = +� +� +� +� +� +� +� +� +� +< ∞ +if m = 1/2 +−3rg′′′(x) +if m = 3/2 +0 +if m = 1, m > 3/2 +∞ +otherwise +∂5 +∂σ2∂x3 proxσ2mr/2 +g +(x)|σ=0 = +� +� +� +� +� +� +� +� +� +< ∞ +if m = 1/2 +−rg(4)(x) +if m = 1 +0 +if m > 1 +∞ +otherwise +D +Moments and Integrals for the Laplace Distribution +D.1 +Moments of Acceptance Ratio for the Laplace Distribution +The indicator functions in the definition of φd identify four different regions: +R1 := +� +(x, z) : |x| ≤ σ2mr/2 ∧ +���� +� +1 − +1 +σ2(m−1)r +� +x + σz +���� ≤ σ2mr/2 +� +, +R2 := +� +(x, z) : |x| > σ2mr/2 ∧ +����x − σ2 +2 sgn(x) + σz +���� ≤ σ2mr/2 +� +, +R3 := +� +(x, z) : |x| ≤ σ2mr/2 ∧ +���� +� +1 − +1 +σ2(m−1)r +� +x + σz +���� > σ2mr/2 +� +, +R4 := +� +(x, z) : |x| > σ2mr/2 ∧ +����x − σ2 +2 sgn(x) + σz +���� > σ2mr/2 +� +, +73 + +with corresponding acceptance ratios +φ1 +d(x, z) = |x| − +���� +� +1 − +1 +σ2(m−1)r +� +x + σz +���� + z2 +2 +− +1 +2σ2 +�� +2 +σ2(m−1)r − +1 +σ4(m−1)r2 +� +x − +� +1 − +1 +σ2(m−1)r +� +σz +�2 +φ2 +d(x, z) = |x| − +����x − σ2 +2 sgn(x) + σz +���� + z2 +2 +− +1 +2σ2 +� +1 +σ2(m−1)rx + +� +1 − +1 +σ2(m−1)r +� �σ2 +2 sgn(x) − σz +��2 +φ3 +d(x, z) = |x| − +���� +� +1 − +1 +σ2(m−1)r +� +x + σz +���� + z2 +2 +− +1 +2σ2 +� +1 +σ2(m−1)rx − σz + σ2 +2 sgn +�� +1 − +1 +σ2(m−1)r +� +x + σz +��2 +φ4 +d(x, z) = |x| − +����x − σ2 +2 sgn(x) + σz +���� + z2 +2 +− +1 +2σ2 +�σ2 +2 sgn(x) − σz + σ2 +2 sgn +� +x − σ2 +2 sgn(x) + σz +��2 +. +Let us denote +A1 := +� +x : 0 ≤ x ≤ σ2mr +2 +� +, +A2 := +� +x : −σ2mr +2 +≤ x ≤ 0 +� +, +A3 := +� +x : x > σ2mr +2 +� +, +A4 := +� +x : x < −σ2mr +2 +� +, +and +B1 := +� +z : 0 ≤ +� +1 − +1 +σ2(m−1)r +� +x + σz ≤ σ2mr +2 +� +, +B2 := +� +z : −σ2mr +2 +≤ +� +1 − +1 +σ2(m−1)r +� +x + σz ≤ 0 +� +, +B3 := +� +z : +� +1 − +1 +σ2(m−1)r +� +x + σz > σ2mr +2 +� +, +B4 := +� +z : +� +1 − +1 +σ2(m−1)r +� +x + σz < −σ2mr +2 +� +, +74 + +and +C1 := +� +(x, z) : 0 ≤ x − σ2 +2 sgn(x) + σz ≤ σ2mr +2 +� +, +C2 := +� +(x, z) : −σ2mr +2 +≤ x − σ2 +2 sgn(x) + σz ≤ 0 +� +C3 := +� +(x, z) : x − σ2 +2 sgn(x) + σz > σ2mr +2 +� +, +C4 := +� +(x, z) : x − σ2 +2 sgn(x) + σz < −σ2mr +2 +� +, +so that +R1 = (A1 ∪ A2) ∩ (B1 ∪ B2), +R2 = (A3 ∪ A4) ∩ (C1 ∪ C2), +R3 = (A1 ∪ A2) ∩ (B3 ∪ B4), +R4 = (A3 ∪ A4) ∩ (C3 ∪ C4). +Proposition 16. Take X a Laplace random variable and Z a standard normal random variable +independent of X, then if σ2 = ℓ2d−2/3, we have +lim +d→+∞ dE [φd(X, Z)] = − +ℓ3 +3 +√ +2π. +Proof. Taking expectations of φi +d1Ri for i = 1, . . . , 4 and exploiting the symmetry of the laws of X +75 + +and Z, we can write +E +� +φ1 +d(X, Z)1R1(X, Z) +� += 2E +�� +1 +σ2(m−1)rX − σZ +� +1A1(X)1B1(X, Z) +� ++ 2E +�� +2X − +1 +σ2(m−1)rX + σZ +� +1A1(X)1B2(X, Z) +� ++ 2E +�� +Z2 +2 − +1 +2σ2 +�� +2 +σ2(m−1)r − +1 +σ4(m−1)r2 +� +X − +� +1 − +1 +σ2(m−1)r +� +σZ +�2� +×1A1(X)1B1∪B2(X, Z)] , +E +� +φ2 +d(X, Z)1R2(X, Z) +� += 2E +��σ2 +2 − σZ +� +1A3(X)1C1(X, Z) +� ++ 2E +�� +2X − σ2 +2 + σZ +� +1A3(X)1C2(X, Z) +� ++ 2E +�� +Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX + +� +1 − +1 +σ2(m−1)r +� �σ2 +2 − σZ +��2� +×1A3(X)1C1∪C2(X, Z)] , +E +� +φ3 +d(X, Z)1R3(X, Z) +� += 2E +�� +1 +σ2(m−1)rX − σZ +� +1A1(X)1B3(X, Z) +� ++ 2E +�� +2X − +1 +σ2(m−1)rX + σZ +� +1A1(X)1B4(X, Z) +� ++ 2E +�� +Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX − σZ + σ2 +2 +�2� +1A1(X)1B3(X, Z) +� ++ 2E +�� +Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX − σZ − σ2 +2 +�2� +1A1(X)1B4(X, Z) +� +, +E +� +φ4 +d(X, Z)1R4(X, Z) +� += 2E +�� +2X − σ2 +2 + σZ +� +1A3(X)1C4(X, Z) +� +. +Using the integrals in Appendix D.4 and Lebesgue’s dominated convergence theorem, we find that +76 + +for α = β = 1/3 and r ≥ 0 +lim +d→+∞ dE +� +φ1 +d(X, Z) +� += 0 +lim +d→+∞ dE +� +φ2 +d(X, Z) +� += −2 ℓ3r +4 +√ +2π +� 0 +−∞ +e−z2/2zdz = +ℓ3r +2 +√ +2π +lim +d→+∞ dE +� +φ3 +d(X, Z) +� += 3ℓ3r +8 +√ +2π +� 0 +−∞ +e−z2/2zdz − +ℓ3r +8 +√ +2π +� +∞ +0 +e−z2/2zdz = − ℓ3r +2 +√ +2π +lim +d→+∞ dE +� +φ4 +d(X, Z) +� += +ℓ3 +6 +√ +2π +� 0 +−∞ +e−z2/2z3dz = − +ℓ3 +3 +√ +2π, +which gives +lim +d→+∞ dE [φd(X, Z)] = +lim +d→+∞ d +� +E +� +φ1 +d(X, Z) +� ++ E +� +φ2 +d(X, Z) +� ++ E +� +φ3 +d(X, Z) +� ++ E +� +φ4 +d(X, Z) +�� += − +ℓ3 +3 +√ +2π. +For α = 1/3, β = m/3 for m > 1 and r ≥ 0 we have +lim +d→+∞ dE +� +φ1 +d(X, Z) +� += 0 +lim +d→+∞ dE +� +φ2 +d(X, Z) +� += 0 +lim +d→+∞ dE +� +φ3 +d(X, Z) +� += 0 +lim +d→+∞ dE +� +φ4 +d(X, Z) +� += +ℓ3 +6 +√ +2π +� 0 +−∞ +e−z2/2z3dz = − +ℓ3 +3 +√ +2π, +which gives +lim +d→+∞ dE [φd(X, Z)] = +lim +d→+∞ d +� +E +� +φ1 +d(X, Z) +� ++ E +� +φ2 +d(X, Z) +� ++ E +� +φ3 +d(X, Z) +� ++ E +� +φ4 +d(X, Z) +�� += − +ℓ3 +3 +√ +2π. +Proposition 17. Take X a Laplace random variable and Z a standard normal random variable +independent of X, then if σ2 = ℓ2d−2/3 +lim +d→+∞ d Var (φd(X, Z)) = +2ℓ3 +3 +√ +2π. +Proof. As a consequence of the previous Proposition we have +lim +d→+∞ dE [φd(X, Z)]2 = 0. +77 + +Then, because Rj ∩ Ri = ∅ for all j ̸= i, we have that +E +� +φ(X, Z)2� += E +� +φ1 +d(X, Z)2R1(X, Z) +� ++ E +� +φ2 +d(X, Z)2R2(X, Z) +� ++ E +� +φ3 +d(X, Z)2R3(X, Z) +� ++ E +� +φ4 +d(X, Z)2R4(X, Z) +� +, +and, exploiting again the symmetry of the laws of X and Z, we have +E +� +φ1 +d(X, Z)2R1(X, Z) +� += 2E +� +� +� +1 +σ2(m−1)rX − σZ + Z2 +2 − +1 +2σ2 +�� +2 +σ2(m−1)r − +1 +σ4(m−1)r2 +� +X − +� +1 − +1 +σ2(m−1)r +� +σZ +�2�2 +×1A1(X)1B1(X, Z)] , ++ 2E +� +� +� +2X − +1 +σ2(m−1)rX − σZ + Z2 +2 − +1 +2σ2 +�� +2 +σ2(m−1)r − +1 +σ4(m−1)r2 +� +X − +� +1 − +1 +σ2(m−1)r +� +σZ +�2�2 +×1A1(X)1B2(X, Z)] , +E +� +φ2 +d(X, Z)21R2(X, Z) +� += 2E +� +� +� +σ2 +2 − σZ + Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX + +� +1 − +1 +σ2(m−1)r +� �σ2 +2 − σZ +��2�2 +1A3(X)1C1(X, Z) +� +� ++ 2E +� +� +� +2X − σ2 +2 + σZ + Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX + +� +1 − +1 +σ2(m−1)r +� �σ2 +2 − σZ +��2�2 +×1A1(X)1C2(X, Z)] , +E +� +φ3 +d(X, Z)21R3(X, Z) +� += 2E +� +� +� +1 +σ2(m−1)rX − σZ + Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX − σZ + σ2 +2 +�2�2 +1A1(X)1B3(X, Z) +� +� ++ 2E +� +� +� +2X − +1 +σ2(m−1)rX + σZ + Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX − σZ − σ2 +2 +�2�2 +1A1(X)1B4(X, Z) +� +� , +E +� +φ4 +d(X, Z)21R4(X, Z) +� += 2E +�� +2X − σ2 +2 + σZ +�2 +1A3(X)1C4(X, Z) +� +. +Proceeding as for Proposition 16, using the integrals in Appendix D.4 and Lebesgue’s dominated +convergence theorem we can then show that for α = 1/3, β = m/3 for m ≥ 1 and r ≥ 0 +lim +d→+∞ d Var (φd(X, Z)) = +2ℓ3 +3 +√ +2π. +78 + +Proposition 18. Take X a Laplace random variable and Z a standard normal random variable +independent of X, then if σ2 = ℓ2d−2/3 we have +lim +d→+∞ dE +� +φd(X, Z)3� += 0. +Proof. Following the same structure of the previous propositions we have that +E +� +φ(X, Z)3� += E +� +φ1 +d(X, Z)3R1(X, Z) +� ++ E +� +φ3 +d(X, Z)2R2(X, Z) +� ++ E +� +φ3 +d(X, Z)3R3(X, Z) +� ++ E +� +φ4 +d(X, Z)3R4(X, Z) +� +, +exploiting again the symmetry of the laws of X and Z, using the integrals in Appendix D.4, the +dominated convergence theorem we can then show that +lim +d→+∞ dE +� +φd(X, Z)3� += 0. +D.2 +Bound on Second Moment of Acceptance Ratio for the Laplace Dis- +tribution +Lemma 3. Let Z be a standard normal random variable and σ = ℓ/dα for α = 1/3. Then, there +exists a constant C > 0 such that for all a ∈ R and d ∈ N: +E +� +φd(a, Z)2� +≤ C +d2α . +Proof. We consider the case a ≥ 0 and r ≥ σ2(m−1) only, all the other cases follow from identical +arguments. As in the derivation of the moments of φd in Appendix D.1, we distinguish four regions. +We recall that σ = ℓ/dα for α = 1/3 and thus σp+1 ≤ σp for all p ∈ N. Take r ≥ σ−2(m−1), for R1, +79 + +we have, using H¨older’s inequality multiple times, +E +� +φ1 +d(a, Z)21R1(a, Z) +� += E +� +φ1 +d(a, Z)21B1∪B2(a, Z) +� +≤ Cσ2 +� σ2m−1r/2−(1−1/σ2(m−1)r)a +−(1−1/σ2(m−1)r)a/σ +e−z2/2 +√ +2π +�� +1 +σ2m−1ra − z +�2 ++ +� +z2 +2σ − +1 +2σ3 +�� +2 +σ2(m−1)r − +1 +σ4(m−1)r2 +� +a − +� +1 − +1 +σ2(m−1)r +� +σz +�2�2� +� dz ++ Cσ2 +� −(1−1/σ2(m−1)r)a/σ +−σ2m−1r/2−(1−1/σ2(m−1)r)a +e−z2/2 +√ +2π +��2a +σ − +1 +σ2m−1ra + z +�2 ++ +� +z2 +2σ − +1 +2σ3 +�� +2 +σ2(m−1)r − +1 +σ4(m−1)r2 +� +a − +� +1 − +1 +σ2(m−1)r +� +σz +�2�2� +� dz +≤ Cσ2 +� +∞ +−∞ +e−z2/2 +√ +2π +� +4 +� a +σ +�2 ++ 2 +� +1 +σ2m−1ra +�2 ++ 2z2 +� +dz ++ Cσ2 +� σ2m−1r/2−(1−1/σ2(m−1)r)a +−σ2m−1r/2−(1−1/σ2(m−1)r)a +e−z2/2 +√ +2π +× +� +z2 +2σ − +1 +2σ3 +�� +2 +σ2(m−1)r − +1 +σ4(m−1)r2 +� +a − +� +1 − +1 +σ2(m−1)r +� +σz +�2�2 +≤ Cσ2 + Cσ2 +� σ2m−1r/2−(1−1/σ2(m−1)r)a +−σ2m−1r/2−(1−1/σ2(m−1)r)a +e−z2/2 +√ +2π +× +� +z4 +4σ2 + +1 +4σ6 +�� +2 +σ2(m−1)r − +1 +σ4(m−1)r2 +�4 +a4 + +� +1 − +1 +σ2(m−1)r +�4 +σ4z4 +�� +dz +≤ Cσ2, +where we used the fact that the moments of Z are bounded and a ≤ σ2mr/2 for the first term, +and the fact that z ≤ σ2m−1r/2 for the second one. +Proceeding as above, for R3, a > 0 and +80 + +r ≥ σ−2(m−1), we have +E +� +φ3 +d(a, Z)21R3(a, Z) +� += E +� +φ3 +d(a, Z)21B3∪B4(a, Z) +� +≤ Cσ2 +� +∞ +σ2m−1r/2−(1−1/σ2(m−1)r)a/σ +e−z2/2 +√ +2π +�� +1 +σ2m−1ra − z +�2 ++ +� +z2 +2σ − +1 +2σ3 +� +1 +σ2(m−1)ra − σz + σ2 +2 +�2�2� +� dz ++ Cσ2 +� −σ2m−1r/2−(1−1/σ2(m−1)r)a/σ +−∞ +e−z2/2 +√ +2π +��2a +σ − +1 +σ2m−1ra + z +�2 ++ +� +z2 +2σ − +1 +2σ3 +� +1 +σ2(m−1)ra − σz + σ2 +2 +�2�2� +� dz +≤ Cσ2 + Cσ2 +� +∞ +−∞ +e−z2/2 +√ +2π +� 1 +2σ3 +� +a2 +σ4(m−1)r2 + σ4 +4 − σ3z + +a +σ2m−4r − +2az +σ2m−3r +��2 +dz +≤ Cσ2, +where we used again the boundedness of the moments of Z, the fact that a ≤ σ2mr/2 and that +σp+1 ≤ σp. For R2 and a > 0, we have +E +� +φ2 +d(a, Z)21R2(a, Z) +� += E +� +φ2 +d(a, Z)21C1∪C2(a, Z) +� +≤ Cσ2 +� σ/2+σ2m−1r/2−a/σ +σ/2−a/σ +e−z2/2 +√ +2π +��σ2 +2 − σz +�2 ++ +� +z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)ra + +� +1 − +1 +σ2(m−1)r +� �σ2 +2 − σz +��2�2� +� dz ++ Cσ2 +� σ/2−a/σ +σ/2−σ2m−1r/2−a/σ +e−z2/2 +√ +2π +�� +2a − σ2 +2 + σz +�2 ++ +� +z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)ra + +� +1 − +1 +σ2(m−1)r +� �σ2 +2 − σz +��2�2� +� dz. +The first integral is bounded using the moments of Z, while for the third one let us denote +81 + +χ(a, σ, z) := a − σ2/2 + σz, then +� σ/2+σ2m−1r/2−a/σ +σ/2−σ2m−1r/2−a/σ +e−z2/2 +√ +2π +� +z2 +2σ − +1 +2σ3 +� +1 +σ2(m−1)ra + +� +1 − +1 +σ2(m−1)r +� �σ2 +2 − σz +��2�2 +dz += +� σ/2+σ2m−1r/2−a/σ +σ/2−σ2m−1r/2−a/σ +e−z2/2 +√ +2π +� +z2 +2σ − +1 +2σ3 +�χ(a, σ, z) +σ2(m−1)r + σ2 +2 − σz +�2�2 +dz +≤ C +� σ/2+σ2m−1r/2−a/σ +σ/2−σ2m−1r/2−a/σ +e−z2/2 +√ +2π +� +z2 +2σ − +1 +2σ3 +�σ2 +2 − σz +�2�2 +dz ++ C +� σ/2+σ2m−1r/2−a/σ +σ/2−σ2m−1r/2−a/σ +e−z2/2 +√ +2π +�χ(a, σ, z)2 +2σ4m−1r2 +�2 ++ +�χ(a, σ, z) +rσ2m−1 +�σ2 +2 − σz +��2 +dz; +recalling that in R2 we have |χ(a, σ, z)| ≤ σ2mr/2, we obtain that this term is also bounded by +Cσ2. For R4 and a > 0, we have +E +� +φ4 +d(a, Z)21R4(a, Z) +� += E +� +φ4 +d(a, Z)21C4(a, Z) +� += +� σ/2−σ2m−1r/2−a/σ +−∞ +e−z2/2 +√ +2π +� +2a − σ2 +2 + σz +�2 +dz += σ2 +� σ/2−σ2m−1r/2−a/σ +−∞ +e−z2/2 +√ +2π +�2a +σ − σ +2 + z +�2 +dz +Collecting all the terms together, we obtain +E +� +φd(a, Z)2� += +4 +� +i=1 +E +� +φi +d(a, Z)21Ti(a, Z) +� +≤ Cσ2 + Cσ2 +� σ/2−a/σ +−∞ +e−z2/2 +√ +2π +�2a +σ − σ +2 + z +�2 +dz. +Recall that σ = ℓd−1/3. To bound the last integral we use H¨older’s inequality +� σ/2−a/σ +−∞ +e−z2/2 +√ +2π +�2a +σ − σ +2 + z +�2 +dz ≤ C +� σ/2−a/σ +−∞ +e−z2/2 +√ +2π +��σ +2 + z +�2 ++ 4 +�σ +2 − a +σ +�2� +dz +≤ C +� σ/2−a/σ +−∞ +e−z2/2 +√ +2π +��ℓ2 +4 + z2 +� ++ 4 +�σ +2 − a +σ +�2� +dz. +The first term is bounded since the moments of Z are bounded. For the second term we use an +estimate of the Gaussian cumulative distribution function. Let κ(ℓ, d, a) := ℓd−1/3/2 − ad1/3/ℓ. +When z < κ(ℓ, d, a) < 0, we have 1 < z/κ(ℓ, d, a) and therefore +(2π)−1/2κ(ℓ, d, a)2 +� κ(ℓ,d,a) +−∞ +e−z2/2dz ≤ (2π)−1/2κ(ℓ, d, a) +� κ(ℓ,d,a) +−∞ +ze−z2/2dz, += (2π)−1/2κ(ℓ, d, a) exp(−κ(ℓ, d, a)2/2). +82 + +However y �→ ye−y2/2 is bounded over R, therefore (a, d) �−→ (2π)−1/2κ(ℓ, d, a) exp(−κ(ℓ, d, a)2/2) +is bounded over R∗ ++ × N. +If κ(ℓ, d, a) ≥ 0, then we still have κ(ℓ, d, a) < ℓ and thus have the +inequality +(2π)−1/2κ(ℓ, d, a)2 +� κ(ℓ,d,a) +−∞ +e−z2/2dz ≤ (2π)−1/2ℓ2 +� +∞ +−∞ +e−z2/2dz = ℓ2. +The result then follows since σ = ℓd−1/3. +D.3 +Additional Integrals for the Laplace Distribution +We collect here two auxiliary Lemmata which are used in the proof of Proposition 3. +Lemma 4. Take X a Laplace random variable and Z a standard normal random variable indepen- +dent of X. Let ˜X := X − +1 +σ2(m−1)rX1|X|≤σ2mr/2 − σ2 +2 sgn(X)1|X|>σ2mr/2 + σZ, then, for σ = ℓd−α +with α = 1/3, +E +� +1{sgn(X)̸=sgn( ˜ +X)} +� +→ 0 +if d → ∞. +Proof. Using the same strategy of Appendix D.1 and the symmetry of the laws of X, Z, we find +that +E +� +1{sgn(X)̸=sgn( ˜ +X)} +� += 2E [1A1(X)1B2(X, Z)] + 2E [1A3(X)1C2(X, Z)] ++ 2E [1A1(X)1B4(X, Z)] + 2E [1A3(X)1C4(X, Z)] . +Using the same strategy used to obtain the moments of φd in Appendix D.1, we find that +E [1A1(X)1B2(X, Z)] = o(1), +in addition +E [1A3(X)1C2(X, Z)] = +1 +2 +√ +2π +� σ/2−σ2m−1r/2 +−∞ +e−z2/2 +� σ2/2−σz+σ2mr/2 +σ2/2−σz +e−xdx dz + o(1) += +1 +2 +√ +2π +� σ/2−σ2m−1r/2 +−∞ +e−z2/2 +�σ2r +2 δm1 + ... +� +dz + o(1), +where δm1 is a Dirac’s delta, and +E [1A1(X)1B4(X, Z)] + E [1A3(X)1C4(X, Z)] += +1 +2 +√ +2π +� σ/2−σ2m−1r +−∞ +e−z2/2 +� σ2/2−σz−σ2mr/2 +0 +e−xdx dz + o(1) += +1 +2 +√ +2π +� σ/2−σ2m−1r +−∞ +e−z2/2 [−σz + ...] dz + o(1). +83 + +Since σ = ℓd−1/3 and the remainder terms of the Taylor expansions are bounded, Lebesque’s +dominated convergence theorem gives +E [1A3(X)1C2(X, Z)] → 0, +E [1A1(X)1B4(X, Z)] + E [1A3(X)1C4(X, Z)] → 0 +as d → ∞. +Lemma 5. Take X a Laplace random variable and Z a standard normal random variable indepen- +dent of X. Then, +dαE [|Z| |φd (X, Z)|] → 0 +for α = 1/3. +Proof. Using Cauchy-Schwarz’s inequality we have that +E [|Z| |φd (X, Z)|] ≤ E +� +Z2�1/2 E +� +φd(X, Z)2�1/2 ; +the first expectation is equal to one, and the second one converges to zero at rate d1/2 by Proposi- +tion 17. The result follows straightforwardly. +D.4 +Integrals for Moment Computations +We distinguish the case m = 1/2 and m ≥ 1 since the integration bounds significantly differ in +these two cases. For values between 1/2 and 1 the integrals are not finite. The expectations below +are obtained by integrating w.r.t. x and using a Taylor expansion about σ = 0 to obtain the leading +order terms. Using the Lagrange form of the remainder for the Taylor expansions, we find that the +remainder terms are all of the form σ1/α+1f(γ(σ, z))/(1/α + 1)! where γ(σ, z) is a point between +the limits of integration w.r.t. x and f : x �→ p(x)e−x, where p is a polynomial. Therefore, using +the boundedness of the remainder and Lebesgue’s dominated convergence theorem, the integrals +w.r.t. z of the remainder terms all converge to 0. +D.4.1 +First Moment +For simplicity, we only consider the case for r ≥ σ−2(m−1), the other case follows analogously. +Region R1 +Let us consider φ1 +d first. We have +A1 ∩ B1 = +� +� +� +� +� +� +� +0 ≤ x ≤ σ2mr +2 +if 0 ≤ z ≤ σ +2 +0 ≤ x ≤ +� +σ2mr +2 +− σz +� � +1 − +1 +σ2(m−1)r +�−1 +if σ +2 ≤ z ≤ σ2m−1r +2 +−σz +� +1 − +1 +σ2(m−1)r +�−1 ≤ x ≤ σ2mr +2 +if σ +2 − σ2m−1r +2 +≤ z ≤ 0 +, +A1 ∩ B2 = +� +� +� +� +� +� +� +0 ≤ x ≤ σ2mr +2 +if − σ2m−1r +2 +≤ z ≤ σ +2 − σ2m−1r +2 +0 ≤ x ≤ −σz +� +1 − +1 +σ2(m−1)r +�−1 +if σ +2 − σ2m−1r +2 +≤ z ≤ 0 +− +� +σ2mr +2 ++ σz +� � +1 − +1 +σ2(m−1)r +�−1 ≤ x ≤ σ2mr +2 +if σ +2 − σ2m−1r ≤ z ≤ − σ2m−1r +2 +, +84 + +and +A1 ∩ (B1 ∪ B2) = +� +� +� +� +� +� +� +� +� +� +� +� +� +0 ≤ x ≤ σ2mr +2 +if − σ2m−1r +2 +≤ z ≤ σ +2 +0 ≤ x ≤ +� +σ2mr +2 +− σz +� � +1 − +1 +σ2(m−1)r +�−1 +if σ +2 ≤ z ≤ σ2m−1r +2 +− +� +σ2mr +2 ++ σz +� � +1 − +1 +σ2(m−1)r +�−1 ≤ x ≤ σ2mr +2 +if σ +2 − σ2m−1r ≤ z ≤ − σ2m−1r +2 +. +The corresponding expectations are +E +�� +X +σ2(m−1)r − σZ +� +1A1(X)1B1(X, Z) +� += +1 +2 +√ +2π +� σ2m−1r/2 +σ/2 +e−z2/2 � +z4−2mσ4−2mξ(r) + . . . +� +dz ++ +1 +2 +√ +2π +� 0 +σ/2−rσ2m−1/2 +e−z2/2 � +z4−2mσ4−2mξ(r) + . . . +� +dz ++ o(1), +E +�� +2X − +X +σ2(m−1)r + σZ +� +1A1(X)1B2(X, Z) +� += +1 +2 +√ +2π +� 0 +σ/2−σ2m−1r/2 +e−z2/2 � +z4−2mσ4−2mξ(r) + . . . +� +dz ++ +1 +2 +√ +2π +� −σ2m−1r/2 +σ/2−σ2m−1r +e−z2/2 � +z4−2mσ4−2mξ(r) + . . . +� +dz ++ o(1), +where ξ : [0, +∞) → R is a function of r only which might change from one line to the other, and +E +�� +Z2 +2 − +1 +2σ2 +�� +2 +σ2(m−1)r − +1 +σ4(m−1)r2 +� +X − +� +1 − +1 +σ2(m−1)r +� +σZ +�2� +1A1(X)1B1∪B2(X, Z) +� += +1 +2 +√ +2π +� σ/2 +−σ2m−1r/2 +e−z2/2 [+ . . . ] dz ++ +1 +2 +√ +2π +� σ2m−1r/2 +σ/2 +e−z2/2 [+ . . . ] dz ++ +1 +2 +√ +2π +� −σ2m−1r/2 +σ/2−σ2m−1r +e−z2/2 � +z2σ2ξ(r) + . . . +� +dz, +where ξ : [0, +∞) → R is a function of r only which might change from one line to the other. +85 + +Region R2 +For φ2 +d, we have +A3 ∩ C1 = +� +σ2/2 − σz ≤ x ≤ −σz + σ2/2 + σ2mr/2 +if z < σ/2 − σ2m−1r/2 +σ2mr/2 < x ≤ −σz + σ2/2 + σ2mr/2 +if σ/2 − σ2m−1r/2 ≤ z ≤ σ/2 +, +A3 ∩ C2 = +� +σ2/2 − σz − σ2mr/2 ≤ x ≤ σ2/2 − σz +if z < σ/2 − σ2m−1r +σ2mr/2 < x ≤ σ2/2 − σz +if σ/2 − σ2m−1r < z < σ/2 − σ2m−1r/2 +and +A3 ∩ (C1 ∪ C2) = +� +� +� +� +� +σ2mr/2 ≤ x ≤ σ2mr/2 + σ2/2 − σz +if σ/2 − σ2m−1r ≤ z ≤ σ/2 +−σ2mr/2 + σ2/2 − σz < x ≤ σ2mr/2 + σ2/2 − σz +if z < σ/2 − σ2m−1r +. +The corresponding expectations are +E +��σ2 +2 − σZ +� +1A3(X)1C1(X, Z) +� += +1 +2 +√ +2π +� σ/2−σ2m−1r/2 +−∞ +e−z2/2 � +−rz +2 σ2m+1 + . . . +� +dz ++ +1 +2 +√ +2π +� σ/2 +σ/2−σ2m−1r/2 +e−z2/2 � +z2σ2 + . . . +� +dz, +E +�� +2X − σ2 +2 + σZ +� +1A3(X)1C2(X, Z) +� += +1 +2 +√ +2π +� σ/2−σ2m−1r +−∞ +e−z2/2 � +−rz +2 σ2m+1 + . . . +� +dz ++ +1 +2 +√ +2π +� σ/2−σ2m−1r/2 +σ/2−σ2m−1r +e−z2/2 � +−rz +2 σ2m+1 + . . . +� +dz, +and +E +�� +Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX + +� +1 − +1 +σ2(m−1)r +� �σ2 +2 − σZ +��2� +1A3(X)1C1∪C2(X, Z) +� += +1 +2 +√ +2π +� σ/2−σ2m−1r +−∞ +e−z2/2 �rz +2 σ2m+1 + . . . +� +dz ++ +1 +2 +√ +2π +� σ/2 +σ/2−σ2m−1r +e−z2/2 +� +−z3 +2rσ3−2m + . . . +� +dz. +Region R3 +For φ3 +d, we have, in the case r ≥ σ−2(m−1), +A1 ∩ B3 = +� +0 ≤ x ≤ σ2mr +2 +if z > σ2m−1r +2 +� +σ2mr +2 +− σz +� � +1 − +1 +σ2(m−1)r +�−1 < x ≤ σ2mr +2 +if σ +2 ≤ z ≤ σ2m−1r +2 +, +A1 ∩ B4 = +� +0 ≤ x ≤ σ2mr +2 +if z < σ +2 − σ2m−1r +0 ≤ x < +� +σ2mr +2 +− σz +� � +1 − +1 +σ2(m−1)r +�−1 +if σ +2 − σ2m−1r ≤ z ≤ − σ2m−1r +2 +. +86 + +The corresponding expectations are +E +�� +1 +σ2(m−1)rX − σZ + Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX − σZ + σ2 +2 +�2� +1A1(X)1B3(X, Z) +� += +1 +2 +√ +2π +� +∞ +σ2m−1r/2 +e−z2/2 � +−rz +8 σ2m+1 + . . . +� +dz ++ +1 +2 +√ +2π +� σ2m−1r/2 +σ/2 +e−z2/2 � +z3−2mσ3−2mξ(r) + . . . +� +dz, +E +�� +2X − +1 +σ2(m−1)rX + σZ + Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX − σZ − σ2 +2 +�2� +1A1(X)1B4(X, Z) +� += +1 +2 +√ +2π +� σ/2−σ2m−1r/2 +−∞ +e−z2/2 +�3 +8σ2m+1 + . . . +� +dz ++ +1 +2 +√ +2π +� −σ2m−1r/2 +σ/2−σ2m−1r/2 +e−z2/2 � +z3σ3−2mξ(r) + . . . +� +dz, +where ξ : [0, +∞) → R is a function of r only which might change from one line to the other. +Region R4 +Finally, for φ4 +d we have +A3 ∩ C4 = +� +z < σ +2 − σ2m−1r, σ2mr +2 +< x ≤ σ2 +2 − σz − σ2mr +2 +� +, +and +E +�� +2X − σ2 +2 + σZ +� +1A3(X)1C4(X, Z) +� += +1 +2 +√ +2π +� σ/2−σ2m−1r +−∞ +e−z2/2 +�z3 +6 σ3 + . . . +� +dz. +D.4.2 +Second Moment +For simplicity, we only consider the case for r ≥ σ−2(m−1), the other case follows analogously. +Region R1 +For φ1 +d we have +E +� +φ1 +d(X, Z)21A1(X)1B1(X, Z) +� += +1 +2 +√ +2π +� σ2m−1r/2 +σ/2 +e−z2/2 � +z3σ3ξ(r) + . . . +� +dz ++ +1 +2 +√ +2π +� 0 +σ/2−rσ2m−1/2 +e−z2/2 � +z3σ3ξ(r) + . . . +� +dz ++ o(1), +E +� +φ1 +d(X, Z)21A1(X)1B2(X, Z) +� += +1 +2 +√ +2π +� 0 +σ/2−σ2m−1r/2 +e−z2/2 � +z3σ3ξ(r) + . . . +� +dz ++ +1 +2 +√ +2π +� −σ2m−1r/2 +σ/2−σ2m−1r +e−z2/2 � +z3σ3ξ(r) + . . . +� +dz ++ o(1), +87 + +where ξ : [0, +∞) → R is a function of r only which might change from one line to the other. +Region R2 +For φ2 +d we have +E +� +φ2 +d(X, Z)21A3(X)1C1(X, Z) +� += +1 +2 +√ +2π +� σ/2−σ2m−1r/2 +−∞ +e−z2/2 +�rz2 +2 σ2m+2 + . . . +� +dz ++ +1 +2 +√ +2π +� σ/2 +σ/2−σ2m−1r/2 +e−z2/2 � +−z3σ3 + . . . +� +dz, +E +� +φ2 +d(X, Z)21A3(X)1C2(X, Z) +� += +1 +2 +√ +2π +� σ/2−σ2m−1r +−∞ +e−z2/2 � +z2σ2m+2 + . . . +� +dz ++ +1 +2 +√ +2π +� σ/2−σ2m−1r/2 +σ/2−σ2m−1r +e−z2/2 +� +−rz2 +2 σ2m+2 + . . . +� +dz. +Region R3 +For φ3 +d we have +E +� +� +� +1 +σ2(m−1)rX − σZ + Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX − σZ + σ2 +2 +�2�2 +1A1(X)1B3(X, Z) +� +� += +1 +2 +√ +2π +� +∞ +σ2m−1r/2 +e−z2/2 +�rz2 +24 σ2m+2 + . . . +� +dz ++ +1 +2 +√ +2π +� σ2m−1r/2 +σ/2 +e−z2/2 +�rz2 +24 σ2m+2 + . . . +� +dz, +E +� +� +� +2X − +1 +σ2(m−1)rX + σZ + Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX − σZ − σ2 +2 +�2�2 +1A1(X)1B4(X, Z) +� +� += +1 +2 +√ +2π +� σ/2−σ2m−1r/2 +−∞ +e−z2/2 +�7rz2 +24 σ2m+2 + . . . +� +dz ++ +1 +2 +√ +2π +� −σ2m−1r/2 +σ/2−σ2m−1r/2 +e−z2/2 +�7rz2 +24 σ2m+2 + . . . +� +dz, +Region R4 +For φ4 +d we have +E +�� +2X − σ2 +2 + σZ +�2 +1A3(X)1C4(X, Z) +� += +1 +2 +√ +2π +� σ/2−σ2m−1r +−∞ +e−z2/2 +� +−z3 +3 σ3 + . . . +� +dz. +D.4.3 +Third Moment +Having established that the only possible scaling is given by α = 1/3, β = m/3 with m ≥ 1, we +now proceed to bound the third moment of φd in this case. For simplicity, we only consider the +case for r ≥ 1, the other case follows analogously. +88 + +Since m ≥ 1, we find that E +� +φ1 +d(X, Z)31R1(X, Z) +� += o(1) as d → ∞ since the limits of integra- +tion all converge to 0. Then, using H¨older’s inequality for φ2 +d, we have +E +� +φ2 +d(X, Z)3� +≤ CE +��σ2 +2 − σZ +�3 +1A3(X)1C1(X, Z) +� ++ CE +�� +2X − σ2 +2 + σZ +�3 +1A3(X)1C2(X, Z) +� ++ CE +� +� +� +Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX + +� +1 − +1 +σ2(m−1)r +� �σ2 +2 − σZ +��2�3 +×1A3(X)1C1∪C2(X, Z)] += +C +2 +√ +2π +� σ/2−σ2m−1r/2 +−∞ +e−z2/2 +� +−rz3 +2 σ5 + ... +� +dz ++ +C +2 +√ +2π +� σ/2−σ2m−1r +−∞ +e−z2/2 +� +−rz3 +2 σ5 + ... +� +dz ++ +C +2 +√ +2π +� σ/2−σ2m−1r +−∞ +e−z2/2 � +z3σ5ξ(r) + ... +� +dz + o(1), +where ξ : [0, +∞) → R is a function of r only which might change from one line to the other. For +89 + +φ3 +d, we have, using again H¨older’s inequality, +E +� +� +� +1 +σ2(m−1)rX − σZ + Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX − σZ + σ2 +2 +�2�3 +1A1(X)1B3(X, Z) +� +� +≤ CE +�� +1 +σ2(m−1)rX − σZ +�3 +1A1(X)1B3(X, Z) +� ++ CE +� +� +� +Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX − σZ + σ2 +2 +�2�3 +1A1(X)1B3(X, Z) +� +� += +1 +2 +√ +2π +� +∞ +σ2m−1r/2 +e−z2/2 +� +−z3r +2 σ5 + ... +� +dz ++ +1 +2 +√ +2π +� +∞ +σ2m−1r/2 +e−z2/2 � +z3σ5ξ(r) + ... +� +dz + o(1) +E +� +� +� +2X − +1 +σ2(m−1)rX + σZ + Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX − σZ − σ2 +2 +�2�3 +1A1(X)1B4(X, Z) +� +� +≤ CE +�� +2X − +1 +σ2(m−1)rX + σZ +�3 +1A1(X)1B4(X, Z) +� ++ CE +� +� +� +Z2 +2 − +1 +2σ2 +� +1 +σ2(m−1)rX − σZ − σ2 +2 +�2�3 +1A1(X)1B4(X, Z) +� +� += +1 +2 +√ +2π +� +∞ +σ2m−1r/2 +e−z2/2 +�z3r +2 σ5 + ... +� +dz ++ +1 +2 +√ +2π +� +∞ +σ2m−1r/2 +e−z2/2 � +z3σ5ξ(r) + ... +� +dz + o(1), +where ξ : [0, +∞) → R is a function of r only which might change from one line to the other. +Finally, for φ4 +d we have +E +�� +2X − σ2 +2 + σZ +�3 +1A3(X)1C4(X, Z) +� += +1 +2 +√ +2π +� σ/2−σ2m−1r +−∞ +e−z2/2 +�z3 +10σ5 + ... +� +dz. +90 + diff --git a/KdE0T4oBgHgl3EQfigFl/content/tmp_files/load_file.txt b/KdE0T4oBgHgl3EQfigFl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d95e01ac8d9dea0c50d37cee3a8acaf22ced127 --- /dev/null +++ b/KdE0T4oBgHgl3EQfigFl/content/tmp_files/load_file.txt @@ -0,0 +1,3813 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf,len=3812 +page_content='Optimal Scaling Results for a Wide Class of Proximal MALA Algorithms Francesca R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Crucinio∗1, Alain Durmus2, Pablo Jim´enez3, and Gareth O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Roberts4 1CREST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ENSAE Paris 2Centre de Math´ematiques Appliqu´ees,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Ecole Polytechnique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Institut Polytechnique de Paris 3Sorbonne Universit´e and Universit´e Paris Cit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Laboratoire de Probabilit´es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Statistique et Mod´elisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' F-75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' France 4Department of Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' University of Warwick Abstract We consider a recently proposed class of MCMC methods which uses proximity maps in- stead of gradients to build proposal mechanisms which can be employed for both differentiable and non-differentiable targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' These methods have been shown to be stable for a wide class of targets, making them a valuable alternative to Metropolis-adjusted Langevin algorithms (MALA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and have found wide application in imaging contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The wider stability properties are obtained by building the Moreau-Yoshida envelope for the target of interest, which depends on a parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In this work, we investigate the optimal scaling problem for this class of algorithms, which encompasses MALA, and provide practical guidelines for the implementation of these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Proximal MALA Algorithms 5 3 Optimal scaling of Proximal MALA 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 Differentiable targets .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 Laplace target .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='CO] 6 Jan 2023 6 Proof of the Result for the Laplace distribution 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 Proof of Theorem 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 66 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 67 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3 Derivatives of the Proximity Map for Differentiable Targets .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 71 D Moments and Integrals for the Laplace Distribution 73 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 Moments of Acceptance Ratio for the Laplace Distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 73 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 Bound on Second Moment of Acceptance Ratio for the Laplace Distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 79 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3 Additional Integrals for the Laplace Distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 88 1 Introduction Gradient-based Markov chain Monte Carlo methods have proved to be very successful at sampling from high-dimensional target distributions [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The key to their success is that in many cases their mixing time appears to be better than their competitor algorithms which do not use gradient information (see for example [34]), while their implementation has similar computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Indeed, gradients of target densities can often be computed with computational complexity (in dimension d) which scales no worse than evaluation of the target density itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Gradient-based MCMC methods are mainly motivated from stochastic processes constructed to have the target density as limiting distribution [25, 8, 6, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Our analysis will concentrate on 2 Metropolis Adjusted Langevin Algorithm (MALA) and its proximal variants which are based on the Langevin diffusion dLt = dBt + ∇ log π(Lt) 2 dt , (1) where π denotes the target density with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' It is well-known that un- der appropriate conditions, (1) defines a continuous-time Markov process associated with a Markov semigroup which is reversible with respect to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' From this observation, it has been suggested to use a Euler-Maruyama (EM) approximation of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' This scheme has been popularized in statistics by [20] and referred to as the Unadjusted Langevin Algorithm (ULA) in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Due to the time- discretization, ULA does not have π as stationary distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To address this problem, [39] and independently Besag in his contribution to [20] proposed to add a Metropolis acceptance step at each iteration of the EM scheme, leading to the Metropolis Adjusted Langevin Algorithm (MALA) following [36] who also derive basic stability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The accept/reject step in this algorithm confers two significant advantages: it ensures that the resulting algorithm has exactly the correct invariant distribution, while step sizes can be chosen larger than in the unadjusted case as there is not need to make step size small to reduce discretization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' On the other hand, MALA algorithms are typically hard to analyze theoretically (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' [7, 13, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' However, [34] (see also [5, 32]) have established that MALA has better convergence properties than the Random Walk Metropolis (RWM) algorithm with respect to the dimension d from an optimal scaling perspective (see also [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Whereas gradient-based methods have been successively applied and offer interesting features, they are typically less robust than their vanilla alternatives (for example see [36]) while intuition suggests, and existing underpinning theory requires, that target densities need to be sufficiently smooth for the gradients to be aiding Markov chain convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Moreover, while gradient-based MCMC have been successful for smooth densities, there is no reason to believe that they should be effective for densities which are not differentiable at a subset D ⊆ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For non-smooth densities, [30] proposes modified gradient-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Their proposed P-MALA algorithm is inspired by the proximal algorithms popular in the optimization literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The main idea is to approximate the (possibly non differentiable but) log-concave target density π ∝ exp(−G) by substituting the potential G with its Moreau-Yoshida envelope Gλ (see (3) below for its definition), to obtain a distribution πλ whose level of smoothness is controlled by the proximal parameter λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Given this smooth approximation to π one can then build proposals based on time discretizations of the Langevin diffusion targeting πλ [30, 14]: ξk+1 = ξk − σ2 2 ∇Gλ(ξk) + σZk+1 , (2) where σ2 > 0 is a fixed stepsize and (Zk)k∈N∗ is a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' zero-mean Gaussian random variables with identity covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Our aims in this paper are broadly to provide theo- retical underpinning for a slightly larger family of proximal MALA algorithms, analyze how these methods scale with dimension, and to give insights and practical guidance into how they should be implemented supported by the theory we establish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proximal optimization and MCMC methods proved to be particularly well-suited for image estimation, where penalties involving the sparsity inducing norms are common [30, 14, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Similar targets are also common in sparse regression contexts [2, 19, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In these situations, the set of non-differentiability points for the target density π is a null set for the Lebesgue measure, and, 3 following [12], we shall focus on this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' However, in contrast to the conclusions of [12] for RWM, we shall demonstrate that optimal scaling of proximal MALA may be affected by non-smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' More precisely, in this work, we first extend the results of [31] and consider a wider range of proximal MALA algorithms, as well as a wider class of finite dimensional target distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We let both the steps size σ2 and the regularization parameter λ depend on the dimension d of the target and find that the scaling properties of proximal MALA depend on the relative speed at which λ and σ converge to 0 as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We start by considering a class of sufficiently differentiable target distributions π to which MALA can also be applied, to allow direct comparison between MALA and proximal MALA and thus between a gradient-based method and one which approximates the gradient through proximal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' When λ goes to 0 at least as fast as σ2, we find that the scaling properties of proximal MALA are equivalent to those of MALA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ2 should decay as d−1/3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' see Theorem 1–(b), Theorem 1–(c) and Theorem 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' when λ converges to 0 more slowly than σ2, proximal MALA is less efficient than MALA with σ2 decaying as d−1/2 (Theorem 1–(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' As in some cases the proximal operator for a given distribution π is cheaper to compute than ∇ log π [29, 11, 30], we anticipate that proximal MALA with an appropriately tuned λ might provide a cheaper alternative to MALA retaining similar scaling properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We then turn to the optimal scaling of proximal MALA applied to the Laplace distribution π(x) ∝ e−|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We focus on this particular non-smooth target since it is the most widely used in applications of proximal MALA, including image deconvolution [30, 14, 43], LASSO, and sparse regression [2, 19, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We establish that non-differentiability of the target even at one point leads to a different optimal scaling than MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In particular, the step size has to scale as d−2/3 and not as d−1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' This appears to be a new optimal scaling for Metropolis MCMC algorithms which is between the one of RWM and MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' From the conclusion for smooth target distributions, we restrict our study to the choice of λ going to 0 at least as fast as σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The proof of the result for the differentiable case extends that of [34] for MALA, while the structure of the proof for the Laplace target is similar to that of [12] and constitutes the main element of novelty in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' As a special case of the result for the Laplace distribution, we also obtain the optimal scaling for MALA on Laplace targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We point out that the strategy adopted in the proof of this result is not unique to the Laplace distribution, and could be applied to other distributions provided that the required integrals can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To sum up, our main contributions are: 1) We extend the result of [31] beyond the Gaussian case, covering all finite dimensional (suffi- ciently) differentiable targets, and show that, in some cases, proximal MALA affords the same scaling properties of MALA if the proximal parameter λ is chosen appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 2) Motivated by applications in imaging and sparse regression applications, we study the scaling of proximal MALA methods for the Laplace target, and show that for values of λ decaying sufficiently fast, the optimal scaling of proximal MALA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' the choice for σ2, is different from the one for MALA on differentiable targets and is of order d−2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 3) We use the insights obtained with the aforementioned results to provide practical guidelines for the selection of the proximal parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Organization of the paper The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In Section 2, we rigorously introduce the class of proximal MALA algorithms that are studied and discuss related works on optimal scaling for MCMC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 we state the main result for differentiable targets, showing that the scaling properties of proximal MALA depend on the relative speed at 4 which λ goes to 0 with respect to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 we obtain a scaling limit for proximal MALA when π is a Laplace distribution, as a special case of our result we also obtain the scaling properties of a sub-gradient version of MALA for this target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We collect in Section 4 the main practical takeaways from these results and discuss possible extensions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Finally, in Section 6 we prove the result for the Laplace distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The proof of the result for differentiable targets is postponed to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 2 Proximal MALA Algorithms We now introduce the general class of proximal MALA algorithms, first studied in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' This class of algorithms aims at sampling from a density with respect to the Lebesgue measure on Rd of the form π(x) = exp(−G(x))/ � Rd exp(−G(˜x))d˜x, with G satisfying the following assumption A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The function G : Rd → R is convex, proper and lower semi-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The main idea behind proximal MALA is to approximate the (possibly non differentiable) target density π by approximating the potential G with its Moreau-Yoshida envelope Gλ : Rd → R defined for λ > 0 by Gλ(x) = min u∈Rd[G(u) + ∥u − x∥2/(2λ)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (3) Since G is supposed to be convex, by [38, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='26], the Moreau-Yoshida envelope is well- defined, convex and continuously differentiable with ∇Gλ(x) = λ−1(x − proxλ G(x)) , proxλ G(x) = arg min u∈Rd[G(u) + ∥u − x∥2/(2λ)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (4) The proximity operator x �→ proxλ G(x) behaves similarly to a gradient mapping and moves points in the direction of the minimizers of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In the limit λ → 0 the quadratic penalty dominates (4) and the proximity operator coincides with the identity operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' proxλ G(x) = x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' in the limit λ → ∞, the quadratic penalty term vanishes and (4) maps all points to the set of minimizers of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' It was shown in [14, Proposition 1] that, under A0, � Rd exp(−Gλ(x))dx < ∞, and therefore the probability density πλ ∝ exp(−Gλ) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In addition, it has been shown that ∥π − πλ∥TV → 0 as λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Based on this observation and since as we have emphasized πλ is now continuously differentiable, it has been suggested in [30, 14] to use the discretization of the Langevin diffusion associated with πλ given by (2), which can be rewritten using (4) as ξk+1 = � 1 − σ2 2λ � ξk + σ2 2λ proxλ G(ξk) + σZk+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (5) Similarly to other MCMC methods based on discretizations of the Langevin diffusion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' [36]), one can build unadjusted schemes which target πλ, expecting draws from these schemes to be close to draws from π for small enough λ, or add a Metropolis-Hastings step to ensure that the resulting algorithm targets π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Unadjusted proximal MCMC methods have been analyzed in [14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' in this paper we focus on Metropolis adjusted proximal MCMC methods and study their scaling properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' More precisely, at each step k and given the current state of the Markov chain Xk, a candidate Yk+1 is generated from the transition density associated to (5), (x, y) �→ q(x, y) = ϕ(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' [1−σ2/(2λ)]x+σ2 proxλ G(x)/2λ, σ2 Id), where ϕ(· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' u, Σ) stands for the d-dimension Gaussian density with mean u and covariance matrix Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Given Xk and Yk+1, Then, the next state is set as: Xk+1 = Yk+1bk+1 + Xk(1 − bk+1) , bk+1 = 1R+ �π(Yk+1)q(Yk+1, Xk) π(Xk)q(Xk, Yk+1) ∧ 1 − Uk+1 � , (6) 5 where (Ui)i∈N∗ is a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' uniform random variables on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The value of λ characterizes how close the distribution πλ is to the original target π and therefore how good the proposal is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Small values of λ provide better approximations to π and therefore better proposals (see [14, Proposition 1]), while larger values of λ provide higher levels of smoothing for non-differentiable distributions (see [30, Figure 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In the case λ = σ2/2 we obtain the special case of proximal MALA referred to as P-MALA in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The main contribution of this paper is to analyze the optimal scaling for proximal MALA defined by (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Optimal scaling and related works We briefly summarize here some examples of MCMC algorithms and their optimal scaling results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' a full review is out of the scope of this paper and we only mention algorithms to which we will compare proximal MALA in the development of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Popular examples of Metropolis MCMC are RWM and MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' RWM uses as a proposal the transition density (x, y) �→ ϕ(y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' x, σ2 Id), where σ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The MALA scheme uses as proposal (x, y) �→ ϕ(y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' x + (σ2/2)∇ log π(x), σ2 Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' As we will show in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1, proximal MALA can be considered as an extension of MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' A natural question to address when implementing Metropolis adjusted algorithms is how to set the parameter σ2 (variance parameter for RWM, step size parameter for MALA) to maximize the efficiency of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Small values of σ2 result in higher acceptance probability and cause sticky behaviour, while large values of σ2 result in a high number of rejections with the chain (Xk)k≥0 moving slowly [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Optimal scaling studies aim to address this question by investigating how σ2 should behave with respect to the dimension d of the support of π in the high dimensional setting d → ∞, to obtain the best compromise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The standard optimal scaling set-up considers the case of d-dimensional targets πd which are product form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' πd(xd) = d � i=1 π(xd i ) , (7) where xd i stands for the i-th component of xd and π is a one-dimensional probability density with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under appropriate assumptions on the regularity of π, and assuming that the MCMC algorithm is initialized at stationarity, the optimal value of σ2 scales as ℓ/dα with ℓ > 0, α = 1 for RWM [33] and α = 1/3 for MALA [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' By setting α to these values, it is then possible to show that each as d → ∞ each 1-dimensional component of the Markov chain defined by RWM and MALA, appropriately rescaled in time, converges to the Langevin diffusion dLt = h(ℓ)1/2dBt − h(ℓ) 2 [log π]′(x)dt , where (Bt)t≥0 is a standard Brownian motion and h(ℓ), referred to as speed function of the diffusion, is a function of the parameter ℓ > 0 that we may tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Indeed, it is well-known that (Lh(ℓ)t)t≥0 is a solution of the Langevin diffusion (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' As a result, we may identify the values of ℓ maximizing h(ℓ) for the algorithms at hand to approximate the fastest version of the Langevin diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The optimal values for ℓ results in an optimal average acceptance probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='234 for RWM and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='574 for MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 6 The scaling properties allow to get an intuition of the efficiency of the corresponding algorithms: RWM requires O(d) steps to achieve convergence on a d-dimensional target, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' its efficiency is O(d−1), while MALA has efficiency O(d−1/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' While these results are asymptotic in d, the insights obtained by considering the limit case d → ∞ prove to be useful in practice [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In the context of non-smooth and even discontinuous target distributions, studying the simpler RWM algorithm applied to a class of distributions on compact intervals, [27, 28] show that the lack of smoothness effects the optimal scaling of RWM with respect to dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' More precisely, they show that for a class of discontinuous densities which includes the uniform distribution on [0, 1], the optimal scaling of RWM is of order O(d−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' On the other hand, in the case where the set of non-differentiability D of π is a null set with respect to the Lebesgue measure, [12] shows that under appropriate conditions, including Lp differentiability, the optimal scaling of RWM is of order O(d−1) still.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The scaling properties of proximal MALA have been partially investigated in [31], which shows that P-MALA, obtained when λ = σ2/2, has the same scaling properties of MALA for the finite dimensional Gaussian density and for a class of infinite dimensional target measures (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 therein, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 3 Optimal scaling of Proximal MALA We consider the same set up of [34] and briefly recalled above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Given a real-valued function g : R → R satisfying A0 we consider the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' d-dimensional target specified by (7) with π(x) ∝ exp(−g(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (8) Since for any xd, G(xd) = �d i=1 g(xd i ), we have by [29, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1] proxλ G(xd) = (proxλ g(xd 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , proxλ g(xd d))⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' It follows that the distribution of the proposal (10) with target πd in (7)-(8) is also product form qd(xd, yd) = �d i=1 q(xd i , yd i ) , q(xd i , yd i ) = 1 (2πσ2)1/2 exp � −(yd i −xd i +σ2g′[proxλ g (xd i )]/2) 2 2σ2 � , with λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For any dimension d ∈ N∗, we denote by (Xd k)k∈N the Markov chain defined by the Metropolis recursion (6) with target distribution πd and proposal density qd and associated to the sequence of candidate moves Y d k+1 = � 1 − σ2 2λ � Xd k + σ2 2λ proxλ G(Xd k) + σZd k+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (9) As mentioned in the introduction, the focus of this work is on investigating the optimal depen- dence of the proposal variance σ2 on the dimension d of the target π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In this section, we make the dependence of the proposal variance on the dimension explicit and let σ2 d = ℓ2/d2α and λd = c2/2d2β for some α, β > 0 and some constants c, ℓ independent on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Thus, we can write λd as a function of σd, λd = σ2m d r/2, where we defined r = c2/ℓ2m > 0 and m = β/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' By writing λd as a function of σd we can decouple the effect of the constants c, ℓ from that of the dependence on d (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The value of m controls the relative speed at which σd and λd converge to 0 as d → ∞, when m = 1, 7 σd and λd decay to 0 at the same rate, for m > 1 the decay of λd is faster than that of σd and for m < 1 the decay of λd is slower than that of σd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The parameter r allows to refine the comparison between σd and λd as β = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In the limit r → 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' λd/σd → 0 if α = β), the proposal (10) coincides with that of MALA, in the case m = 1, r = 1 we get the P-MALA algorithm studied in [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Note that for all other values of r, m we have a family of proposals whose behaviour depends on r and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 Differentiable targets We start with the case where π is continuously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Since MALA can be applied to this class of targets, the results obtained in this section allow direct comparison of proximal MALA algorithms with MALA and thus between gradient-based algorithms (MALA) and algorithms that use proximal operator-based approximations of the gradient (proximal MALA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' If G = − log π is continuously differentiable, using [3, Corollary 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6], proxλ G(x) = −λ∇G(proxλ G(x)) + x, and (5) reduces to ξk+1 = ξk − σ2 2 ∇G(proxλ G(ξk)) + σZk+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (10) Hence, the value of λ controls how close to ξk is the point at which the gradient is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For λ → 0, the proximal MALA proposal becomes arbitrarily close to that of MALA, while, as λ increases (10) moves away from MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Our main result, Theorem 1 below, shows that the relative speed of decay (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' m) influences the optimal scaling of the resulting proximal MALA algorithm, while the constant r influences the speed function of the limiting diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We make the following assumptions on the regularity of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' g is a C8-function whose derivatives are bounded by some polynomial: there exists k0 ∈ N such that sup x∈R max i∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',8}[g(i)(x)/(1 + |x|k0)] < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Note that under A0 and A1, [14, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1] implies that � R xk exp(−g(x))dx < ∞ for any k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We also assume that the sequence of proximal MALA algorithms is initialized at stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For any d ∈ N∗, Xd 0 has distribution πd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The assumptions above closely resemble those of [34] used to obtain the optimal scaling results for MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In particular, A1 ensures that we can approximate the log-acceptance ratio in (6) with a Taylor expansion, while A2 avoids technical complications due to the transient phase of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We discuss how the latter assumption could be relaxed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For technical reasons, and to allow direct comparisons with the results established in [34] for MALA, we will also consider the following regularity assumption A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The function g′ is Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We denote by Ld t the linear interpolation of the first component of the discrete time Markov chain (Xd k)k≥0 obtained with the generic proximal MALA algorithm described above Ld t = (⌈d2αt⌉ − d2αt)Xd ⌊d2αt⌋,1 + (d2αt − ⌊d2αt⌋)Xd ⌈d2αt⌉,1 , (11) 8 where ⌊·⌋ and ⌈·⌉ denote the lower and upper integer part functions, respectively, and denote by Xd k,1 the first component of Xd k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The following result shows that in the limit d → ∞ the properties of proximal MALA depend on the relative speed at which σ2 d = ℓ2/d2α and λd = c2/2d2β converge to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Recall that we set r = c2/ℓ2m > 0 and under A2, consider for any d ∈ N∗, ad(ℓ, r) = E � πd(Y d 1 )qd(Y d 1 , Xd 0) πd(Xd 0)qd(Xd 0, Y d 1 ) ∧ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (12) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Assume A0, A1 and A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For any d ∈ N∗, let σ2 d = ℓ2/d2α and λd = c2/2d2β with α, β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Then, the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (a) If α = 1/4, β = 1/8 and r > 0, we have limd→+∞ ad(ℓ, r) = 2Φ � −ℓ2K1(r)/2 � , where Φ is the distribution function of a standard normal and K2 1(r) = r2 4 E �� g′′(Xd 0,1)g′(Xd 0,1) �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' If in addition, A3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (b) If α = 1/6, β = 1/6 and r > 0, we have limd→+∞ ad(ℓ, r) = 2Φ � −ℓ3K2(r)/2 � , where Φ is the distribution function of a standard normal and K2 2(r) = �r 8 + r2 4 � E � g′′(Xd 0,1)g′(Xd 0,1) 2� + � 1 16 + r 8 � E � g′′(Xd 0,1)3� + 5 48E � g′′′(Xd 0,1)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (c) If α = 1/6, β > 1/6 and r > 0, we have limd→+∞ ad(ℓ, r) = 2Φ � −ℓ3K2(0)/2 � , where Φ is the distribution function of a standard normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In addition, in all these cases, as d → ∞ the process (Ld t )t≥0 converges weakly to the Langevin diffusion dLt = h(ℓ, r)1/2dBt − h(ℓ, r) 2 g′(x)dt , (13) where (Bt)t≥0 denotes standard Brownian motion and h(ℓ, r) = ℓ2a(ℓ, r) is the speed of the diffusion, setting a(ℓ, r) = limd→∞ ad(ℓ, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' If α = 1/4, β = 1/8, for any r > 0, ℓ �→ h(ℓ, r) is maximized at the unique value of ℓ such that a(ℓ, r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='452;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' while if α = 1/6, β = m/6 with m ≥ 1 and r > 0, ℓ �→ h(ℓ, r) is maximized at the unique value of ℓ such that a(ℓ, r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The proof follows that of [34, Theorem 1, Theorem 2] and is postponed to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The theorem above shows that the relative speed at which λd converges to 0 influences the scaling of the resulting proximal algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In case (c), m > 1 and λd decays with d at a faster rate than σ2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' This causes the proximity map (4) to collapse onto the identity and therefore the proposal (10) is arbitrarily close to that of MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The resulting scaling limit also coincides with that of MALA established in [34, Theorem 1, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 9 If λd and σ2 d decay at the same rate (case (b)), the amount of gradient information provided by the proximity map is controlled by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Comparing our result for case (b) with [34, Theorem 1] we find that K2 2(0) = 1 16E � g′′(Xd 0,1)3� + 5 48E � g′′′(Xd 0,1)2� = K2 MALA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' thus, we have K2 2(r) = K2 2(0) + �r 8 + r2 4 � E � g′′(Xd 0,1)2g′(Xd 0,1)2� + r 8E � g′′(Xd 0,1)3� = K2 MALA + �r 8 + r2 4 � E � g′′(Xd 0,1)2g′(Xd 0,1)2� + r 8E � g′′(Xd 0,1)3� ≥ K2 MALA , since the convexity of g implies that g′′ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In particular, K2 2(r) is an increasing function of r achieving its minimum when r → 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' MALA), see Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In case (a), m = 1/2 and λd decays more slowly than σ2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' As a consequence, the gradient information provided by the proximity map is smaller than in cases (b)–(c), and the resulting scaling differs from that of MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The value of K2 1(r) is increasing in r and the speed of the corresponding diffusion also depends on r (see Figure 1(a) gray lines and Figure 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Example 1 (Gaussian target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Take g(x) = x2/2, proxg λ(x) = x/(1 + λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In this case, g′ is Lipschitz continuous and we have K2 1(r) = r2/4, K2 2(r) = � 1 + 4r + 4r2� /16 and K2 2(0) = K2 MALA = 1/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The corresponding speeds are given in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Optimizing for m = 1, r = 0 (MALA) and m = 1, r = 1 (P-MALA) we obtain hMALA(ℓ, r) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5639, hP-MALA(ℓ, r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='7519, achieved with ℓMALA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6503 and ℓP-MALA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1443, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For Gaussian targets, MALA is geometrically ergodic [13], and therefore the optimal choice in terms of speed of convergence is MALA which is obtained for r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The result for r = 1 and m = 1 are also given in [31, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Example 2 (Target with light tails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Take g(x) = x4, which gives a normalized distribution with normalizing constant 2Γ(5/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The proximity map is proxλ g(x) = 1 2 � 3� 9λ2x + √ 54λ4x2 + 3λ3 32/3λ − 1 3� 27λ2x + 3 √ 54λ4x2 + 3λ3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In this case g′ is not Lipschitz continuous and therefore we only consider (a), for which we have K2 1(r) = 144r2Γ(11/4)/Γ(5/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The corresponding speed is given in Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 Laplace target As discussed in the introduction, proximal MALA has been widely used to quantify uncertainty in imaging applications, in which target distributions involving the ℓ1 norm are particularly common [30, 14, 1, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Here, we consider πL d to be the product of d i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Laplace distributions as in (7), πL d (xd) = d � i=1 πL(xd i ), for xd ∈ Rd, where πL(x) = 2−1 exp(−|x|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (14) 10 r Speed (a) Gaussian target r Speed (b) Light tail target Figure 1: Value of K for i = 1, 2 and speed of the corresponding Langevin diffusion as a function of r for a Gaussian target and a light tail target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We denote by h1 the speed obtained in case (a), by h2 that obtained in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In case (c) both K3 and the speed h3 are constant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' r and coincide with that of MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For the Gaussian target we report the results for case (a)–(c) while for the light tail target we only report (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For this particular choice of one-dimensional target distribution, the corresponding potential G is x �→ |x| and satisfies A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Then, the proximity map is given by the soft thresholding operator [29, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3] proxλ G(x) = (x − sgn(x)λ)1{|x| > λ} , (15) where sgn : R → {−1, 1} is the sign function, given by sgn(x) = −1 if x ≤ 0, and sgn(x) = 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' This operator is a continuous but not continuously differentiable map whose non- differentiability points are the extrema of the interval [−λ, λ] and are controlled by the value of the proximity parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Plugging (15) in (9), the proximal MALA algorithm applied to πL d proposes component-wise for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , d Y d k+1,i = Xd k,i − σ2 d 2 sgn(Xd k,i)1{|Xd k,i| > λd} − σ2 d 2λd Xd k,i1{|Xd k,i| ≤ λd} + σdZd k+1,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (16) For Xd k,i close to 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' the point of non-differentiability) the proximal MALA proposal is a biased random walk around Xd k,i, while outside the region [−λd, λd] the proposal coincides with that of MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' As λd → 0 the region in which the proximal MALA proposal coincide with that of MALA increases and when λd ≈ 0 the region [−λd, λd] in which the proposal corresponds to a biased random walk is negligible, as confirmed by the asymptotic acceptance rate in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We also consider the case λd = 0 for any d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Then, the proposal (16) becomes the proposal for the subgradient version of MALA: Y d k+1,i = Xd k,i − (σ2 d/2) sgn(Xd k,i) + σdZd k+1,i, referred to as sG-MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The proof of the optimal scaling for the Laplace distribution follows the structure of that of [12] for Lp-mean differentiable distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We start by characterizing the asymptotic acceptance 11 ratio of a generic proximal MALA algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' contrary to Theorem 1 for differentiable targets, in the limit d → ∞ the properties of proximal MALA do not depend on the relative speed at which σ2 d = ℓ2/d2α and λd = c2/2d2β converge to 0, as long as λd decays at least at the same rate as σ2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In this regime, the region in which the proposal (16) corresponds to a biased random walk proposal is negligible, and therefore we obtain the same scaling obtained with λd = 0 and corresponding to sG-MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Assume A2 and consider the sequence of target distributions {πL d }d∈N∗ given in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For any d ∈ N∗, let σ2 d = ℓ2/d2α and λd = c2/2d2β with α = 1/3 and β = m/3 for m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Then, we have limd→∞ ad(ℓ, r) = aL(ℓ) = 2Φ(−ℓ3/2/(72π)1/4), where (ad(ℓ, r))d∈N∗ is defined in (12), with r = c2/ℓ2m, and Φ is the distribution function of a standard normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The proof is postponed to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Note that the asymptotic average acceptance rate aL(ℓ) does not depend on r and as a result on c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Having identified the possible scaling for proximal MALA with Laplace target, we are now ready to show weak convergence to the appropriate Langevin diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To this end, we adapt the proof strategy followed in [22] and [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' As for the differentiable case, consider the linear interpolation (Ld t )t≥0 of the first component of the Markov chain (Xd k)k≥0 given in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For any d ∈ N∗, denote by νd the law of the process (Ld t )t≥0 on the space of continuous functions from R+ to R, C(R+, R), endowed with the topology of uniform convergence over compact sets and its corresponding σ-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We first show that the sequence (νd)d∈N∗, admits a weak limit point as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Assume A2 and consider the sequence of target distributions {πL d }d∈N∗ given in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For any d ∈ N∗, let σ2 d = ℓ2/d2α and λd = c2/2d2β with α = 1/3 and β = m/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The sequence (νd)d∈N∗ is tight in M1 (C(R+, R)), the set of probability measures acting on C(R+, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' See Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' By Prokhorov’s theorem, the tightness of (νd)d∈N∗ implies existence of a weak limit point ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In our next result, we give a sufficient condition to show that any limit point of (νd)d∈N∗ coincides with the law of a solution of: dLt = [hL(ℓ)]1/2dBt − hL(ℓ) 2 sgn(Lt)dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (17) To this end, we consider the martingale problem (see [42]) associated with (17), that we now present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let us denote by C∞ c (R, R) the subset of functions of C(R, R) which are infinitely many times differentiable and with compact support, and define the generator of (17) for V ∈ C∞ c (R, R) by LV (x) = hL(ℓ) 2 [V ′′(x) − sgn(x)V ′(x)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (18) Denote by (Wt)t≥0 the canonical process on C(R+, R), Wt : {ws}s≥0 �→ wt and the corresponding filtration by (Ft)t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' A probability measure ν is said to solve the martingale problem associated 12 with (17) with initial distribution πL, if the pushforward of ν by W0 is πL and if for all V ∈ C∞ c (R, R), the process � V (Wt) − V (W0) − � t 0 LV (Wu)du � t≥0 is a martingale with respect to ν and the filtration (Ft)t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The following proposition gives a sufficient condition to prove that ν is a solution of the martingale problem: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Suppose that for any V ∈ C∞ c (R, R), m ∈ N, ρ : Rm → R bounded and continuous, and for any 0 ≤ t1 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ≤ tm ≤ s ≤ t: lim d→+∞ Eνd �� V (Wt) − V (Ws) − � t s LV (Wu)du � ρ(Wt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=', Wtm) � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Then any limit point of (νd)d∈N∗ on M1 (C(R+, R)) is a solution to the martingale problem associated with (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' See Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Finally, we use this sufficient condition to establish that any limit point of (νd)d∈N∗ is a solution of the martingale problem for (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Uniqueness in law of solutions of (17) allows to conclude that (Ld t )t≥0 converges weakly to the Langevin diffusion (17), which establishes our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The sequence of processes {(Ld t )t≥0 : d ∈ N∗} converges in distribution towards (Lt)t≥0, solution of (17) as d → ∞, with hL(ℓ) = ℓ2aL(ℓ) and aL defined in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In addition, hL is maximized at the unique value of ℓ such that aL(ℓ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' See Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 4 Practical Implications and Numerical Simulations The optimal scaling results in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 provide some guidance on the choice of the parameters σ and λ of proximal MALA algorithms, suggesting that smaller values of λ provide better efficiency in terms of number of steps necessary to convergence (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' However, a number of other factors must be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' First, as shown in [26, 37, 36, 21] the convergence properties of Metropolis adjusted algorithms are influenced by the shape of the target distribution and, in particular, by its tail behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Secondly, when comparing proximal MALA algorithms with gradient-based methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' MALA) one must take into account the cost of obtaining the gradients, whether this comes from automatic differentiation algorithms or from evaluating a potentially complicated gradient function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' On the other hand, proximity mappings can be quickly found or approximated solving convex optimization problems which have been widely studied in the convex optimization literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' [29, Chapter 6], [11] and [30, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In terms of convergence properties, we are usually interested in the family of distributions for which the discrete time Markov chain produced by our algorithm is geometrically ergodic, together with the optimal scaling results briefly recalled in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Normally, the ergodicity results are given by considering the one-dimensional class of distributions E(β, γ) introduced in [36] and defined for γ > 0 and 0 < β < ∞ by E(β, γ) : � π : R → [0, +∞) : π(x) ∝ exp � −γ|x|β� , |x| > x0 for some x0 > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 13 As observed by [24], there usually is a trade-off between ergodicity and optimal scaling results, algorithms providing better optimal scaling results tend to be geometrically ergodic for a smaller set of targets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' MALA w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' RWM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' As suggested by Theorem 1, the scaling properties of proximal MALA on differentiable targets are close to those of MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' This leads to a natural comparison between the two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' First, we observe that A0 rules out targets for which G is not convex and therefore restricts the families E(β, γ) to β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To compare MALA with proximal MALA we therefore focus on distributions with β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' It is shown in [36] that MALA is geometrically ergodic for targets in E(β, γ) with 1 ≤ β ≤ 2 (with some caveat for β = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Theorem 1–(b) and (c) show that in this case proximal MALA has the same scaling properties of MALA but in case (b) the asymptotic speed of convergence decays as the constant r increases (Figure 1(a)), with the maximum achieved for r → 0, for which proximal MALA collapses onto MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Since MALA is geometrically ergodic, and achieves better (or equivalent) scaling properties than proximal MALA, it would be natural to prefer MALA to proximal MALA for this set of targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' However, if the gradient is costly to obtain, one might instead consider to use proximal MALA with a small λ, to retain scaling properties as close as possible to that of MALA but to reduce the computational cost of evaluating the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In the case of differentiable targets with light-tails (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' β > 2), MALA is known not to be geometrically ergodic [36, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2] while the ergodicity properties of proximal MALA have only been partially studied in [30, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2] for the case λ = σ2/2 (P-MALA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' As shown in [30, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1], given a distribution π ∈ E(β, γ) with β ≥ 1, the distribution πλ obtained using the potential (3) belongs to E(β′, γ′), where β′ = min(β, 2) and γ′ depending on λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' This suggests that proximal MALA is likely to be ergodic for appropriate choices of λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' a first result in this direction is given in [30, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2] for the P-MALA case λ = σ2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Theorem 1–(c) restricts the sets of available λs showing that for light-tail distributions (for which A3 does not hold) λ should decay at half the speed of σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Studying the ergodicity properties of proximal MALA in function of the parameter λ is, of course, an interesting problem that we leave for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For the Laplace distribution, Theorem 2 shows that the value of λ does not influence the asymptotic acceptance ratio of proximal MALA, as long as λ decays with d at least as fast as σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The scaling properties and the asymptotic speed h(ℓ) in Theorem 3 do not depend on λ and coincide with that of the sG-MALA (obtained for λ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Hence, in terms of optimal scaling, there does not seem to be a difference between proximal MALA and sG-MALA for the Laplace distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 Numerical Experiments To illustrate the results established in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 we consider here a small collection of simulation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The aim of these studies is to empirically confirm the optimal scalings identified in Theorem 1 and 2, investigate the dimension d at which the asymptotic acceptance ratio limd→∞ ad(ℓ, r) well approximates the empirical average acceptance ratio and, consequently, for which dimensions d we can expect the optimal asymptotic acceptances in Theorem 1 and 2 to guarantee maximal speed h(ℓ, r) (approximated by the expected squared jumping distance, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' [18]) for the corresponding diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We summarize here our findings, a more detailed discussion can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For the differentiable case, we consider the Gaussian distribution in Example 1 and four algorith- mic settings which correspond to the three cases identified in Theorem 1 and MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The different values of r and m influence the dimension required to observe convergence to the theoretical limit 14 in Theorem 1: for r → 0 and m = 1 (MALA) and m = 1/2, r = 1 (corresponding to Theorem 1–(a)) the theoretical limit is already achieved for d of order 102, while in the cases m = 3, r = 2 and m = r = 1 (corresponding to Theorem 1–(c) and (b), respectively) our simulation result match the theoretical limit only for d of order 105 or higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The results for the Laplace case are similar, with the case m > 1 requiring a higher d to observe convergence to the theoretical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In general, we find that the optimal average acceptance ratios in Theorem 1 guarantee maximal speed h(ℓ, r) for d sufficiently large (for small d the optimal acceptance ratio often differs from the optimal asymptotic one, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' [40, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 5 Discussion In this work we analyze the scaling properties of a wide class of proximal MALA algorithms intro- duced in [30, 14] for smooth targets and for the Laplace distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We show that the scaling properties of proximal MALA are influenced by the relative speed at which the proximal parameter λd and the proposal variance σd decay to 0 as d → ∞ and suggest practical ways to choose λd as a function of σd to guarantee good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In the case of smooth targets, we provide a detailed comparison between proximal MALA and MALA, showing that proximal MALA scales no better than MALA (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In particular, Theorem 1–(a) shows that if λd is too large w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σd then the efficiency of proximal MALA is of order O(d−1/2) and therefore worse than the O(d−1/3) of MALA, suggesting that λd should be chosen to decay approximately as σd, if possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' If λd decays sufficiently fast, then MALA and proximal MALA have similar scaling properties and, in the case in which the proximity map is cheaper to compute that the gradient, one can build proximal MALA algorithms which are as efficient as MALA in terms of scaling but more computationally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In the case of the Laplace distribution, we show that the scaling of proximal MALA is O(d−2/3) for any λd decaying sufficiently fast w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σd and, in the limit λd ≈ 0, we obtain a novel optimal scaling result for MALA on Laplace targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' As discussed in Section 4, our analysis provides some guidance on the choice of the parameters that need to be specified to implement proximal MALA, but this analysis should be complemented by an exploration of the ergodicity properties of proximal MALA to obtain a comprehensive descrip- tion of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We conjecture that for sufficiently large values of λ, proximal MALA applied to light tail distributions will be exponentially ergodic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' establishing exactly how large should λ be to guarantee fast convergence is an interesting question that we leave for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Obtaining these results would open the doors to adaptive tuning strategies for proximal MALA, which are likely to produce better results than those given by the strategies currently used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The set up under which we carried out our analysis closely resembles that of [34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' we anticipate that A2 could be relaxed following similar ideas as those in [10, 22] and that our analysis could be extended to d-dimensional targets πd possessing some dependence structure following the approach of [40, 4, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Finally, the analysis carried out for the Laplace distribution could be extended to other piecewise smooth distributions provided that the moments necessary for the proof in Section 6 can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 15 6 Proof of the Result for the Laplace distribution In this section we prove the results in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 which give the scaling properties of proximal MALA (and sG-MALA) for the Laplace distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We collect technical results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' moment computations, bounds, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=') in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We recall that σ2 d = ℓ2/d2α and λd = c2/2d2β for some α, β > 0 and some constants c, ℓ independent on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Thus, we can write λd as a function of σd, λd = σ2m d r/2, where we define r = c2/ℓ2m > 0 and m = β/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In order to study the scaling limit of proximal MALA with Laplace target, consider the mapping bd : R2 → R given by bd : (x, z) �→ z − σd 2 sgn(x)1 � |x| > σ2m d r/2 � − 1 σ2m−1 d rx1 � |x| ≤ σ2m d r/2 � , (19) which allows us to write the proposal as Y d 1,i = Xd 0,i + σdbd(Xd 0,i, Zd 1,i) , for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We consider also the function φd : R2 → R, given by φd : (x, z) �→ log π(x + σdbd(x, z))q(x + σdbd(x, z), x) π(x)q(x, x + σdbd(x, z)) (20) = |x| − |x + σdbd(x, z)| + z2 2 − 1 2σ2 d �σ2 d 2 sgn [x + σdbd(x, z)] 1 � |x + σdbd(x, z)| > σ2m d r 2 � − σdbd(x, z) + 1 σ2(m−1) d r [x + σdbd(x, z)] 1 � |x + σdbd(x, z)| ≤ σ2m d r 2 ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We introduce, for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , d}, φd,i = φd(Xd 0,i, Zd 1,i) for the sake of conciseness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' This allows us to rewrite ad(ℓ, r), defined in (12), in the following way, ad(ℓ, r) = E � exp � d � i=1 φd,i � ∧ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (21) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under A2, the families of random variables (bd(Xd 0,i, Zd 1,i))i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',d} and (φd,i)i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',d} are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 Proof of Theorem 2 The proof of Theorem 2 uses the first three moments of φd,1, whose computation is postponed to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1, and is an application of Lindeberg’s central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To identify the optimal scaling for the Laplace distribution, we look for those values of α such that �d i=1 E[φd,i] and Var(�d i=1 φd,i) converge to a finite value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using Remark 1, we have that, d � i=1 E [φd,i] = d E [φd,1] and Var � d � i=1 φd,i � = d Var (φd,1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (22) 16 Then, using the integrals in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1, we find that the only value of α for which (22) converge to a finite value with the variance strictly positive is α = 1/3 as confirmed empirically in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Having identified α = 1/3, we can then proceed applying Lindeberg’s CLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We start by showing that the acceptance ratio converges to a Gaussian dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Define µd = E[φd,1] and Fd,i = σ((Xd 0,j, Zd 1,j), 1 ≤ j ≤ i), the natural filtration for (Xd 0,i, Zd 1,i)d∈N,1≤i≤d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The square-integrable martingale sequence � �Sd,i = i � j=1 Wd,i, Fd,i � � d∈N∗,1≤i≤d where Wd,i = φd,i − µd, forms a triangular array, to which we can apply the corresponding CLT (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' [41, Theorem 4, page 543]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In particular, we have that, lim d→∞ d � i=1 E � W 2 d,i | Fd,i−1 � = lim d→∞ d Var (φd,1) = 2ℓ3 3 √ 2π , as shown in Proposition 17 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' It remains to verify Lindeberg’s condition: for ε > 0, lim d→∞ dE � W 2 d,11 {|Wd,1| > ε} � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In order to verify Lindeberg’s condition we verify the stronger Lyapunov condition: there exists ϵ > 0 such that lim d→∞ dE � W 2+ϵ d,1 � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Pick ϵ = 1 and expand the cube using µd = E[φd,i], E � W 3 d,1 � = E � φ3 d,i � − 3µdE � φ2 d,i � + 2µ3 d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (23) By Proposition 16 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1, we have limd→∞ dµ3 d = 0, limd→∞ µd = 0, and, by Proposi- tion 17 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1, lim d→∞ dE � φ2 d,i � = 2ℓ3 3 √ 2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Finally, for the remaining term in (23) we use Proposition 18 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 to show that limd→∞ dE[φ3 d,i] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The above and the fact that, by Proposition 16 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1, lim d→∞ dµd = − ℓ3 3 √ 2π , show, by Lindeberg’s CLT, that the acceptance ratio converges in law to a normal random variable �Z with mean −ℓ3/(3 √ 2π) and variance 2ℓ3/(3 √ 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To conclude the proof, we plug this convergence into (21), since x �→ ex ∧ 1 is a continuous and bounded mapping, we have that lim d→∞ exp � d � i=1 φd,i � ∧ 1 d= e � Z ∧ 1 and lim d→∞ ad(ℓ, r) = E � e � Z ∧ 1 � , where the limit does not depend on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Defining aL(ℓ) = limd→∞ ad(ℓ, r) and using [33, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4], we have the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 Proof of Proposition 1 We are interested in the law νd of the linear interpolant (Ld t )t≥0, defined in (11), of the first component of the chain (Xd k)k∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let us recall the definition of the chain: assumption A2 gives the initial distribution πd, then, for any k ∈ N, the proposal Y d k+1 = (Y d k+1,i)1≤i≤d is defined in (16) with σ2 d = ℓ2/d2α, λd = σ2m d r/2 with α = 1/3 and m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' With the notations introduced in Section 6, we can rewrite (16), for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , d}, Y d k+1,i = Xd k,i + σdbd(Xd k,i, Zd k+1,i) , (24) where bd is defined in (19) with r = c2/ℓ2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' From there, we apply the acceptance-rejection step described in (6), we additionally define the acceptance event Ad k+1 = � bd k+1 = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We can now expand the expression of the linear interpolant Ld t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' for t ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' using (6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (11) and the definition of Ad k+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Ld t = � � � � � � � � � � � � � � � Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 + (d2αt − ⌊d2αt⌋) � σdZd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 − σ2 d 2 sgn(Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1) � 1Ad ⌈d2αt⌉ if |Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1| > σ2m d r 2 Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 + (d2αt − ⌊d2αt⌋) � σdZd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 − 1 σ2(m−1) d rXd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � 1Ad ⌈d2αt⌉ otherwise ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (25) or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' equivalently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Ld t = � � � � � � � � � � � � � � � Xd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 − (⌈d2αt⌉ − d2αt) � σdZd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 − σ2 d 2 sgn(Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1) � 1Ad ⌈d2αt⌉ if |Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1| > σ2m d r 2 Xd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 − (⌈d2αt⌉ − d2αt) � σdZd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 − 1 σ2(m−1) d rXd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � 1Ad ⌈d2αt⌉ otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In order to prove Proposition 1, we consider Kolmogorov’s criterion for tightness (see [23, The- orem 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='7]): the sequence (νd)d≥1 is tight if E � (Ld t − Ld s)4� ≤ γ(t)(t − s)2 , for some non-decreasing positive function γ, all 0 ≤ s ≤ t and all d ∈ N∗ and the sequence (Ld 0)d∈N∗ is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The latter condition is straightforward to check, since by A2 the distribution of Ld 0 = Xd 0,1 is πL for all d ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Consider E � (Ld t − Ld s)4� , if ⌊d2αs⌋ = ⌊d2αt⌋, the inequality follows straight- forwardly recalling that the moments of normal distributions are bounded: in the case |Xd ⌊d2αt⌋,1| = |Xd ⌊d2αs⌋,1| > σ2m d r/2 it follows directly from the boundedness of the sgn function, while in the case |Xd ⌊d2αt⌋,1| = |Xd ⌊d2αs⌋,1| ≤ σ2m d r/2 we exploit the boundedness of Xd ⌊d2αt⌋,1 itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For all 0 ≤ s ≤ t such that ⌈d2αs⌉ ≤ ⌊d2αt⌋, we can distinguish three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 18 Case 1 If |Xd ⌊d2αt⌋,1| > σ2m d r/2 and |Xd ⌊d2αs⌋,1| > σ2m d r/2, then Ld t − Ld s = Xd ⌊d2αt⌋,1 − Xd ⌈d2αs⌉,1 + (d2αt − ⌊d2αt⌋) � σdZd ⌈d2αt⌉,1 − σ2 d 2 sgn(Xd ⌊d2αt⌋,1) � 1Ad ⌈d2αt⌉ + (⌈d2αs⌉ − d2αs) � σdZd ⌈d2αs⌉,1 − σ2 d 2 sgn(Xd ⌊d2αs⌋,1) � 1Ad ⌈d2αs⌉ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using H¨older’s inequality and the fact that 0 ≤ d2αt − ⌊d2αt⌋ ≤ 1 (and similarly for s) we have E � (Ld t − Ld s)4� ≤ CE �� Xd ⌊d2αt⌋,1 − Xd ⌈d2αs⌉,1 �4� + C (d2αt − ⌊d2αt⌋)2 d4α E �� ℓZd ⌈d2αt⌉,1 �4 + ℓ4 24d4α � + C (⌈d2αs⌉ − d2αs)2 d4α E �� ℓZd ⌈d2αs⌉,1 �4 + ℓ4 24d4α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Recalling that the moments of Zd are bounded and that d2αs ≤ ⌈d2αs⌉ ≤ ⌊d2αt⌋ ≤ d2αt, it follows E � (Ld t − Ld s)4� ≤ C � (t − s)2 + E �� Xd ⌊d2αt⌋,1 − Xd ⌈d2αs⌉,1 �4�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (26) Case 2 If |Xd ⌊d2αt⌋,1| > σ2m d r/2 and |Xd ⌊d2αs⌋,1| ≤ σ2m d r/2 or |Xd ⌊d2αt⌋,1| ≤ σ2m d r/2 and |Xd ⌊d2αs⌋,1| > σ2m d r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We only describe the argument for the first case, the second case follows from analogous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Take Ld t − Ld s = Xd ⌊d2αt⌋,1 + (d2αt − ⌊d2αt⌋) � σdZd ⌈d2αt⌉,1 − σ2 d 2 sgn(Xd ⌊d2αt⌋,1) � 1Ad ⌈d2αt⌉ − Xd ⌈d2αs⌉,1 − (⌈d2αs⌉ − d2αs) � σdZd ⌈d2αs⌉,1 − 1 σ2(m−1) d r Xd ⌊d2αs⌋,1 � 1Ad ⌈d2αs⌉ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proceeding as above, we find that E � (Ld t − Ld s)4� ≤ C � (t − s)2 + E �� Xd ⌊d2αt⌋,1 − Xd ⌈d2αs⌉,1 �4� +(⌈d2αs⌉ − d2αs)4E � � � 1 σ2(m−1) d r Xd ⌊d2αs⌋ �4� � � � , and recalling that |Xd ⌊d2αs⌋,1| ≤ σ2m d r/2 we have that |Xd ⌊d2αs⌋,1|/(rσ2(m−1) d ) ≤ σ2 d/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using this and the same arguments as above, we have E � (Ld t − Ld s)4� ≤ C � (t − s)2 + E �� Xd ⌊d2αt⌋,1 − Xd ⌈d2αs⌉,1 �4�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (27) 19 Case 3 If |Xd ⌊d2αt⌋,1| ≤ σ2m d r/2 and |Xd ⌊d2αs⌋,1| ≤ σ2m d r/2, then Ld t − Ld s = Xd ⌊d2αt⌋,1 + (d2αt − ⌊d2αt⌋) � σdZd ⌈d2αt⌉,1 − 1 σ2(m−1) d r Xd ⌊d2αt⌋,1 � 1Ad ⌈d2αt⌉ − Xd ⌈d2αs⌉,1 + (⌈d2αs⌉ − d2αs) � σdZd ⌈d2αs⌉,1 − 1 σ2(m−1) d r Xd ⌊d2αs⌋,1 � 1Ad ⌈d2αs⌉ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using the boundedness of moments of Gaussian distributions and of Xd ⌊d2αt⌋,1, Xd ⌊d2αs⌋,1, we have E � (Ld t − Ld s)4� ≤ C � (t − s)2 + E �� Xd ⌊d2αt⌋,1 − Xd ⌈d2αs⌉,1 �4�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (28) Putting (26), (27) and (28) together and using Lemma 1 below we obtain E �� Ld t − Ld s �4� ≤ C � (t − s)2 + 4 � p=2 � ⌊d2αt⌋ − ⌈d2αs⌉ �p d2αp � ≤ C(t − s)2 + C 4 � p=2 d2αp (t − s)p d2αp ≤ C � 2 + t + t2� (t − s)2 , which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We are now ready to state and prove Lemma 1: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' There exists C > 0 such that for any k1, k2 ∈ N with 0 ≤ k1 < k2, E �� Xd k2,1 − Xd k1,1 �4� ≤ C 4 � p=2 (k2 − k1)p d2αp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Recalling the definition of the proposal in (24) and the notations of (19) we can write E �� Xd k2,1 − Xd k1,1 �4� = E � � � k2 � k=k1+1 σdbd � Xd k−1,1, Zd k,1 � 1Ad k �4� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Then, we expand all acceptance or rejection terms between k1 and k2 and use H¨older’s inequality to obtain E �� Xd k2,1 − Xd k1,1 �4� ≤ σ4 dE � � � k2 � k=k1+1 bd � Xd k−1,1, Zd k,1 � �4� � + σ4 dE � � � k2 � k=k1+1 bd � Xd k−1,1, Zd k,1 � 1(Ad k)c �4� � , 20 where bd is defined in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using again H¨older’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' for the first term we have E � � � k2 � k=k1+1 bd � Xd k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � �4� � ≤ C � � �E � � � k2 � k=k1+1 Zd k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 �4� � + σ4 d 24 E � � � k2 � k=k1+1 sgn � Xd k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � 1 � |Xd k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1| > σ2m d r/2 � �4� � +σ4 d 24 E � � � k2 � k=k1+1 1 σ2m−1 d rXd k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='11 � |Xd k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1| ≤ σ2m d r/2 � �4� � � � � ≤ C � 3(k2 − k1)2 + 2σ4 d 24 (k2 − k1)4 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (29) where the last line follows using the moments of Zd k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 and the boundedness of Xd k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 in the set {|Xd k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1| ≤ σ2m d r/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using a Binomial expansion of the rejection term, we obtain E � � � k2 � k=k1+1 bd � Xd k−1,1, Zd k,1 � 1(Ad k) c �4� � = � E � 4 � i=1 bd � Xd mi−1,1, Zd mi,1 � 1(Admi) c � , (30) where the sum is over the quadruplets (mi)1≤i≤4 with mi ∈ {k1 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , k2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We separate the terms in the sum according to their cardinality, let us denote, for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , 4}, Ij = � (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , m4) ∈ {k1 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , k2}4 : # {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , m4} = j � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and define, for any (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , m4) ∈ {k1 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , k2}4, � Xd 0 = Xd 0 and for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , d}, � Xd k+1,i = � Xd k,i + 1{m1−1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',m4−1}c(k)1�Ad k+1σdbd � � Xd k,i, Zd k+1,i � , where �Ad k+1 = � Uk+1 ≤ exp � d � i=1 φd � � Xd k,i, Zd k+1,i ��� , (31) and φd in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Denote by F the σ-algebra generated by the process � Xd and observe that on the event 4� j=1 � Ad mj �c , Xd is equal to � Xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We consider now the terms in the sum (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (i) If (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , m4) ∈ I4, then the mis are all distinct and E � � 4 � j=1 bd � Xd mj−1,1, Zd mj,1 � 1� Admj �c ������ F � � = E � � 4 � j=1 bd � � Xd mj−1,1, Zd mj,1 � 1� �Admj �c ������ F � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 21 However, {bd( � Xd mj−1,1, Zd mj,1)1(�Ad mj )c}j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',4 are independent conditionally on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Thus, E � � 4 � j=1 bd � � Xd mj−1,1, Zd mj,1 � 1� �Admj �c ������ F � � = 4 � j=1 E � bd � � Xd mj−1,1, Zd mj,1 � 1� �Admj �c ����F � = 4 � j=1 E � bd � � Xd mj−1,1, Zd mj,1 � × � 1 − exp � d � i=1 φd � � Xd mj−1,i, Zd mj,i ��� + �����F � , by integrating the uniform variables Umj in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Recalling the definition of bd in (19),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' we can bound the expectation above with �����E � bd � � Xd mj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd mj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � � 1 − exp � d � i=1 φd � � Xd mj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd mj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i ��� + �����F ������ (32) ≤ ���E ��σd 2 sgn � � Xd mj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � 1 � | � Xd mj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1| > σ2m d r/2 � − 1 σ2m−1 d r � Xd mj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='11 � | � Xd mj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1| ≤ σ2m d r/2 �� × � 1 − exp � d � i=1 φd � � Xd mj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd mj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i ��� + �����F ������ + �����E � Zd mj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � 1 − exp � d � i=1 φd � � Xd mj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd mj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i ��� + �����F ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For the first one, we use the boundedness of the sgn function and of � Xd mj−1,1 in the set {| � Xd mj−1,1| ≤ σ2m d r/2} to obtain ���E ��σd 2 sgn � � Xd mj−1,1 � 1 � | � Xd mj−1,1| > σ2m d r/2 � − 1 σ2m−1 d r � Xd mj−1,11 � | � Xd mj−1,1| ≤ σ2m d r/2 �� × � 1 − exp � d � i=1 φd � � Xd mj−1,i, Zd mj,i ��� + �����F ������ ≤ σd 2 E ������ � 1 − exp � d � i=1 φd � � Xd mj−1,i, Zd mj,i ��� + ����� �����F � ≤ σd 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (33) We can write the second term as E � Zd mj,1 � 1 − exp � d � i=1 φd � � Xd mj−1,i, Zd mj,i ��� + �����F � = E � G � � Xd mj−1,1, d � i=2 φd � � Xd mj−1,i, Zd mj,i �������F � , 22 where we define G(a, b) = E � Z (1 − exp (φd (a, Z) + b))+ � with Z a standard Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Be- cause the function x �→ (1 − exp(x))+ is 1-Lipschitz, we have, using Cauchy-Schwarz and Lemma 3 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2, ��E � Z (1 − exp (φd (a, Z) + b))+ � − E � Z (1 − exp (b))+ ��� ≤ E [|Z| |φd (a, Z)|] ≤ E � Z2�1/2 E � φd (a, Z)2�1/2 ≤ E � φd (a, Z)2�1/2 ≤ Cd−α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' However, E � Z (1 − exp (b))+ � = E [Z] (1 − exp (b))+ = 0, and therefore �����E � G � � Xd mj−1,1, d � i=2 φd � � Xd mj−1,i, Zd mj,i �������F ������ ≤ Cd−α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (34) Combining equations (32), (33) and (34) and recalling that σd = ℓd−α, we have �����E � bd � � Xd mj−1,1, Zd mj,1 � � 1 − exp � d � i=1 φd � � Xd mj−1,i, Zd mj,i ��� + �����F ������ ≤ Cd−α , (35) from which follows that � (m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',m4)∈I4 �����E � 4 � i=1 bd � Xd mi−1,1, Zd mi,1 � 1(Admi) c ������ ≤ � (m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',m4)∈I4 E � � 4 � j=1 C dα � � ≤ �k2 − k1 4 � C d4α ≤ C (k2 − k1)4 d4α , (36) using that |I4| = �k2−k1 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (ii) If (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='., m4) ∈ I3, only three of the mis take distinct values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' proceeding as in case (i), we have ������ E � � 3 � j=1 bd � Xd mj−1,1, Zd mj,1 �1+δ1,j 1� Admj �c ������ F � � ������ = 3 � j=1 �����E � bd � � Xd mj−1,1, Zd mj,1 �1+δ1,j � 1 − exp � d � i=1 φd � � Xd mj−1,i, Zd mj,i ��� + �����F ������ , where δ1,j denotes a Dirac’s delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For the terms j ̸= 1, we use (35), while for the term j = 1 23 we bound the indicator function by 1 to obtain ������ E � � 3 � j=1 bd � Xd mj−1,1, Zd mj,1 �1+δ1,j 1� Admj �c ������ F � � ������ ≤ ����E � bd � � Xd m1−1,1, Zd m1,1 �2����F ����� 3 � j=2 C dα ≤ � 3 + 2σ2 d 22d2α � C2 d2α ≤ C 1 d2α , where the second-to-last inequality follows using the same approach taken for (29) and recall- ing that σd = ℓd−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Hence, � (m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',m4)∈I3 �����E � 4 � i=1 bd � Xd mi−1,1, Zd mi,1 � 1(Admi) c ������ (37) ≤ C �k2 − k1 3 � 1 d2α ≤ C (k2 − k1)3 d2α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (iii) If (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='., m4) ∈ I2, we have two different cases: the mis take the two values twice or three mis have the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For the first one, we have, bounding the indicator function with 1, E � �E � � 2 � j=1 bd � Xd mj−1,1, Zd mj,1 �2 1� Admj �c ������ F � � � � ≤ E � � 2 � j=1 E � bd � � Xd mj−1,1, Zd mj,1 �2����F �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Since, conditionally on F, the random variables inside the expectation are Gaussians with bounded mean and variance 1, we have, using the same approach taken for (29), E � � 2 � j=1 E � bd � � Xd mj−1,1, Zd mj,1 �2����F �� � ≤ � 1 + 2σ2 d 22 �2 ≤ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The second case follows similarly ������ E � �E � � 2 � j=1 bd � Xd mj−1,1, Zd mj,1 �1+2δ1,j 1� Admj �c ������ F � � � � ������ ≤ E � �E � � 2 � j=1 ���bd � � Xd mj−1,1, Zd mj,1 ���� 1+2δ1,j ������ F � � � � ≤ C , 24 where δ1,j denotes a Dirac’s delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Therefore, � (m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',m4)∈I2 �����E � 4 � i=1 � bd(Xd mi−1,1, Zd mi,1 � 1(Admi) c ������ (38) ≤ C ��4 2 � + �4 3 �� �k2 − k1 2 � ≤ C(k2 − k1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (iv) If (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='., m4) ∈ I1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' all mis take the same value), we bound the indicator function by 1 and, using the same approach taken for (29), we find E � bd � Xd m1−1,1, Zd m1,1 �4 1(Adm1) c � ≤ C � 3 + 2σ4 d 24 � ≤ C , since σd = ℓd−α and d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Hence, � (m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',m4)∈I1 �����E � 4 � i=1 bd � Xd m1−1,1, Zd m1,1 � 1(Ad mi) c ������ ≤ C �k2 − k1 1 � = C(k2 − k1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (39) The result follows combining (36), (37), (38) and (39) in (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3 Proof of Proposition 2 We start by proving the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let ν be a limit point of the sequence of laws (νd)d≥1 of {(Ld t )t≥0 | d ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Then for any t ≥ 0, the pushforward measure of ν by Wt is πL(dx) = exp(−|x|)dx/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using (25),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' we have E ����Ld t − Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ��� � ≤ E �����(d2αt − ⌊d2αt⌋) � σdZd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 − σ2 d 2 sgn(Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1) � 1|Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1|>σ2m d r/21Ad ⌈d2αt⌉ ���� � + E ������(d2αt − ⌊d2αt⌋) � σdZd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 − 1 σ2(m−1) d r Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � 1|Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1|≤σ2m d r/21Ad ⌈d2αt⌉ ����� � ≤ (d2αt − ⌊d2αt⌋) � σdE ����Zd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1) ��� � + σ2 d 2 E ����sgn(Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1) ��� � + 1 σ2(m−1) d r E ����Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ��� 1|Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1|≤σ2m d r/2 �� ≤ (d2αt − ⌊d2αt⌋) � ℓ dα E � (Zd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1)2�1/2 + ℓ2 2d2α + 1 σ2(m−1) d r E �σ2m d r 2 �� ≤ (d2αt − ⌊d2αt⌋) � ℓ dα + ℓ2 2d2α + ℓ2 2d2α � ≤ C dα ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 25 where we used Cauchy-Schwarz inequality and the fact that the moments of Zd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The above guarantees that, lim d→∞ E ����Ld t − Xd ⌊d2αt⌋,1 ��� � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' As (νd)d≥1 converges weakly towards ν, for any Lipschitz bounded function ψ : R → R, lim d→∞ E � ψ � Xd ⌊d2αt⌋,1 �� = lim d→∞ E � ψ � Ld t �� = Eν [ψ(Wt)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The result follows since Xd ⌊d2αt⌋,1 is distributed according to πL(dx) = exp(−|x|)dx/2 for any t ≥ 0 and d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We are now ready to prove Proposition 2: Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let ν be a limit point of (νd)d≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We start by showing that if for any V ∈ C∞ c (R, R), m ∈ N, any bounded and continuous mapping ρ : Rm → R and any 0 ≤ t1 ≤ · · · ≤ tm ≤ s ≤ t, ν satisfies Eν �� V (Wt) − V (Ws) − � t s LV (Wu)du � ρ(Wt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , Wtm) � = 0 , (40) then ν is a solution to the martingale problem associated with L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let Fs denote the σ-algebra generated by {ρ(Wt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , Wtm) : m ∈ N, ρ : Rm → R bounded and continuous, and 0 ≤ t1 ≤ · · · ≤ tm ≤ s} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Then, Eν � V (Wt) − V (Ws) − � t s LV (Wu)du ����Fs � = 0 , showing that the process � V (Wt) − V (W0) − � t 0 LV (Wu)du � t≥0 is a martingale w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ν and the filtration (Ft)t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To prove (40), it is enough to show that for any V ∈ C∞ c (R, R), m ∈ N and any bounded and continuous mapping ρ : Rm → R and any 0 ≤ t1 ≤ · · · ≤ tm ≤ s ≤ t, the mapping Ψs,t : w �−→ � V (wt) − V (ws) − � t s LV (wu)du � ρ (wt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , wtm) , is continuous on a ν-almost sure subset of C(R+, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let W = {w ∈ C(R+, R) : wu ̸= 0 for almost any u ∈ [s, t]} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Since w ∈ Wc if and only if � t s 1{0}(wu)du > 0, using Lemma 2 and the Fubini–Tonelli’s theorem, Eν �� t s 1{0}(Wu)du � = � t s Eν � 1{0}(Wu) � du = � t s πL({0})du = 0 , 26 and we have that ν(Wc) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Since w �→ wu is continuous for any u ≥ 0, so are w �→ V (wu) and w �→ ρ(wt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , wtm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Thus, it is enough to prove that the mapping w �→ � t s LV (wu)du is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let (wn)n≥0 be a sequence in C(R+, R) that converges to w ∈ W in the uniform topology on compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let u be such that wu ̸= 0, therefore, since the sgn function is continuous in a neighbourhood of wu, limn→∞ LV (wn u) = LV (wu), thus limn→∞ LV (wn u) = LV (wu) for almost any u ∈ [s, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Finally, using the boundedness of the sequence (LV (wn u))n≥0 and Lebesgue’s dominated convergence theorem, lim n→∞ � t s LV (wn u)du = � t s LV (wu)du , which proves that the mappings Ψs,t are continuous on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 Proof of Theorem 3 Let us introduce, for any n ∈ N, Fd n,1 = σ({Xd k,1, 0 ≤ k ≤ n}), the σ-algebra generated by the first components of {Xd k | 0 ≤ k ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We also introduce for any V ∈ C∞ c (R, R) M d n(V ) = ℓ dα n−1 � k=0 V ′(Xd k,1) × � bd � Xd k,1, Zd k+1,1 � 1Ad k+1 − E � bd � Xd k,1, Zd k+1,1 � 1Ad k+1 ���Fd k,1 �� (41) + ℓ2 2d2α n−1 � k=0 V ′′(Xd k,1) × � bd � Xd k,1, Zd k+1,1 �2 1Ad k+1 − E � bd � Xd k,1, Zd k+1,1 �2 1Ad k+1 ���Fd k,1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' where bd is defined in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The proof of Theorem 3 follows using the sufficient condition in Proposition 2, the tightness of the sequence (νd)d≥1 established in Proposition 1 and Proposition 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using Proposition 1, Proposition 2 and Proposition 3 below, it is enough to show that for any V ∈ C∞ c (R, R), m ≥ 1, any 0 ≤ t1 ≤ · · · ≤ tm ≤ s ≤ t and any bounded and continuous mapping ρ : Rm → R, lim d→∞ E �� M d ⌈d2αt⌉(V ) − M d ⌈d2αs⌉(V ) � ρ(Ld t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=', Ld tm) � = 0 , where, for any n ≥ 1, M d n(V ) is given by (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' However, this is straightforwardly obtained by taking successively the conditional expectations with respect to Fd k,1 for k = ⌈d2αt⌉, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , ⌈d2αs⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For any 0 ≤ s ≤ t, V ∈ Cc(R, R) we have lim d→∞ E �����V � Ld t � − V � Ld s � − � t s LV � Ld u � du − � M d ⌈d2αt⌉ (V ) − M d ⌈d2αs⌉ (V ) ����� � = 0 , (42) where (Ld t )t≥0 is defined in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 27 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The process (Ld t )t≥0 is piecewise linear, thus it has finite variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For any τ ≥ 0, we define dLd τ = d2ασdbd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1) � 1Ad ⌈d2ατ⌉dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Thus, recalling that σd = ℓd−α and using the fundamental theorem of integral calculus for piecewise C1 maps V � Ld t � − V � Ld s � = ℓdα � t s V ′ � Ld τ � bd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 � 1Ad ⌈d2ατ⌉dτ , (43) where bd is defined in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' A Taylor expansion of V ′ with Lagrange remainder about Xd ⌊d2ατ⌋,1 gives V ′ � Ld τ � = V ′ � Xd ⌊d2ατ⌋,1 � + ℓ dα � d2ατ − ⌊d2ατ⌋ � V ′′ � Xd ⌊d2ατ⌋,1 � bd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 � 1Ad ⌈d2ατ⌉ + ℓ2 2d2α � d2ατ − ⌊d2ατ⌋ �2 V (3) (χτ) bd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 � 1Ad ⌈d2ατ⌉ , where for any point τ ∈ [s, t], there exists χτ ∈ [Xd ⌊d2ατ⌋,1, Y d τ,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Substituting the above into (43) we obtain V � Ld t � − V � Ld s � = ℓdα � t s V ′ � Xd ⌊d2ατ⌋,1 � bd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 � 1Ad ⌈d2ατ⌉dτ + ℓ2 � t s � d2ατ − ⌊d2ατ⌋ � V ′′ � Xd ⌊d2ατ⌋,1 � bd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 �2 1Ad ⌈d2ατ⌉dτ (44) + ℓ3 2dα � t s � d2ατ − ⌊d2ατ⌋ �2 V (3) (χτ) bd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 �3 1Ad ⌈d2ατ⌉dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Since V (3) is bounded, using Fubini-Tonelli’s theorem and recalling the definition of bd in (19), we have that ℓ3 2dα E ����� � t s � d2ατ − ⌊d2ατ⌋ �2 V (3) (χτ) bd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 �3 1Ad ⌈d2ατ⌉dτ ���� � ≤ C ℓ3 2dα � t s E �����Zd ⌈d2ατ⌉,1 ��� + ℓ 2dα �3� dτ −→ d→∞ 0 , since the moments of Zd ⌈d2ατ⌉,1 are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For the second term in (44), we observe that most of the integrand is piecewise constant since the process Xd ⌊d2ατ⌋,1 evolves in discrete time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Then, for any integer d2αs ≤ k ≤ d2αt − 1, � (k+1)/d2α k/d2α � d2ατ − ⌊d2ατ⌋ � V ′′ � Xd ⌊d2ατ⌋,1 � bd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 �2 1Ad ⌈d2ατ⌉dτ = 1 2d2α V ′′ � Xd k,1 � bd � Xd k,1, Zd k+1,1 �2 1Ad k+1 = 1 2 � (k+1)/d2α k/d2α V ′′ � Xd ⌊d2ατ⌋,1 � bd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 �2 1Ad ⌈d2ατ⌉dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 28 Thus, we can write I = � t s � d2ατ − ⌊d2ατ⌋ � V ′′ � Xd ⌊d2ατ⌋,1 � bd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 �2 1Ad ⌈d2ατ⌉dτ = I1 + I2 , where we define I2 = 1 2 � t s V ′′ � Xd ⌊d2ατ⌋,1 � bd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 �2 1Ad ⌈d2ατ⌉dτ , and I1 = �� ⌈d2αs⌉/d2α s + � t ⌊d2αt⌋/d2α � � d2ατ − ⌊d2ατ⌋ − 1 2 � V ′′ � Xd ⌊d2ατ⌋,1 � × bd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 �2 1Ad ⌈d2ατ⌉dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In addition, we have I1 = 1 2d2α � d2αs − ⌊d2αs⌋ � � ⌈d2αs⌉ − d2αs � V ′′ � Xd ⌊d2αs⌋,1 � bd � Xd ⌊d2αs⌋,1, Zd ⌈d2αs⌉,1 �2 1Ad ⌈d2αs⌉ + 1 2d2α � d2αt − ⌊d2αt⌋ � � ⌈d2αt⌉ − d2αt � V ′′ � Xd ⌊d2αt⌋,1 � bd � Xd ⌊d2αt⌋,1, Zd ⌈d2αt⌉,1 �2 1Ad ⌈d2αt⌉ , and, since V ′′ and the moments of Zd ⌈d2αt⌉,1 are bounded, limd→∞ E [|I1|] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Thus, lim d→∞ E ���V � Ld t � − V � Ld s � − Is,t ��� = 0 , where Is,t = � t s � ℓdαV ′ � Xd ⌊d2ατ⌋,1 � bd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 � (45) + ℓ2 2 V ′′ � Xd ⌊d2ατ⌋,1 � bd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 �2 1Ad ⌈d2ατ⌉ � dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Next, we use (18) and write � t s LV � Ld τ � dτ = � t s hL(ℓ) 2 � V ′′ � Xd ⌊d2ατ⌋1 � − sgn � Xd ⌊d2ατ⌋,1 � V ′ � Xd ⌊d2ατ⌋,1 �� dτ − T d 3 , (46) where we define T d 3 = � t s � LV � Xd ⌊d2ατ⌋,1 � − LV � Ld τ �� dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' we write the difference M d ⌈d2αt⌉(V ) − M d ⌈d2αs⌉(V ) as the integral of a piecewise constant 29 function M d ⌈d2αt⌉(V ) − M d ⌈d2αs⌉(V ) = Is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='t (47) − � t s � ℓdαV ′ � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � E � bd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � 1Ad ⌈d2ατ⌉ ���Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � +ℓ2 2 V ′′ � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � E � bd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 �2 1Ad ⌈d2ατ⌉ ����Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 �� dτ − T d 4 − T d 5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' where T d 4 and T d 5 account for the difference between the sum in (41) and the integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and are defined as T d 4 = − ℓ dα � ⌈d2αt⌉ − d2αt � V ′ � Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � � bd � Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � 1Ad ⌈d2αt⌉ −E � bd � Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � 1Ad ⌈d2αt⌉ ���Fd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 �� − ℓ2 2d2α � ⌈d2αt⌉ − d2αt � V ′′ � Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � � bd � Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 �2 1Ad ⌈d2αt⌉ −E � bd � Xd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2αt⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 �2 1Ad ⌈d2αt⌉ ����Fd ⌊d2αt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and T d 5 = −T d 4 with t substituted by s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Putting (45),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (46) and (47) together we obtain Is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='t − � t s LV � Ld τ � dτ − � M d ⌈d2αt⌉(V ) − M d ⌈d2αs⌉(V ) � = T d 1 + T d 2 + T d 3 + T d 4 + T d 5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' where T d 1 takes into account all the terms involving V ′(Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and T d 2 the terms involving V ′′(Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1): T d 1 = � t s V ′ � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � × � ℓdαE � bd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � 1Ad ⌈d2ατ⌉ ���Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � + hL(ℓ) 2 sgn � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 �� dτ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' T d 2 = � t s V ′′ � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � × �ℓ2 2 E � bd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 �2 1Ad ⌈d2ατ⌉ ����Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � − hL(ℓ) 2 � dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To obtain (42) it is then sufficient to prove that for any 1 ≤ i ≤ 5, limd→∞ E ���T d i ��� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Since V ′, V ′′ are bounded and bd is bounded in expectation because the moments of Zd ⌈d2ατ⌉,1 are bounded, it is easy to show that limd→∞ E ���T d i ��� = 0 for i = 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For T d 3 , we write T d 3 = 30 hL(ℓ)(T d 3,1 − T d 3,2)/2, where T d 3,1 = � t s � V ′′ � Xd ⌊d2ατ⌋,1 � − V ′′ � Ld τ �� dτ , T d 3,2 = � t s � sgn � Xd ⌊d2ατ⌋,1 � V ′ � Xd ⌊d2ατ⌋,1 � − sgn � Ld τ � V ′ � Ld τ �� dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using Fubini-Tonelli’s theorem, the convergence of Xd ⌊d2ατ⌋,1 to Ld τ in Lemma 2 and Lebesgue’s dominated convergence theorem we obtain E ���T d 3,1 ��� ≤ � t s E ����V ′′ � Xd ⌊d2ατ⌋,1 � − V ′′ � Ld τ ���� � dτ −→ d→∞ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We can further decompose T d 3,2 as T d 3,2 = � t s � sgn � Xd ⌊d2ατ⌋,1 � − sgn � Ld τ �� V ′ � Xd ⌊d2ατ⌋,1 � dτ + � t s sgn � Ld τ � � V ′ � Xd ⌊d2ατ⌋,1 � − V ′ � Ld τ �� dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proceeding as for T d 3,1, it is easy to show that the second integral converges to 0 as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We then bound the first integral by E ����� � t s � sgn � Xd ⌊d2ατ⌋,1 � − sgn � Ld τ �� V ′ � Xd ⌊d2ατ⌋,1 � dτ ���� � ≤ C � t s E ����sgn � Xd ⌊d2ατ⌋,1 � − sgn � Ld τ ���� � dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' However, since {sgn(Xd ⌊d2ατ⌋,1) ̸= sgn(Ld τ)} ⊂ {sgn(Xd ⌊d2ατ⌋,1) ̸= sgn(Xd ⌈d2ατ⌉,1)}, using Lemma 4 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3 we have that E ����sgn � Xd ⌊d2ατ⌋,1 � − sgn � Ld τ ���� � = 2E � 1� sgn � Xd ⌊d2ατ⌋,1 � ̸=sgn(Ldτ ) � � = 2E � 1� sgn � Xd ⌊d2ατ⌋,1 � ̸=sgn � Xd ⌈d2ατ⌉,1 �� � −→ d→∞ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The above and the dominated converge theorem show that E ����� � t s � sgn � Xd ⌊d2ατ⌋,1 � − sgn � Ld τ �� V ′ � Xd ⌊d2ατ⌋,1 � dτ ���� � −→ d→∞ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Consider then T d 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' recalling that the derivatives of V are bounded,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' we have E ���T d 1 ��� ≤ � t s CE ����ℓdαE � bd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � 1Ad ⌈d2ατ⌉ ���Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � + hL(ℓ) 2 sgn � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ����� � dτ ≤ � t s C � E ����D(1) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='τ ��� � + E ����D(1) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='τ ��� �� dτ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 31 where we define D(1) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='τ = ℓdαE � Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='11Ad ⌈d2ατ⌉ ���Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' D(1) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='τ = hL(ℓ) 2 sgn � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � − ℓdα �σd 2 sgn(Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1)1|Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1|>σ2m d r/2 + 1 σ2m−1 d rXd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='11|Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1|≤σ2m d r/2 � × E � 1Ad ⌈d2ατ⌉ ���Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let us start with D(1) 1,τ: D(1) 1,τ = ℓdαE � Zd ⌈d2ατ⌉,1 � 1 ∧ exp � d � i=1 φd � Xd ⌊d2ατ⌋,i, Zd ⌈d2ατ⌉,i ��������Fd ⌊d2ατ⌋,1 � , where φd is given in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Then, by independence of the components of Zd ⌈d2ατ⌉, we have E � Zd ⌈d2ατ⌉,1 � 1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,i, Zd ⌈d2ατ⌉,i ��������Fd ⌊d2ατ⌋,1 � = E � Zd ⌈d2ατ⌉,1 � E � 1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,i, Zd ⌈d2ατ⌉,i �������Fd ⌊d2ατ⌋,1 � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' This allows us to write E � |D(1) 1,τ| � ≤ ℓdαE � |Zd ⌈d2ατ⌉,1| �����1 ∧ exp � d � i=1 φd � Xd ⌊d2ατ⌋,i, Zd ⌈d2ατ⌉,i �� − 1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,i, Zd ⌈d2ατ⌉,i ������� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' However, x �→ 1 ∧ exp(x) is a 1-Lipschitz function, thus E � |D(1) 1,τ| � ≤ ℓdαE � |Zd ⌈d2ατ⌉,1| ���φd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 ���� � , and D(1) 1,τ → 0 as d → ∞ by Lemma 5 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For D(1) 2,τ, we observe that − σd 2 1|Xd ⌊d2ατ⌋,1|≤σ2m d r/2 ≤ 1 σ2m−1rXd ⌊d2ατ⌋,11|Xd ⌊d2ατ⌋,1|≤σ2m d r/2 ≤ σd 2 1|Xd ⌊d2ατ⌋,1|≤σ2m d r/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (48) Distinguishing between Xd ⌊d2ατ⌋,1 < 0 and Xd ⌊d2ατ⌋,1 ≥ 0, it follows that |D(1) 2,τ| ≤ ���sgn � Xd ⌊d2ατ⌋,1 ���� × ���� hL(ℓ) 2 − ℓdα �σd 2 1|Xd ⌊d2ατ⌋,1|>σ2m d r/2 + σd 2 1|Xd ⌊d2ατ⌋,1|≤σ2m d r/2 � E � 1Ad ⌈d2ατ⌉ ���Fd ⌊d2ατ⌋,1 ����� ≤ 1 2 ���hL(ℓ) − ℓ2E � 1Ad ⌈d2ατ⌉ ���Fd ⌊d2αr⌋,1⌋ ���� , 32 where we recall that σd = ℓd−α with α = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using the triangle inequality we obtain 2E � |D(1) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='τ| � ≤ E ������hL(ℓ) − ℓ2E � 1 ∧ exp � d � i=1 φd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i �������Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ������ � ≤ E ������hL(ℓ) − ℓ2E � 1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i �������Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ������ � + ℓ2E ������1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i �� − 1 ∧ exp � d � i=1 φd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i ������� � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' where we used Jensen’s inequality to remove the conditional expectation in the last term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Recalling that x �→ 1 ∧ exp(x) is 1-Lipschitz, we can then bound the second term ℓ2E ������1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,i, Zd ⌈d2ατ⌉,i �� − 1 ∧ exp � d � i=1 φd � Xd ⌊d2ατ⌋,i, Zd ⌈d2ατ⌉,i ������� � ≤ ℓ2E ����φd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 ���� � , (49) ≤ ℓ2E � φd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 �2�1/2 , where the final expectation converges to zero as d → ∞ by Proposition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For the remaining term in D(1) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' since (Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i)2≤i≤n is independent of Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' we have ℓ2E � 1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i �������Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � = ℓ2E � 1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' using again the fact that x �→ 1 ∧ exp(x) is 1-Lipschitz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' we have �����hL(ℓ) − ℓ2E � 1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i �������� ≤ �����hL(ℓ) − ℓ2E � 1 ∧ exp � d � i=1 φd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='i �������� + ℓ2E ����φd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ���� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The last term goes to 0 as shown in (49), and, as hL(ℓ) = ℓ2aL(ℓ), with aL(ℓ) = lim d→∞ E � 1 ∧ exp � d � i=1 φd,i �� , 33 by Theorem 2, we obtain lim d→∞ �����hL(ℓ) − ℓ2E � 1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,i, Zd ⌈d2ατ⌉,i ������� � = 0 , showing that D(1) 2,τ → 0 as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To obtain convergence of T d 1 , we observe that for any τ ∈ [s, t], D(1) 1,τ and D(1) 2,τ follow the same distributions as D(1) 1,s and D(1) 2,s, since for any k ∈ N, Xd k has distribution πL d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Therefore, the convergence towards zero of E[|D(1) 1,τ|] and E[|D(1) 2,τ|] is uniform for τ ∈ [s, t], which gives us T d 1 → 0 as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Finally, consider T d 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using analogous arguments to those used for T d 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' we obtain E � |T d 2 | � ≤ C � t s ℓ2 2 E �����E � bd � Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 �2 1Ad ⌈d2ατ⌉ ����Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � − aL(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' r) ���� � dτ ≤ C � t s ℓ2 2 � E � |D(2) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='τ| � + E � |D(2) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='τ| � E � |D(2) 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='τ| �� dτ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' where we define D(2) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='τ = E �� Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 �2 1Ad ⌈d2ατ⌉ ����Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � − aL(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' r) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' D(2) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='τ = �σd 2 sgn(Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1)1|Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1|>σ2m d r/2 + 1 σ2m−1 d rXd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='11|Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1|≤σ2m d r/2 �2 × E � 1Ad ⌈d2ατ⌉ ���Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' D(2) 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='τ = 2 �σd 2 sgn(Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1)1|Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1|>σ2m d r/2 + 1 σ2m−1 d rXd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='11|Xd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1|≤σ2m d r/2 � × E � Zd ⌈d2ατ⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='11Ad ⌈d2ατ⌉ ���Fd ⌊d2ατ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using (48), Cauchy-Schwarz’s inequality and the fact that the moments of Zd ⌈d2ατ⌉,1 are bounded we have E � |D(2) 2,τ| � ≤ σ2 d 4 −→ d→∞ 0 , E � |D(2) 3,τ| � ≤ Cσd −→ d→∞ 0 , since σd = ℓd−α with α = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The remaining term is bounded similarly to D(1) 2,τ, using the fact that x �→ 1 ∧ exp(x) is 1-Lipschitz, we have E � |D(2) 3,τ| � ≤ E ������E �� Zd ⌈d2ατ⌉,1 �2 � 1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,i, Zd ⌈d2ατ⌉,i ��������Fd ⌊d2ατ⌋,1 � − aL(ℓ, r) ����� � + E �� Zd ⌈d2ατ⌉,1 �2 ���φd � Xd ⌊d2ατ⌋,1, Zd ⌈d2ατ⌉,1 ���� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 34 The second expectation is bounded as (49) using Cauchy-Schwarz’s inequality and Proposition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For the first expectation, we use the conditional independence of the components of Zd ⌈d2ατ⌉ and write E �� Zd ⌈d2ατ⌉,1 �2 � 1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,i, Zd ⌈d2ατ⌉,i ��������Fd ⌊d2ατ⌋,1 � = E �� Zd ⌈d2ατ⌉,1 �2� E �� 1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,i, Zd ⌈d2ατ⌉,i ���� = E �� 1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,i, Zd ⌈d2ατ⌉,i ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' It follows that E[|D(2) 3,τ|] → 0 as d → ∞ since, by Theorem 2, �����E �� 1 ∧ exp � d � i=2 φd � Xd ⌊d2ατ⌋,i, Zd ⌈d2ατ⌉,i ���� − aL(ℓ, r) ����� → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Combining the results for T d i , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , 5 we obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Acknowledgments F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' acknowledge support from the EPSRC (grant # EP/R034710/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' acknowledges further support from the EPSRC (grant # EP/R018561/1) and the Alan Turing Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' A.' metadata={'source': 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130(10):6094–6132, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' [46] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zhou, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Chi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proximal MCMC for Bayesian inference of constrained and regularized estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='07378, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' A Proof of Theorem 1 The proof of Theorem 1 follows that of [34, Theorem 1, Theorem 2] and consists of four propositions showing convergence of the log-acceptance probability to a normal random variable and (weak) convergence of the process (11) to a Langevin diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We start by recalling and defining a number of quantities that we will use in the following proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Recall that σd = ℓ/dα, that λd = σ2m d r/2 where m ≥ 1/2 and r > 0 are to be chosen according to the different cases in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Recalling the expression of the proposal given in (9) and using the simplification given in (10), we define the proposal with starting point xd ∈ Rd, yd(xd, zd) = xd − σ2 d 2 ∇G � proxσ2m d r/2 G (xd) � + σdzd , where zd ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Since G is the d-times tensor product of g, the i-th component of the proposal only depends on the i-th components of xd and zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Thus, we introduce the notation for any x, z ∈ R, yd(x, z) = x − σ2 d 2 g′ � proxσ2m d r/2 g (x) � + σdz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' With these notations, we obtain the proposal for the chain (Xd k)k≥0 using Y d k = yd(Xd k, Zd k+1) = (yd(Xd k,i, Zd k+1,i))i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let us define the generator of the discrete process (Xd k)k≥0 for all 38 V ∈ C∞ c (Rd, R), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' infinitely differentiable R-valued multivariate functions with compact support, and any xd ∈ Rd, LdV (xd) = d2αE �� V (yd(xd, Zd 1)) − V (xd) � πd(yd(xd, Zd 1))qd(yd(xd, Zd 1), xd) πd(xd)qd(xd, yd(xd, Zd 1)) ∧ 1 � = d2αE � � V (yd(xd, Zd 1)) − V (xd) � d � i=1 exp � φd(xd i , Zd 1,i) � ∧ 1 � , where the expectation is w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zd 1 = (Zd 1,i)i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',d}, a d-dimensional standard normal random variable, and where we defined φd(x, z) = log π(yd(x, z))q(yd(x, z), x) π(x)q(x, yd(x, z)) (50) = g(yd(x, z)) − g(x) + log q(yd(x, z), x) − log q(x, yd(x, z)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In the remainder we will work with one-dimensional functions V ∈ C∞ c (R, R) applied to the first component of xd so that LdV (xd) = d2αE �� V (yd(xd 1, Zd 1,1)) − V (xd 1) � πd(yd(xd, Zd 1))qd(yd(xd, Zd 1), xd) πd(xd)qd(xd, yd(xd, Zd 1)) ∧ 1 � = d2αE � � V (yd(xd 1, Zd 1,1)) − V (xd 1) � d � i=1 exp � φd � xd i , Zd 1,i �� ∧ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (51) We also define �Ld to be a variant of Ld in which the first component of the acceptance ratio is omitted: �LdV (xd) = d2αE � � V (yd(xd 1, Zd 1,1)) − V (xd 1) � d � i=2 exp � φd � xd i , Zd 1,i �� ∧ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (52) We further define the generator of the Langevin diffusion (13) LV (x) = h(ℓ, r) 2 [V ′′(x) − g′(x)V ′(x)] , (53) where h(ℓ, r) = ℓ2a(ℓ, r) is the speed of the diffusion and a(ℓ, r) = limd→∞ ad(ℓ, r) is given in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We will make use of the derivatives of g in (8) up to order 8, which we denote by g′, g′′, g′′′ and g(k) for all k = 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We recall that (gλ)′ is Lipschitz continuous with Lipschitz constant λ−1 [38, Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='19] and that (gλ)′(x) = λ−1(proxλ g(x)−x), hence proxλ g is Lipschitz continuous with Lipschitz constant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 Auxiliary Results for the Proof of Case (a) First, we characterize the limit behaviour of the acceptance ratio (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under A0, A1 and A2, if α = 1/4, β = 1/8 and r > 0, then 39 (i) the log-acceptance ratio (50) satisfies φd(x, z) = d−1/2C2(x, z) + d−3/4C3(x, z) + d−1C4(x, z) + C5(x, z, σd) , where C2(x, z) is given in (57), C3 and C4 are polynomials in z and the derivatives of g, such that E[C3(Xd 0,1, Zd 1,1)] = 0 and E[C2(Xd 0,1, Zd 1,1)2] = −2E[C4(Xd 0,1, Zd 1,1)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (ii) there exists sets Fd ⊆ Rd with d2απd(F c d) → 0 such that lim d→∞ sup xd∈Fd E ������ d � i=2 φd(xd i , Zd 1,i) − d−1/2 d � i=2 C2(xd i , Zd 1,i) + ℓ4K1(r)2 8 ����� � = 0 , (54) where K1(r) is given in Theorem 1–(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Take one component of the log-acceptance ratio φd(x, z) = g(yd(x, z)) − g(x) + log q(yd(x, z), x) − log q(x, yd(x, z)) , with yd(x, z) = x−σ2 dg′(proxσ2m d r/2 g (x))/2+σdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We have that φd(x, z) = R1(x, z, σd)+R2(x, z, σd), where R1(x, z, σ) = −g � x − σ2 2 g′ � proxσ2mr/2 g (x) � + σz � + g(x) , R2(x, z, σ) = 1 2z2 − 1 2 � z − σ 2 g′ � proxσ2mr/2 g � x + σz − σ2 2 g′ � proxσ2mr/2 g (x) ��� −σ 2 g′ � proxσ2mr/2 g (x) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (55) Following the approach of [34] we approximate φd(x, z) with a Taylor expansion about σd → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (i) Using a Taylor expansion of order 5, we obtain φd(x, z) = d−1/2C2(x, z) + d−3/4C3(x, z) + d−1C4(x, z) + C5(x, z, σd) , (56) where C2(x, z) = ℓ2 2 (−rzg′′(x)g′(x)) , (57) C3(x, z) and C4(x, z) are given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 and we use the integral form for the re- mainder C5(x, z, σd) = � σd 0 ∂5 ∂σ5 R(x, z, σ) ���� σ=u (σd − u)4 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' du , with u between 0 and σd and the derivatives of R1 and R2 given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In addition, integrating by parts and using the moments of Zd 1,1 we find that E[C2(Xd 0,1, Zd 1,1)] = E[C3(Xd 0,1, Zd 1,1)] = 0 and 2E � C4(Xd 0,1, Zd 1,1) � + E � C2(Xd 0,1, Zd 1,1)2� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 40 (ii) To construct the sets Fd, consider, for j = 3, 4, Fd,j = F 1 d,j ∩ F 2 d,j where we define F 1 d,j = � xd ∈ Rd : ����� d � i=2 E � Cj(xd i , Zd 1,i) − Cj(Xd 0,i, Zd 1,i) � ����� ≤ d5/8 � , and F 2 d,j = � xd ∈ Rd : ����� d � i=2 Vj(xd i ) − E � Vj(Xd 0,i) � ����� ≤ d6/5 � , where Vj(x) := Var(Cj(x, Zd 1,1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using Markov’s inequality and the fact that Cj, Vj are bounded by polynomials since g and its derivatives are bounded by polynomials, it is easy to show that d1/2πd((F 1 d,j)c) → 0 and d1/2πd((F 2 d,j)c) → 0, from which follows d1/2πd(F c d,j) → 0 as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To prove L1 convergence of Cj for j = 3, 4, observe that E � � � d � i=2 Cj(xd i , Zd 1,i) − E � Cj(Xd 0,1, Zd 1,1) � �2� � = d � i=2 Vj(xd i ) + � d � i=2 E � Cj(xd i , Zd 1,i) − Cj(Xd 0,1, Zd 1,1) � �2 , and that, for xd ∈ Fd,j, we have E � � � d � i=2 Cj(xd i , Zd 1,i) − E � Cj(Xd 0,1, Zd 1,1) � �2� � ≤ E � Vj(xd 1) � (d − 1) + d6/5 + d5/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Thus, the third and fourth term in the Taylor expansion (56) converge in L1 to 0 and −ℓ4K2 1(r)/8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Now, consider C5(xd i , Zd 1,i, σd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We can bound ∂5R ∂σ5 (x, z, σ) with the derivatives of g evaluated at x + σ2 2 proxσ2mr/2 g (x) + σz and proxσ2mr/2 g (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under our assumptions, the derivatives of g are bounded by polynomials M0, it follows that there exist polynomials p of the form A � 1 + � proxσ2mr/2 g (x) �N� � 1 + zN� � 1 + xN� � 1 + σN� , for sufficiently large A and sufficiently large even integer N, such that ����g(k) � x + σ2 2 proxσ2mr/2 g (x) + σdz ����� ∨ ���g(k) � proxσ2mr/2 g (x) ���� ≤ p(proxσ2mr/2 g (x), x, z, σd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 41 In addition, | proxσ2mr/2 g (x)| ≤ C(1 + |x|) for some C ≥ 1, and we can bound p(proxσ2mr/2 g (x), x, z, σ) ≤ A � 1 + zN� � 1 + x2N� � 1 + σN� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Therefore, we have E ���C5(xd i , Zd 1,i, σd) ��� ≤ AE � 1 + (Zd 1,i)N� � 1 + (xd i )2N� � σd 0 (1 + uN)(σd − u)4 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' du ≤ AE � 1 + (Zd 1,i)N� � 1 + (xd i )2N� d−5/2 ≤ A � 1 + (xd i )2N� d−5/2 , where the last inequality follows since all the moments of Zd 1 are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let us denote p(x) = A � 1 + x2N� and Fd,5 = � xd ∈ Rd : �����d−1 d � i=1 p(xd i ) − E � p(Xd 0,i) � ����� < 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' By Chebychev’s inequality we have πd(F c d,5) ≤ Var(p(Xd 0,1))d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Additionally, for all xd ∈ Fd,5, d � i=2 E ���C5(xd i , Zd 1,i, σd) ��� ≤ d � i=2 d−5/2 � E � p(Xd 0,1) � + d−1� ≤ d−3/2 � E � p(Xd 0,1) � + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Finally, set Fd = ∩5 j=3Fd,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' On Fd the last three terms of (56) converge uniformly in L1, and (54) follows using the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Next, we compare the generator Ld and �Ld in (51) and (52) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under A0, A1 and A2, if α = 1/4, β = 1/8 and r > 0, there exists sets Sd ⊆ Rd with d2απd(Sc d) → 0 such that for any V ∈ C∞ c (R, R) lim d→∞ sup xd∈Sd ���LdV (xd) − �LdV (xd) ��� = 0 , and lim d→∞ sup xd∈Sd E ������ � exp � d � i=1 φd(xd i , Zd 1,i) � ∧ 1 � − � exp � d � i=2 φd(xd i , Zd 1,i) � ∧ 1 ������ � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The function x �→ exp(x) ∧ 1 is Lipschitz continuous with Lipschitz constant 1, hence ���LdV (xd) − �LdV (xd) ��� ≤ d2αE ���V � yd(xd 1, Zd 1,1) � − V (xd 1) �� |R(xd 1, Zd 1,1, σd)| � , 42 where R(x, z, σ) = R1(x, z, σ) + R2(x, z, σ) as in (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using a Taylor expansion of order 1 about σ = 0 with integral remainder: R(x, z, σ) = R(x, z, 0) + ∂R ∂σ (x, z, σ) ���� σ=0 σ + � σ 0 ∂2R ∂σ2 (x, z, σ) ���� σ=u (σ − u)du , we obtain R(x, z, σ) = � σ 0 ∂2R ∂σ2 (x, z, σ) ���� σ=u (σ − u)du , where ∂2R ∂σ2 (x, z, σ) is bounded by the derivatives of g evaluated at x + σ2 2 proxσ2mr/2 g (x) + σz and proxσ2mr/2 g (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under our assumptions, the derivatives of g are bounded by polynomials M0, it follows that there exist polynomials p of the form A � 1 + � proxσ2mr/2 g (x) �N� � 1 + zN� � 1 + xN� � 1 + σN� , for sufficiently large A and sufficiently large even integer N, such that ����g(k) � x + σ2 2 proxσ2mr/2 g (x) + σz ����� ∨ ���g(k) � proxσ2mr/2 g (x) ���� ≤ p(proxσ2mr/2 g (x), x, z, σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proceeding as in Proposition 4, we can bound p(proxσ2mr/2 g (x), x, z, σ) ≤ A � 1 + zN� � 1 + x2N� � 1 + σN� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Therefore, we have ��R � xd 1, Zd 1,1, σd ��� ≤ A � 1 + (Zd 1,1)N� � 1 + (xd 1)2N� × � σd 0 (1 + uN)(σd − u)du ≤ A � 1 + (Zd 1,1)N� � 1 + (xd 1)2N� σ2 d 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (58) Since V ∈ C∞ c (R, R), there exists a constant C such that ��V � yd(xd 1, Zd 1,1) � − V (xd 1) �� ≤ C ��yd(xd 1, Zd 1,1) − xd 1 �� ≤ C � σd|Zd 1,1| + σ2 d 2 ���g′ � proxσ2m d r/2 g (xd 1) ���� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Recalling that g′(proxλ g(x)) = (gλ)′(x) with (gλ)′ Lipschitz continuous with Lipschitz constant λ−1, we have ���g′ � proxσ2m d r/2 g (xd 1) ���� ≤ 2 σ2m d r(1 + |xd 1|) , 43 and ��V � yd(xd 1, Zd 1,1) � − V (xd 1) �� ≤ C � σd|Zd 1,1| + σ2−2m d r � 1 + |xd 1| �� (59) ≤ Cσd � |Zd 1,1| + 1 r � 1 + |xd 1| �� , since m = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Combining (58) and (59) we obtain d2α ��V � yd(xd 1, Zd 1,1) � − V (xd 1) �� ��R(xd 1, Zd 1,1, σd) �� ≤ Cσd � 1 + (Zd 1,1)N� � 1 + (xd i )2N� � |Zd 1,1| + 1 r (1 + |xd 1|) � , for some C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Set Sd to be the set in which 1 + (xd 1)2N+1 ≤ dα/2, applying Markov’s inequality we obtain d2απd(Sc d) = d2απd �� 1 + (xd 1)2N+1�5 ≥ d5α/2� ≤ d−α/2E �� 1 + (xd 1)2N+1�5� −→ d→∞ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Recalling that |Zd 1,1| and 1 + (Zd 1,1)N are bounded, we have that sup xd∈Sd ���LdV (xd) − �LdV (xd) ��� ≤ Cdα/2 ℓ dα −→ d→∞ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The second results follows from (58) using the same argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The following result considers the convergence to the generator of the Langevin diffusion (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under A0, A1 and A2, if α = 1/4, β = 1/8 and r > 0, there exists sets Td ⊆ Rd with d2απd(T c d) → 0 as d → ∞, such that for any V ∈ C∞ c (R, R) lim d→∞ sup xd∈Td ����d2αE � V � yd(xd 1, Zd 1,1) � − V (xd 1) � − ℓ2 2 (V ′′(xd 1) + g′(xd 1)V ′(xd 1)) ���� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Take yd(xd 1, Zd 1,1) = xd 1 + σ2 d 2 g′ � proxσ2m d r/2 g (xd 1) � + σdZd 1,1 , and use a Taylor expansion of order 2 of W(x, z, σ) = V � x + σ2 2 g′ � proxσ2mr/2 g (x) � + σz � , about σ = 0 with integral remainder: W(x, z, σ) = W(x, z, 0) + ∂W ∂σ (x, z, σ) ���� σ=0 σ + 1 2 ∂2W ∂σ2 (x, z, σ) ���� σ=0 σ2 + � σ 0 ∂3W ∂σ3 (x, z, σ) ���� σ=u (σ − u)2 2 du .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 44 Using the derivatives W(x, z, 0) = V (x) , ∂W ∂σ (x, z, σ) ���� σ=0 = V ′(x)z , ∂2W ∂σ2 (x, z, σ) = V ′′(x)z2 + V ′(x)g′(x) , and recalling that E � Zd 1,1 � = 0, E � (Zd 1,1)2� = 1, we have E � V � yd(xd 1, Zd 1,1) � − V (xd 1) � = σ2 d 2 � V ′′(xd 1) + V ′(xd 1)g′(xd 1) � + E �� σd 0 ∂3W ∂σ3 (xd 1, Zd 1,1, σ) ���� σ=u (σd − u)2 2 du � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proceeding as in the previous proposition, we can bound ���� � σd 0 ∂3W ∂σ3 (xd 1, Zd 1,1, σ) �� σ=u (σd − u)2 2 du ���� ≤ A � 1 + (Zd 1,1)N� � 1 + (xd i )2N� d−3α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Setting Td to be the set in which (1 + (xd 1)2N) ≤ dα/2, the result follows by applying Markov’s inequality as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Before proceeding to stating and proving the last auxiliary result, let us denote by ψ1 : R → [0, +∞) the characteristic function of the distribution N(0, K2 1(r)), where K2 1(r) is given in Theo- rem 1–(a), ψ1(t) = exp(−t2ℓ2K1(r)2/2) , and by ψd 1(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' t) = � exp(itw)Qd 1(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' dw) the characteristic functions associated with the law Qd 1(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ·) = L � d−1/2 d � i=2 C2(xd i , Zd 1,i) � , Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under A0, A1 and A2, if α = 1/4, β = 1/8 and r > 0, there exists a sequence of sets Hd ⊆ Rd such that (i) lim d→∞ d2απd(Hc d) = 0 , (ii) for all t ∈ R lim d→∞ sup xd∈Hd |ψd 1(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' t) − ψ1(t)| = 0 , (iii) d−1/2 d � i=2 C2(xd i , Zd 1,i) L −→ N � 0, ℓ4K2 1(r) 2 � , where L −→ denotes convergence in law, 45 (iv) lim d→∞ sup xd∈Hd �����E � 1 ∧ exp � d−1/2 d � i=2 C2(xd i , Zd 1,i) − ℓ4K2 1(r) 2 �� − 2Φ � −ℓ2K1(r) 2 ������ , where Φ is the distribution function of the standard normal random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (i) Define the functions hj(x) = [−g′′(x)g′(x)]j with j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , 4 and let Hd = Hd,1 ∩Hd,2 where Hd,1 = � xd ∈ Rd : ����� 1 d d � i=2 hj(xd i ) − � R hj(u)π(u)du ����� ≤ d1/3 for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , 4 � , Hd,2 = � xd ∈ Rd : |hj(xd i )| ≤ d2/3 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , d and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' , 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using Chebychev’s inequality, the fact that the derivatives of g are bounded by polynomials and that π has finite moments, we have d1/2πd((Hd,1)c) → 0 as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Similarly, by Markov’s inequality we have d1/2πd((Hd,2)c) → 0 as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (ii) We follow [34, Lemma 3(b)] and decompose |ψd 1(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' t) − ψ1(t)| ≤ �����ψd 1(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' t) − d � i=2 � 1 − t2 2dv(xd i ) ������ (60) + ����� d � i=2 � 1 − t2 2dv(xd i ) � − d � i=2 exp � −t2 v(xd i ) 2d ������ + ����� d � i=2 exp � −t2 v(xd i ) 2d � − exp � −t2 ℓ2K1(r)2 2 ������ where v(xd i ) = Var(C2(xd i , Zd 1,i)) = ℓ4E[C2(xd i , Zd 1,i)2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For the first term, decompose the characteristic function ψd 1(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' t) = �d i=2 θd i (xd i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' t) as the product of the characteristic functions of d−1/2Wi where we define Wi = C2(xd i , Zd 1,i), using [15, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3)] as in the proof of [15, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='10] we obtain ����θd i (xd i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' t) − � 1 − t2 2dv(xd i ) ����� ≤ EZ � |t|3 d3/2 |Wi|3 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ∧ 2|t|2 d |Wi|2 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � ≤ EZ � |t|3 d3/2 |Wi|3 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' |Wi| ≤ d1/2ε � + t2 d EZ � |Wi|2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' |Wi| > d1/2ε � ≤ ε|t|3 6d EZ � |Wi|2� + t2 ε2d2 EZ � |Wi|4� for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For sufficiently large d, we have that t2v(xd i )/(2d) ≤ 1 for x ∈ Hd,2, and we can use [15, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3] �����ψd j (xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' t) − d � i=2 � 1 − t2 2dv(xd i ) ������ ≤ d � i=2 �ε|t|3 6d EZ � |Wi|2� + t2 ε2d2 EZ � |Wi|4�� ≤ εℓ4|t|3 6 (K2 1(r) + D1d−1/3) + t2 ε2d(E � |Wi|4� + D2ℓ8d−1/4) 46 where the last inequality follows from the fact that xd ∈ Hd,2 and D1, D2 are positive con- stants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For any δ > 0 we can chose ε small enough so that the first term in the above is less than δ/2 and we can chose d sufficiently large to make the second term less than δ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Thus, for any δ > 0 we can find ε > 0 and d ∈ N such that �����ψd 1(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' t) − d � i=2 � 1 − t2 2dv(xd i ) ������ < δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' the uniform convergence then follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The second term in (60) converges to 0 uniformly for all xd ∈ Hd,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' while for the third term in (60) we use again [15, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3] ����� d � i=2 exp � −t2 v(xd i ) 2d � − exp � −t2 ℓ2K1(r)2 2 ������ ≤ d � i=2 t4v(xd i )2 4d2 → 0 for all x ∈ Hd,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The result then follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (iii) This is a straightforward consequence of (ii) and the L´evy’s continuity Theorem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' [41, Theorem 1, page 322]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (iv) This statement follows directly from (iii) and [33, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 Auxiliary Results for the Proof of Case (b) First, we characterize the limit behaviour of the acceptance ratio (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The following result is an extension of [34, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under A0, A1, A2 and A3, if α = 1/6, β = 1/6 and r > 0, then (i) the the log-acceptance ratio (50) satisfies φd(xd i , Yi) = d−1/2C3(xi, Zi) + d−2/3C4(xi, Zi) + d−5/6C5(xi, Zi) + d−1C6(xi, Zi) + C7(xi, Zi, σd), where C3(xi, Zi) is given in (61), C4, C5, C6 are polynomials in Zi and the derivatives of g, such that EX [EZ [Cj(X, Z)]] = 0 for j = 3, 4, 5 and EX � EZ � C3(X, Z)2�� = −2EX [EZ [C6(X, Z)]] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (ii) there exists sets Fd ⊆ Rd with d2απd(F c d) → 0 such that lim d→∞ sup xd∈Fd E ������ d � i=2 φd(xd i , Yi) − d−1/2 d � i=2 C3(xd i , Zi) + ℓ6K2(r)2 2 ����� � = 0, where K2(r) is given in Theorem 1–(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 47 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Take one component of the log-acceptance ratio φd(xd i , Yi) = g(Yi) − g(xd i ) + log q(Yi, xd i ) − log q(xd i , Yi) with Yi = xd i − σ2 d 2 g′ � proxσ2m d r/2 g (xd i ) � + σdZi and Zi ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proceeding as in the proof for case (a), we have that φd(xd i , Yi) = R1(xd i , Z, σd) + R2(xd i , Z, σd) where R1, R2 are given in (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Following the approach of [34] we approximate φd(xd i , Yi) with a Taylor expansion about σd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (i) Using a Taylor expansion of order 7, we find that φd(xd i , Yi) = d−1/2C3(xd i , Zi) + d−2/3C4(xd i , Zi) + d−5/6C5(xd i , Zi) + d−1C6(xd i , Zi) + C7(xd i , Zi, σd), where C3(x, z) = ℓ3 6 �1 2g′′′(x)z3 − 3 2g′′(x)g′(x)z (1 + 2r) � , (61) C4(x, z), C5(x, z) and C6(x, z) are given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 and integral form of the remainder C7(xd i , Zi, σd) = � σd 0 ∂7 ∂σ7 R(xd i , Zi, σ)|σ=ϵ (σd − ϵ)6 6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' dϵ, with ϵ between 0 and σd and the derivatives of R1 and R2 are given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In addition, integrating by parts and using the moments of Z we find that EX [EZ [C3(X, Z)]] = EX [EZ [C4(X, Z)]] = EX [EZ [C5(X, Z)]] = 0 and EX [EZ [C6(X, Z)]] = ℓ6 � − 1 16 � r + 2r2� [g′′(X)g′(X)]2 − 1 16 �1 2 + r � g′′(X)3 − 5 96g′′′(X)2 � , which shows that EX � EZ � C3(X, Z)2 + 2C6(X, Z) �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (ii) The proof of this result follows using the same steps as case (a) and is analogous to that of [34, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Next, we compare the generator Ld and �Ld in (51) and (52) respectively, extending [34, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under A0, A1, A2 and A3, if α = 1/6, β = 1/6 and r > 0, there exists sets Sd ⊆ Rd with d2απd(Sc d) → 0 such that for any V ∈ C∞ c (R, R) lim d→∞ sup xd∈Sd ���LdV (xd) − �LdV (xd) ��� = 0 and lim d→∞ sup xd∈Sd E ������ πd(Y )qd(Y , xd) πd(xd)qd(xd, Y ) ∧ 1 − d � i=2 φd(xd i , yi) ∧ 1 ����� � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 48 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The function x �→ exp(x) ∧ 1 is Lipschitz continuous with Lipschitz constant 1, hence ���LdV (xd) − �LdV (xd) ��� ≤ d2αE � |V (Y1) − V (xd 1)||R(xd 1, Z1, σd)| � where R(x, z, σ) = R1(x, z, σ) + R2(x, z, σ) as in (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using a Taylor expansion of order 1 about σ = 0 with integral remainder: R(x, z, σ) = R(x, z, 0) + ∂R ∂σ (x, z, σ)|σ=0σ + � σ 0 ∂2R ∂σ2 (x, z, σ)|σ=ϵ(σ − ϵ)dϵ, we obtain R(xd 1, Z, σd) = � σd 0 ∂2R ∂σ2 (x, z, σ)|σ=ϵ(σd − ϵ)dϵ, where ∂2R ∂σ2 (x, z, σ) is bounded by the derivatives of g evaluated at x + σ2 d 2 proxσ2mr/2 g (x) + σz and proxσ2mr/2 g (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under our assumptions, the derivatives of g are bounded by polynomials M0, it follows that there exist polynomials p of the form A � 1 + � proxσ2mr/2 g (x) �N� � 1 + zN� � 1 + xN� � 1 + σN� , for suffi- ciently large A and sufficiently large even integer N, such that ����g(k) � x + σ2 2 proxσ2mr/2 g (x) + σz ����� ∨ ���g(k) � proxσ2mr/2 g (x) ���� ≤ p(proxσ2mr/2 g (x), x, z, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proceeding as in Proposition 8, we can bound p(proxσ2mr/2 g (x), x, z, σ) ≤ A � 1 + zN� � 1 + x2N� � 1 + σN� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Therefore, we have |R(xd 1, Z1, σd)| ≤ A � 1 + ZN 1 � � 1 + (xd i )2N� × � σd 0 (1 + ϵN)(σd − ϵ)dϵ ≤ A � 1 + ZN 1 � � 1 + (xd 1)2N� σ2 d 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Since V ∈ C∞ c (R, R), there exists a constant C such that |V (Y1) − V (xd 1)| ≤ C|Y1 − xd 1| ≤ C � σd|Z1| + σ2 d 2 |g′ � proxσ2m d r/2 g (xd 1) � | � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under A3, g′ is Lipschitz continuous and we have, for some C ≥ 1, |g′ � proxσ2m d r/2 g (xd 1) � | ≤ C(1 + | proxσ2m d r/2 g (xd 1)|) ≤ C(1 + |xd 1|), where we used the fact that proxλ g is 1-Lipschitz continuous for all λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The result then follows exactly as in [34, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 49 The following result considers the convergence to the generator of the Langevin diffusion (53) and is a generalization of [34, Lemma 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under A0, A1, A2 and A3, if α = 1/6, β = 1/6 and r > 0, there exists sets Td ⊆ Rd with d2απd(T c d) → 0 such that for any V ∈ C∞ c (R, R) lim d→∞ sup xd∈Td ����d2αE � V (Y1) − V (xd 1) � − ℓ2 2 (V ′′(xd 1) + g′(xd 1)V ′(xd 1)) ���� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The proof is identical to that of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Before proceeding to stating and proving the last auxiliary result, let us denote by ψ2 : R → [0, +∞) the characteristic function of the distribution N(0, K2 2(r)), where K2 2(r) is given in Theo- rem 1, ψ2(t) = exp(−t3ℓ2K2(r)2/2), and by ψd 2(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' t) = � exp(itw)Qd 2(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' dw) the characteristic functions associated with the law Qd 2(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ·) = L � d−1/2 d � i=2 C3(xd i , Zi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The following result extends [34, Lemma 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under A0, A1, A2 and A3, if α = 1/6, β = 1/6 and r > 0, there exists a sequence of sets Hd ⊆ Rd such that (i) lim d→∞ d2απd(Hc d) = 0, (ii) for all t ∈ R lim d→∞ sup xd∈Hd |ψd 2(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' t) − ψ2(t)| = 0, (iii) d−1/2 d � i=2 C3(xd i , Zi) L −→ N � 0, ℓ6K2 2(r) 2 � , where L −→ denotes convergence in law, (iv) lim d→∞ sup xd∈Hd �����EZ � 1 ∧ exp � d−1/2 d � i=2 C3(xd i , Zi) − ℓ6K2 2(r) 2 �� − 2Φ � −ℓ3K2(r) 2 ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 50 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (i) The proof is analogous to that of Proposition 7 and follows the same steps of that of [34, Lemma 3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (ii) The proof is analogous to that of Proposition 7 and follows the same steps of that of [34, Lemma 3(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (iii) This is a straightforward consequence of (ii) and the L´evy’s continuity Theorem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' [41, Theorem 1, page 322]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (iv) This statement follows directly from (iii) and [33, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3 Auxiliary Results for the Proof of Case (c) First, we characterize the limit behaviour of the acceptance ratio (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under A0, A1, A2 and A3 and, if α = 1/6, β = m/6 for m > 1 and r > 0, then (i) the log-acceptance ratio (50) satisfies φd(xd i , Yi) = d−1/2C3(xi, Zi) + d−2/3C4(xi, Zi) + d−5/6C5(xi, Zi) + d−1C6(xi, Zi) + C7(xi, Zi, σd), where C3(xi, Zi) is given in (62), C4, C5, C6 are polynomials in Zi and the derivatives of g, such that EX [EZ [Cj(X, Z)]] = 0 for j = 3, 4, 5 and EX � EZ � C3(X, Z)2�� = −2EX [EZ [C6(X, Z)]] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (ii) there exists sets Fd ⊆ Rd with d2απd(F c d) → 0 such that lim d→∞ sup xd∈Fd E ������ d � i=2 φd(xd i , Yi) − d−1/2 d � i=2 C3(xd i , Zi) + ℓ6K2 3 2 ����� � = 0, where K3 = K2(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Take one component of the log-acceptance ratio φd(xd i , Yi) = g(Yi) − g(xd i ) + log q(Yi, xd i ) − log q(xd i , Yi) with Yi = xd i − σ2 d 2 g′ � proxσ2m d r/2 g (xd i ) � + σdZi and Zi ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proceeding as in the proof for case (a), we have that φd(xd i , Yi) = R1(xd i , Z, σd) + R2(xd i , Z, σd) where R1, R2 are given in (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Following the approach of [34] we approximate φd(xd i , Yi) with a Taylor expansion about σd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (i) Using a Taylor expansion of order 7, we find that φd(xd i , Yi) = d−1/2C3(xd i , Zi) + d−2/3C4(xd i , Zi) + d−5/6C5(xd i , Zi) + d−1C6(xd i , Zi) + C7(xd i , Zi, σd), 51 where C3(x, z) = ℓ3 6 �1 2g′′′(x)z3 − 3 2g′′(x)g′(x)z � , (62) C4(x, z), C5(x, z) and C6(x, z) are given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3 and integral form of the remainder C7(xd i , Zi, σd) = � σd 0 ∂7 ∂σ7 R(xd i , Zi, σ)|σ=ϵ (σd − ϵ)6 6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' dϵ, with ϵ between 0 and σd and the derivatives of R1 and R2 are given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In addition, integrating by parts and using the moments of Z we find that EX [EZ [C3(X, Z)]] = EX [EZ [C4(X, Z)]] = EX [EZ [C5(X, Z)]] = 0 and EX [EZ [C6(X, Z)]] = ℓ6 � − 1 32g′′(X)3 − 5 96g′′′(X)2 � , which shows that EX � EZ � C3(X, Z)2 + 2C6(X, Z) �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (ii) The proof of this result follows using the same steps as case (a) and is analogous to that of [34, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Next, we compare the generator Ld and �Ld in (51) and (52) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under A0, A1, A2 and A3, if α = 1/6, β = m/6 for m > 1 and r > 0, there exists sets Sd ⊆ Rd with d2απd(Sc d) → 0 such that for any V ∈ C∞ c (R, R) lim d→∞ sup xd∈Sd ���LdV (xd) − �LdV (xd) ��� = 0 and lim d→∞ sup xd∈Sd E ������ πd(Y )qd(Y , xd) πd(xd)qd(xd, Y ) ∧ 1 − d � i=2 φd(xd i , yi) ∧ 1 ����� � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The proof is identical to that of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The following result considers the convergence to the generator of the Langevin diffusion (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under A0, A1, A2 and A3, if α = 1/6, β = m/6 for m > 1 and r > 0, there exists sets Td ⊆ Rd with d2απd(T c d) → 0 such that for any V ∈ C∞ c (R, R) lim d→∞ sup xd∈Td ����d2αE � V (Y1) − V (xd 1) � − ℓ2 2 (V ′′(xd 1) + g′(xd 1)V ′(xd 1)) ���� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The proof is identical to that of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 52 Before proceeding to stating and proving the last auxiliary result, let us denote by ψ3 : R → [0, +∞) the characteristic function of the distribution N(0, K2 3), where K2 3 = K2 2(0), ψ3(t) = exp(−t2ℓ3K2 3/2), and by ψd 3(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' t) = � exp(itw)Qd 3(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' dw) the characteristic functions associated with the law Qd 3(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ·) = L � d−1/2 d � i=2 C3(xd i , Zi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Under A0, A1, A2 and A3, if α = 1/6, β = m/6 for m > 1 and r > 0, there exists a sequence of sets Hd ⊆ Rd such that (i) lim d→∞ d2απd(Hc d) = 0, (ii) for all t ∈ R lim d→∞ sup xd∈Hd |ψd 3(xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' t) − ψ3(t)| = 0, (iii) d−1/2 d � i=2 C3(xd i , Zi) L −→ N � 0, ℓ6K2 3 2 � , where L −→ denotes convergence in law, (iv) lim d→∞ sup xd∈Hd �����EZ � 1 ∧ exp � d−1/2 d � i=2 C3(xd i , Zi) − ℓ6K2 3 2 �� − 2Φ � −ℓ3K3 2 ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (i) The proof is analogous to that of Proposition 7 and follows the same steps of that of [34, Lemma 3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (ii) The proof is analogous to that of Proposition 7 and follows the same steps of that of [34, Lemma 3(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (iii) This is a straightforward consequence of (ii) and the L´evy’s continuity Theorem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' [41, Theorem 1, page 322]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (iv) This statement follows directly from (iii) and [33, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 53 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 Proof of Theorem 1 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (a) The asymptotic acceptance rate follows by combining Propositions 4– 6 with part (iv) of Proposition 7 as in the proof of [34, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To prove the weak convergence of the process it suffices to show that there exists events F ⋆ d ∈ Rd such that for all t > 0 P � Ld t ∈ F ⋆ d for all 0 ≤ s ≤ t � → 1 and lim d→∞ sup xd∈F ⋆ d ��LdV (xd) − LV (xd) �� for all V ∈ C∞ c (R, R) [17, Chapter 4, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We take F ⋆ d = Fd ∩ Sd ∩ Td ∩ Hd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Then, d2απd ((F ⋆ d )c) → 0 and P � Ld t ∈ F ⋆ d for all 0 ≤ s ≤ t � → 1 for all fixed t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Combining Propositions 4–7 we obtain convergence of the generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To obtain the value of a(ℓ, r) maximizing the speed, we observe that K2 1(r) is a function of the ratio r = c2/ℓ2m = c2/ℓ only, we can take c ∝ ℓ1/2 so that K2 1(r) is constant for given c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using the same substitution as in [34, Theorem 2] we find that h(ℓ, r) is maximized at the unique value of ℓ such that a(ℓ, r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (b) The proof is analogous to that of case (a) replacing Propositions 4, 5, 6 and 7 with Proposi- tions 8, 9, 10 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To obtain the value of a(ℓ, r) maximizing the speed, we observe that K2 2(r) is a function of the ratio r = c2/ℓ2m = c2/ℓ2 only, we can take c ∝ ℓ so that K2 2(r) is constant for given c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using the same substitution as in [34, Theorem 2] we find that h(ℓ, r) is maximized at the unique value of ℓ such that a(ℓ, r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' (c) The proof is analogous to that of case (a) replacing Propositions 4, 5, 6 and 7 with Proposi- tions 12, 13, 14 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To obtain the value of a(ℓ, r) maximizing the speed, we observe that K2 3 is constant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' r, we can use the same substitution as in [34, Theorem 2] we find that h(ℓ, r) is maximized at the unique value of ℓ such that a(ℓ, r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' B Numerical Experiments B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 Differentiable Targets We collect here a number of numerical experiments confirming the results in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To do so, we consider the Gaussian distribution in Example 1 and four algorithmic settings summarized in Table 1 which correspond to the three cases identified in Theorem 1 and MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The first plot in Figure 2–5 show that for values of α different from those identified in Theorem 1 the acceptance ratio ad(ℓ, r) becomes degenerate as d increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For the values of α identified in Theorem 1 we analyze the influence of ℓ on the acceptance ad(ℓ, r) (second plot), obtaining for d → +∞ the expression given by Theorem 1–(a) for Figure 2, the expression in Theorem 1–(b) for Figures 3 and 5 and that in Theorem 1–(c) for Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 54 Case Figure Algorithm α β m r (a) 2 Proximal MALA 1/4 1/8 1/2 1 (b) 3 P-MALA 1/6 1/6 1 1 (c) 4 Proximal MALA 1/6 1/2 3 2 — 5 MALA 1/6 1/6 1 ≈ 0 Table 1: Algorithm setting for the simulation study on the Gaussian target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Finally, we consider the relationship between acceptance ratio ad(ℓ, r) and the speed of the diffusion h(ℓ, r) approximated by the expected squared jumping distance (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' [18]) ESJDd := d2αE � (Xd 0 − Xd 1)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Looking at the last plot in Figure 2–5 we find that, even for relatively small values of d, the shape of the plot of ESJDd as a function of the acceptance ad(ℓ, r) is similar to that of the theoretical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' This suggests that tuning the acceptance ratio to be approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='452 when α = 1/4, β = 1/8 and approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='574 when α = 1/6, β = m/6 with m ≥ 1 should generally guarantee high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 Laplace Target We collect here a number of numerical experiments confirming the results for the Laplace distribu- tion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Similarly to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 we consider three algorithmic settings, summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Figure Algorithm α β m r 6 MALA 1/3 1/3 1 ≈ 0 7 P-MALA 1/3 1/3 1 1 8 Proximal MALA 1/3 1 3 2 Table 2: Algorithm setting for the simulation study on the Laplace target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The first plot in Figures 6–8 shows that for values of α ̸= 1/3 the acceptance ration ad(ℓ, r) becomes degenerate as d increases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' while the second plot shows that, for sufficiently large d, the average acceptance ratio and the ESJDd converge to aL(ℓ) and hL(ℓ) given in Theorems 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In the case m = 3, r = 2 in Figure 8, we find that the behaviour for low values of d significantly differs from the limiting one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For values of d lower than 130 the ESJDd achieves its maximum at a value of the acceptance ad(ℓ, r) different from that predicted by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In practice, this might mean that for low dimensional settings the recommended choice of a(ℓ, c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='360 is far from optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Similar behaviours for small d have also been observed in the case of RWM and MALA (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=', [40, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 55 0 10 20 30 40 50 60 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 = 1 8 = 1 4 = 1 2 d ad(ℓ, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 d = 15 d = 50 d = 300 d = 10000 theoretical limit ℓ ad(ℓ, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 d=15 d=50 d=300 d=10000 theory ad(ℓ, r) ESJDd Figure 2: Case (a): Proximal MALA with Gaussian target and m = 1/2, r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Average ac- ceptance rate for different choices of α (first);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' acceptance rate as a function of ℓ for increasing dimension d (second);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ESJDd as a function of the acceptance rate ad(ℓ, r) (third).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 56 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 = 1 3 = 1 6 = 1 12 d ad(ℓ, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 d = 3 d = 41 d = 300 d = 10000 theoretical limit ℓ ad(ℓ, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 d=41 d=300 d=30000 theory ad(ℓ, r) ESJDd Figure 3: Case (b): Proximal MALA with Gaussian target and m = 1, r = 1 (P-MALA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Average acceptance rate for different choices of α (first);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' acceptance rate as a function of ℓ for increasing dimension d (second);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ESJDd as a function of the acceptance rate ad(ℓ, r) (third).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 57 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 = 1 3 = 1 6 = 1 12 d ad(ℓ, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 d = 100 d = 1000 d = 60000 theoretical limit ℓ ad(ℓ, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 d=100 d=1000 d=60000 theory ad(ℓ, r) ESJDd Figure 4: Case (c): Proximal MALA with Gaussian target and m = 3, r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Average acceptance rate for different choices of α (first);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' acceptance rate as a function of ℓ for increasing dimension d (second);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ESJDd as a function of the acceptance rate ad(ℓ, r) (third).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 58 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 = 1 3 = 1 6 = 1 12 d ad(ℓ, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 d = 3 d = 22 d = 41 theoretical limit ℓ ad(ℓ, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='00 d=22 d=100 d=5000 theory ad(ℓ, r) ESJDd Figure 5: Proximal MALA with Gaussian target and m = 1, r → 0 (MALA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Average acceptance rate for different choices of α (first);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' acceptance rate as a function of ℓ for increasing dimension d (second);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ESJDd as a function of the acceptance rate ad(ℓ, r) (third).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 59 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 = 2 3 = 1 3 = 1 6 d ad(ℓ, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 d = 5 d = 26 d = 100 theoretical limit ℓ ad(ℓ, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 d = 15 d = 80 d = 1000 theoretical limit ad(ℓ, r) ESJDd Figure 6: Proximal MALA with Laplace target and m = 1, r = 0 (sG-MALA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Average acceptance rate for different choices of α (first);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' acceptance rate as a function of ℓ for increasing dimension d (second);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ESJDd as a function of the acceptance rate ad(ℓ, r) (third).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 60 0 10 20 30 40 50 60 70 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 = 2 3 = 1 3 = 1 6 d ad(ℓ, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 d = 5 d = 50 d = 300 d = 10000 theoretical limit ℓ ad(ℓ, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 d = 300 d = 1000 d = 15000 theoretical limit ad(ℓ, r) ESJDd Figure 7: Proximal MALA with Laplace target and m = 1, r = 1 (P-MALA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Average acceptance rate for different choices of α (first);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' acceptance rate as a function of ℓ for increasing dimension d (second);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ESJDd as a function of the acceptance rate ad(ℓ, r) (third).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 61 0 10 20 30 40 50 60 70 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 = 2 3 = 1 3 = 1 6 d ad(ℓ, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 d = 15 d = 32 d = 58 d = 300 theoretical limit ℓ ad(ℓ, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='0 d = 15 d = 50 d = 150 d = 1000 theoretical limit ad(ℓ, r) ESJDd Figure 8: Proximal MALA with Laplace target and m = 3, r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Average acceptance rate for different choices of α (first);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' acceptance rate as a function of ℓ for increasing dimension d (second);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ESJDd as a function of the acceptance rate ad(ℓ, r) (third).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 62 C Taylor Expansions for the Results on Differentiable Tar- gets C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 Coefficients of the Taylor Expansion We collect here the coefficients of the Taylor expansions in Propositions 4, 8 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 Case (a) If α = 1/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' β = 1/8 and r > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' then the log-acceptance ratio (50) satisfies φd(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = d−1/2C2(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) + d−3/4C3(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) + d−1C4(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) + C5(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σd) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' where C2(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = ℓ2 2 (−rzg′′(x)g′(x)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and C3(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = ℓ3 6 �z3 2 g′′′(x) − 3 2z2rg′(x)g′′′(x) − 3 2rz2 [g′′(x)]2 + 3 4zr2g′′′(x) [g′(x)]2 −3 2zg′(x)g′′(x) + 3 2r [g′(x)]2 g′′(x) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' C4(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = ℓ4 24 � g(4)(x) � z4 − zr3 2 [g′(x)]3 − 3z3rg′(x) + 3 2z2r2 [g′(x)]2 � + g′′′(x) � −6z2g′(x) − 9rz3g′′(x) + 9z2r2g′(x)g′′(x) + 9rz [g′(x)]2 −9 2zr3 [g′(x)]2 g′′(x) − 3 2r2 [g′(x)]3 � + 12rzg′(x) [g′′(x)]2 + 3g′′(x) [g′(x)]2 − 3z2 [g′′(x)]2 + 3z2r2 [g′′(x)]3 −6r2 [g′(x)g′′(x)]2 − 3zr3g′(x) [g′′(x)]3� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and we use the integral form for the remainder C5(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σd) = � σd 0 ∂5 ∂σ5 R(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ) ���� σ=u (σd − u)4 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' du , with u between 0 and σd and the derivatives of R1 and R2 given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 Case (b) If α = 1/6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' β = 1/6 and r > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' the the log-acceptance ratio (50) satisfies φd(xd i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Yi) = d−1/2C3(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zi) + d−2/3C4(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zi) + d−5/6C5(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zi) + d−1C6(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zi) + C7(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σd),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 63 where C3(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = ℓ3 6 �1 2g′′′(x)z3 − 3 2g′′(x)g′(x)z (1 + 2r) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and C4(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = ℓ4 24 � z4g(4)(x) − 6z2g′′′(x)g′(x)(1 + r) −3z2 [g′′(x)]2 (1 + 2r) + 3g′′(x) [g′(x)]2 (1 + 2r) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' C5(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = ℓ5 120 �3 2z5g(5)(x) − 15z3g(4)(x)g′(x)(1 + r) + 15z [g′(x)]2 g′′′(x) �3 2 + 3r + r2 � +15z(1 + 4r + 2r2)g′(x) [g′′(x)]2 − 15z3g′′(x)g′′′(x)(1 + 3r) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' C6(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = ℓ6 720 � 2z6g(6)(x) − 30 (1 + r) z4g′(x)g(5)(x) + 45 � 2 + 4r + r2� z2 [g′(x)]2 g(4)(x) + 90 � r + r2� z2 [g′′(x)]3 − 15 � 2 + 6r + 3r2� g′′′(x) [g′(x)]3 − 30 (1 + 4r) z4g′′(x)g(4)(x) + 45 � 3 + 16r + 6r2� z2g′(x)g′′(x)g′′′(x) −45 2 (1 + 4r) z4 [g′′′(x)]2 − 45 2 � 1 + 8r + 8r2� [g′(x)g′′(x)]2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and integral form of the remainder C7(xd i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σd) = � σd 0 ∂7 ∂σ7 R(xd i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ)|σ=ϵ (σd − ϵ)6 6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' dϵ, with ϵ between 0 and σd and the derivatives of R1 and R2 are given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3 Case (c) If α = 1/6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' β = m/6 for m > 1 and r > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' then the log-acceptance ratio (50) satisfies φd(xd i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Yi) = d−1/2C3(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zi) + d−2/3C4(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zi) + d−5/6C5(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zi) + d−1C6(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zi) + C7(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σd),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' where C3(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = ℓ3 6 �1 2g′′′(x)z3 − 3 2g′′(x)g′(x)z � 64 and C4(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = ℓ4 24 � � � � � � � � � z4g(4)(x) − 6z2g′′′(x)g′(x) − 3z2 [g′′(x)]2 + 3g′′(x) [g′(x)]2 −12zrg′(x)g′′(x) if m = 3/2 z4g(4)(x) − 6z2g′′′(x)g′(x) − 3z2 [g′′(x)]2 + 3g′′(x) [g′(x)]2 otherwise ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' C5(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = ℓ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2z5g(5)(x) − 15z3g(4)(x)g′(x) + 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 z [g′(x)]2 g′′′(x) + 15zg′(x) [g′′(x)]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−15z3g′′(x)g′′′(x) − 30z2r [g′′(x)]2 + 30r [g′(x)]2 g′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−30rz2g′(x)g′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='if m = 3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2z5g(5)(x) − 15z3g(4)(x)g′(x) + 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 z [g′(x)]2 g′′′(x) + 15zg′(x) [g′′(x)]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−15z3g′′(x)g′′′(x) − 60zrg′(x)g′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='if m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2z5g(5)(x) − 15z3g(4)(x)g′(x) + 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 z [g′(x)]2 g′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+15zg′(x) [g′′(x)]2 − 15z3g′′(x)g′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='otherwise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' C6(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = ℓ6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='720 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2z6g(6)(x) − 30z4g′(x)g(5)(x) + 90z2 [g′(x)]2 g(4)(x) − 30g′′′(x) [g′(x)]3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 [g′(x)g′′(x)]2 − 30z4g′′(x)g(4)(x) + 135z2g′(x)g′′(x)g′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 z4 [g′′′(x)]2 − 90z3rg′(x)g(4)(x) + 540rzg′(x) [g′′(x)]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+180rzg′(x)g′′(x) + 270rzg′′′(x) [g′(x)]2 − 45zrg′′(x)g′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+90rz3g′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='if m = 3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2z6g(6)(x) − 30z4g′(x)g(5)(x) + 90z2 [g′(x)]2 g(4)(x) − 30g′′′(x) [g′(x)]3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 [g′(x)g′′(x)]2 − 30z4g′′(x)g(4)(x) + 135z2g′(x)g′′(x)g′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 z4 [g′′′(x)]2 − 180z2rg′(x)g′′′(x) − 180z2r [g′′(x)]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+180rg′′(x) [g′(x)]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='if m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2z6g(6)(x) − 30z4g′(x)g(5)(x) + 90z2 [g′(x)]2 g(4)(x) − 30g′′′(x) [g′(x)]3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 [g′(x)g′′(x)]2 − 30z4g′′(x)g(4)(x) + 135z2g′(x)g′′(x)g′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 z4 [g′′′(x)]2 − 360zrg′(x)g′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='if m = 5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2z6g(6)(x) − 30z4g′(x)g(5)(x) + 90z2 [g′(x)]2 g(4)(x) − 30g′′′(x) [g′(x)]3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 [g′(x)g′′(x)]2 − 30z4g′′(x)g(4)(x) + 135z2g′(x)g′′(x)g′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 z4 [g′′′(x)]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='otherwise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and integral form of the remainder C7(xd i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σd) = � σd 0 ∂7 ∂σ7 R(xd i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ)|σ=ϵ (σd − ϵ)6 6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' dϵ, with ϵ between 0 and σd and the derivatives of R1 and R2 are given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 65 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 Taylor Expansions of the Log-acceptance Ratio C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 R1 Recall that R1(x, z, σ) = −g � x − σ2 2 g′ � proxσ2mr/2 g (x) � + σz � + g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We compute the derivatives of R1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ evaluated at 0: R1(x, z, 0) = 0, ∂R1 ∂σ (x, z, σ)|σ=0 = −g′(x)z, ∂2R1 ∂σ2 (x, z, σ)|σ=0 = −z2g′′(x) + [g′(x)]2 , ∂3R1 ∂σ3 (x, z, σ)|σ=0 = −z3g′′′(x) + 3g′(x)g′′(x) � z + ∂ ∂σ proxσ2mr/2 g (x)|σ=0 � , ∂4R1 ∂σ4 (x, z, σ)|σ=0 = −z4g(4)(x) + 6z2g′′′(x)g′(x) − 3g′′(x) [g′(x)]2 + 12z [g′′(x)]2 ∂ ∂σ proxσ2mr/2 g (x)|σ=0 + 6g′(x) × � g′′(x) ∂2 ∂σ2 proxσ2mr/2 g (x)|σ=0 + � ∂ ∂σ proxσ2mr/2 g (x)|σ=0 �2 g′′′(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' for m > 1/2 we will also use ∂5R1 ∂σ5 (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ)|σ=0 = −z5g(5)(x) + 10z3g(4)(x)g′(x) − 15zg′′′(x) [g′(x)]2 + 30z [g′′(x)]2 ∂2 ∂2σ proxσ2mr/2 g (x)|σ=0 + 10g′(x)g′′(x) ∂3 ∂3σ proxσ2mr/2 g (x)|σ=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ∂6R1 ∂σ6 (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ)|σ=0 = −z6g(6)(x) + 15z4g(5)(x)g′(x) − 45z2g(4)(x) [g′(x)]2 + 15g′′′(x) [g′(x)]3 − 90g′(x) [g′′(x)]2 ∂2 ∂σ2 proxσ2mr/2 g (x)|σ=0 − 60zg′′(x) ∂3 ∂σ3 proxσ2mr/2 g (x)|σ=0 + 90g′′(x)g′′′(x) ∂2 ∂σ2 proxσ2mr/2 g (x)|σ=0 + 15g′(x) × � g′′(x) ∂4 ∂σ4 proxσ2mr/2 g (x)|σ=0 + 3 � ∂2 ∂σ2 proxσ2mr/2 g (x)|σ=0 �2 g′′′(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 66 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 R2 Recall that R2(x, z, σ) = 1 2z2 − 1 2 � z − σ 2 g′ � proxσ2mr/2 g � x + σz − σ2 2 g′ � proxσ2mr/2 g (x) ��� − σ 2 g′ � proxσ2mr/2 g (x) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 67 We compute the derivatives of R2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ evaluated at 0: R2(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 0) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ∂R2 ∂σ (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ)|σ=0 = zg′(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ∂2R2 ∂σ2 (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ)|σ=0 = − [g′(x)]2 + zg′′(x) � z + 2 ∂ ∂σ proxσ2mr/2 g (x)|σ=0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ∂3R2 ∂σ3 (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ)|σ=0 = −3g′(x)g′′(x) � z + 2 ∂ ∂σ proxσ2mr/2 g (x)|σ=0 � + 3 2z �� z + ∂ ∂σ proxσ2mr/2 g (x)|σ=0 �2 g′′′(x) + � −g′(x) + 2z ∂2 ∂σ∂x proxσ2mr/2 g (x)|σ=0 + ∂2 ∂σ2 proxσ2mr/2 g (x)|σ=0 � g′′(x) + � ∂ ∂σ proxσ2mr/2 g (x)|σ=0 �2 g′′′(x) + ∂2 ∂σ2 proxσ2mr/2 g (x)|σ=0g′′(x) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ∂4R2 ∂σ4 (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ)|σ=0 = −3 [g′′(x)]2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='z + 2 ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 6g′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='z + ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−g′(x) + 2z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 + ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g′′′(x) + ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0g′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 2zg(4)(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='z + ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 6zg′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='z + ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−g′(x) + 2z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 + ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 2zg′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−3g′′(x) ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 − 3g′(x) ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+3z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ∂x2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 + 3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ ∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 2zg(4)(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 2zg′′(x) ∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 6z ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0g′′′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 68 We then proceed to get the derivatives needed for m > 1/2: ∂5R2 ∂σ5 (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ)|σ=0 = −15zg′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='z2g′′′(x) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−g′(x) + 2 ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 5g′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2z3g(4)(x) + 6zg′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−g′(x) + ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+2g′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 + ∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+2g′′(x) ∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2g(5)(x)z5 + 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2zg′′(x) ∂4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ4 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 15g(4)(x)z3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−g′(x) + ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 zg′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−g′(x) + ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 zg′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 10z2g′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 + ∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2zg′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ∂4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ4 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 − 6g′′(x) ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−6g′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 + 6z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2∂x2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+4z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂6R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ6 (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ)|σ=0 = −45 2 � z2g′′′(x) + � −g′(x) + 2 ∂2 ∂σ2 proxσ2mr/2 g (x)|σ=0 � g′′(x) �2 − 15zg′′(x) � 2z3g(4)(x) + 6zg′′′(x) � −g′(x) + ∂2 ∂σ2 proxσ2mr/2 g (x)|σ=0 � + 2g′′(x) � 3z ∂3 ∂σ2∂x proxσ2mr/2 g (x)|σ=0 + ∂3 ∂σ3 proxσ2mr/2 g (x)|σ=0 � +2g′′(x) ∂3 ∂σ3 proxσ2mr/2 g (x)|σ=0 � + 6g′(x)A(5) − zA(6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='69 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='A(5) = −5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2g(5)(x)z4 − 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2g′′(x) ∂4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ4 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 15g(4)(x)z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−g′(x) + ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 g′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−g′(x) + ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 g′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 10zg′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 + ∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='− 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2g′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ∂4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ4 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 − 6g′′(x) ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−6g′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+6z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2∂x2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 + 4z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='A(6) = −3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='10g′′′(x) ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+g′′(x) ∂5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ5 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 + g(6)(x)z5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 10g(5)(x)z3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−g′(x) + ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 15zg(4)(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−g′(x) + ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 10g(4)(x)z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 + 3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 10g′′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−g′(x) + ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 + 3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 5g′′′(x)z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−6g′′(x) ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 − 6g′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+6z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2∂x2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 + 3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ ∂4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ4 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ g′′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−10g′′(x) ∂3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 − 10g′(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 5z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ4∂x proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 + 10z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ3∂x2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 10z3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2∂x3 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−30g′(x)z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ2∂x2 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 + ∂5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='∂σ5 proxσ2mr/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='(x)|σ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 70 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3 Derivatives of the Proximity Map for Differentiable Targets Recall that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' in the differentiable case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' proxσ2mr/2 g (x) = − σ2mr 2 g′(proxσ2mr/2 g (x)) + x then ∂ ∂σ proxσ2mr/2 g (x)|σ=0 = � � � � � − r 2g′(x) if m = 1/2 0 if m > 1/2 ∞ otherwise ∂2 ∂σ2 proxσ2mr/2 g (x)|σ=0 = � � � � � � � � � r2 2 g′(x)g′′(x) if m = 1/2 −rg′(x) if m = 1 0 if m > 1 ∞ otherwise ∂3 ∂σ3 proxσ2mr/2 g (x)|σ=0 = � � � � � � � � � − 3r3 8 g′′′(x) [g′(x)]2 − 3r3 4 g′(x) [g′′(x)]2 if m = 1/2 −3rg′(x) if m = 3/2 0 if m = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' m > 3/2 ∞ otherwise ∂4 ∂σ4 proxσ2mr/2 g (x)|σ=0 = � � � � � � � � � � � � � � � < ∞ if m = 1/2 6r2g′(x)g′′(x) if m = 1 −12rg′(x) if m = 2 0 if m = 3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' m > 2 ∞ otherwise ∂5 ∂σ5 proxσ2mr/2 g (x)|σ=0 = � � � � � � � � � < ∞ if m = 1/2 −60rg′(x) if m = 5/2 0 if m = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' m = 3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' m = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' m > 5/2 ∞ otherwise and ∂ ∂x proxσ2mr/2 g (x)|σ=0 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ∂(k) ∂x(k) proxσ2mr/2 g (x)|σ=0 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 71 for all integers k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For the mixed derivatives we have ∂2 ∂σ∂x proxσ2mr/2 g (x)|σ=0 = � � � � � − r 2g′′(x) if m = 1/2 0 if m > 1/2 ∞ otherwise ∂3 ∂σ2∂x proxσ2mr/2 g (x)|σ=0 = � � � � � � � � � r2 2 [g′′(x)]2 + r2 2 g′(x)g′′′(x) if m = 1/2 −rg′′(x) if m = 1 0 if m > 1/2 ∞ otherwise ∂4 ∂σ3∂x proxσ2mr/2 g (x)|σ=0 = � � � � � � � � � < ∞ if m = 1/2 −3rg′′(x) if m = 3/2 0 if m = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' m > 3/2 ∞ otherwise ∂5 ∂σ4∂x proxσ2mr/2 g (x)|σ=0 = � � � � � � � � � � � � � � � < ∞ if m = 1/2 6r2 � g′(x)g′′′(x) + [g′′(x)]2� if m = 1 −12rg′′(x) if m = 2 0 if m = 3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' m > 2 ∞ otherwise 72 and ∂3 ∂σ∂x2 proxσ2mr/2 g (x)|σ=0 = � � � � � − r 2g′′′(x) if m = 1/2 0 if m > 1/2 ∞ otherwise ∂4 ∂σ∂x3 proxσ2mr/2 g (x)|σ=0 = � � � � � < ∞ if m = 1/2 0 if m > 1/2 ∞ otherwise ∂4 ∂σ2∂x2 proxσ2mr/2 g (x)|σ=0 = � � � � � � � � � < ∞ if m = 1/2 −rg′′′(x) if m = 1 0 if m > 1/2 ∞ otherwise ∂5 ∂σ3∂x2 proxσ2mr/2 g (x)|σ=0 = � � � � � � � � � < ∞ if m = 1/2 −3rg′′′(x) if m = 3/2 0 if m = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' m > 3/2 ∞ otherwise ∂5 ∂σ2∂x3 proxσ2mr/2 g (x)|σ=0 = � � � � � � � � � < ∞ if m = 1/2 −rg(4)(x) if m = 1 0 if m > 1 ∞ otherwise D Moments and Integrals for the Laplace Distribution D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 Moments of Acceptance Ratio for the Laplace Distribution The indicator functions in the definition of φd identify four different regions: R1 := � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) : |x| ≤ σ2mr/2 ∧ ���� � 1 − 1 σ2(m−1)r � x + σz ���� ≤ σ2mr/2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' R2 := � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) : |x| > σ2mr/2 ∧ ����x − σ2 2 sgn(x) + σz ���� ≤ σ2mr/2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' R3 := � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) : |x| ≤ σ2mr/2 ∧ ���� � 1 − 1 σ2(m−1)r � x + σz ���� > σ2mr/2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' R4 := � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) : |x| > σ2mr/2 ∧ ����x − σ2 2 sgn(x) + σz ���� > σ2mr/2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 73 with corresponding acceptance ratios φ1 d(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = |x| − ���� � 1 − 1 σ2(m−1)r � x + σz ���� + z2 2 − 1 2σ2 �� 2 σ2(m−1)r − 1 σ4(m−1)r2 � x − � 1 − 1 σ2(m−1)r � σz �2 φ2 d(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = |x| − ����x − σ2 2 sgn(x) + σz ���� + z2 2 − 1 2σ2 � 1 σ2(m−1)rx + � 1 − 1 σ2(m−1)r � �σ2 2 sgn(x) − σz ��2 φ3 d(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = |x| − ���� � 1 − 1 σ2(m−1)r � x + σz ���� + z2 2 − 1 2σ2 � 1 σ2(m−1)rx − σz + σ2 2 sgn �� 1 − 1 σ2(m−1)r � x + σz ��2 φ4 d(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) = |x| − ����x − σ2 2 sgn(x) + σz ���� + z2 2 − 1 2σ2 �σ2 2 sgn(x) − σz + σ2 2 sgn � x − σ2 2 sgn(x) + σz ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let us denote A1 := � x : 0 ≤ x ≤ σ2mr 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' A2 := � x : −σ2mr 2 ≤ x ≤ 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' A3 := � x : x > σ2mr 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' A4 := � x : x < −σ2mr 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and B1 := � z : 0 ≤ � 1 − 1 σ2(m−1)r � x + σz ≤ σ2mr 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' B2 := � z : −σ2mr 2 ≤ � 1 − 1 σ2(m−1)r � x + σz ≤ 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' B3 := � z : � 1 − 1 σ2(m−1)r � x + σz > σ2mr 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' B4 := � z : � 1 − 1 σ2(m−1)r � x + σz < −σ2mr 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 74 and C1 := � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) : 0 ≤ x − σ2 2 sgn(x) + σz ≤ σ2mr 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' C2 := � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) : −σ2mr 2 ≤ x − σ2 2 sgn(x) + σz ≤ 0 � C3 := � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) : x − σ2 2 sgn(x) + σz > σ2mr 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' C4 := � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) : x − σ2 2 sgn(x) + σz < −σ2mr 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' so that R1 = (A1 ∪ A2) ∩ (B1 ∪ B2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' R2 = (A3 ∪ A4) ∩ (C1 ∪ C2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' R3 = (A1 ∪ A2) ∩ (B3 ∪ B4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' R4 = (A3 ∪ A4) ∩ (C3 ∪ C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Take X a Laplace random variable and Z a standard normal random variable independent of X, then if σ2 = ℓ2d−2/3, we have lim d→+∞ dE [φd(X, Z)] = − ℓ3 3 √ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Taking expectations of φi d1Ri for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 4 and exploiting the symmetry of the laws of X 75 and Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' we can write E � φ1 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)1R1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � = 2E �� 1 σ2(m−1)rX − σZ � 1A1(X)1B1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � + 2E �� 2X − 1 σ2(m−1)rX + σZ � 1A1(X)1B2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � + 2E �� Z2 2 − 1 2σ2 �� 2 σ2(m−1)r − 1 σ4(m−1)r2 � X − � 1 − 1 σ2(m−1)r � σZ �2� ×1A1(X)1B1∪B2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' E � φ2 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)1R2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � = 2E ��σ2 2 − σZ � 1A3(X)1C1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � + 2E �� 2X − σ2 2 + σZ � 1A3(X)1C2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � + 2E �� Z2 2 − 1 2σ2 � 1 σ2(m−1)rX + � 1 − 1 σ2(m−1)r � �σ2 2 − σZ ��2� ×1A3(X)1C1∪C2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' E � φ3 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)1R3(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � = 2E �� 1 σ2(m−1)rX − σZ � 1A1(X)1B3(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � + 2E �� 2X − 1 σ2(m−1)rX + σZ � 1A1(X)1B4(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � + 2E �� Z2 2 − 1 2σ2 � 1 σ2(m−1)rX − σZ + σ2 2 �2� 1A1(X)1B3(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � + 2E �� Z2 2 − 1 2σ2 � 1 σ2(m−1)rX − σZ − σ2 2 �2� 1A1(X)1B4(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' E � φ4 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)1R4(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � = 2E �� 2X − σ2 2 + σZ � 1A3(X)1C4(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using the integrals in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 and Lebesgue’s dominated convergence theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' we find that 76 for α = β = 1/3 and r ≥ 0 lim d→+∞ dE � φ1 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � = 0 lim d→+∞ dE � φ2 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � = −2 ℓ3r 4 √ 2π � 0 −∞ e−z2/2zdz = ℓ3r 2 √ 2π lim d→+∞ dE � φ3 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � = 3ℓ3r 8 √ 2π � 0 −∞ e−z2/2zdz − ℓ3r 8 √ 2π � +∞ 0 e−z2/2zdz = − ℓ3r 2 √ 2π lim d→+∞ dE � φ4 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � = ℓ3 6 √ 2π � 0 −∞ e−z2/2z3dz = − ℓ3 3 √ 2π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' which gives lim d→+∞ dE [φd(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)] = lim d→+∞ d � E � φ1 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � + E � φ2 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � + E � φ3 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � + E � φ4 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) �� = − ℓ3 3 √ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For α = 1/3, β = m/3 for m > 1 and r ≥ 0 we have lim d→+∞ dE � φ1 d(X, Z) � = 0 lim d→+∞ dE � φ2 d(X, Z) � = 0 lim d→+∞ dE � φ3 d(X, Z) � = 0 lim d→+∞ dE � φ4 d(X, Z) � = ℓ3 6 √ 2π � 0 −∞ e−z2/2z3dz = − ℓ3 3 √ 2π, which gives lim d→+∞ dE [φd(X, Z)] = lim d→+∞ d � E � φ1 d(X, Z) � + E � φ2 d(X, Z) � + E � φ3 d(X, Z) � + E � φ4 d(X, Z) �� = − ℓ3 3 √ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proposition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Take X a Laplace random variable and Z a standard normal random variable independent of X, then if σ2 = ℓ2d−2/3 lim d→+∞ d Var (φd(X, Z)) = 2ℓ3 3 √ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' As a consequence of the previous Proposition we have lim d→+∞ dE [φd(X, Z)]2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 77 Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' because Rj ∩ Ri = ∅ for all j ̸= i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' we have that E � φ(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)2� = E � φ1 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)2R1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � + E � φ2 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)2R2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � + E � φ3 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)2R3(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � + E � φ4 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)2R4(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' exploiting again the symmetry of the laws of X and Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' we have E � φ1 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)2R1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � = 2E � � � 1 σ2(m−1)rX − σZ + Z2 2 − 1 2σ2 �� 2 σ2(m−1)r − 1 σ4(m−1)r2 � X − � 1 − 1 σ2(m−1)r � σZ �2�2 ×1A1(X)1B1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' + 2E � � � 2X − 1 σ2(m−1)rX − σZ + Z2 2 − 1 2σ2 �� 2 σ2(m−1)r − 1 σ4(m−1)r2 � X − � 1 − 1 σ2(m−1)r � σZ �2�2 ×1A1(X)1B2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' E � φ2 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)21R2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � = 2E � � � σ2 2 − σZ + Z2 2 − 1 2σ2 � 1 σ2(m−1)rX + � 1 − 1 σ2(m−1)r � �σ2 2 − σZ ��2�2 1A3(X)1C1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � � + 2E � � � 2X − σ2 2 + σZ + Z2 2 − 1 2σ2 � 1 σ2(m−1)rX + � 1 − 1 σ2(m−1)r � �σ2 2 − σZ ��2�2 ×1A1(X)1C2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' E � φ3 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)21R3(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � = 2E � � � 1 σ2(m−1)rX − σZ + Z2 2 − 1 2σ2 � 1 σ2(m−1)rX − σZ + σ2 2 �2�2 1A1(X)1B3(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � � + 2E � � � 2X − 1 σ2(m−1)rX + σZ + Z2 2 − 1 2σ2 � 1 σ2(m−1)rX − σZ − σ2 2 �2�2 1A1(X)1B4(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' E � φ4 d(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)21R4(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � = 2E �� 2X − σ2 2 + σZ �2 1A3(X)1C4(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proceeding as for Proposition 16, using the integrals in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 and Lebesgue’s dominated convergence theorem we can then show that for α = 1/3, β = m/3 for m ≥ 1 and r ≥ 0 lim d→+∞ d Var (φd(X, Z)) = 2ℓ3 3 √ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 78 Proposition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Take X a Laplace random variable and Z a standard normal random variable independent of X, then if σ2 = ℓ2d−2/3 we have lim d→+∞ dE � φd(X, Z)3� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Following the same structure of the previous propositions we have that E � φ(X, Z)3� = E � φ1 d(X, Z)3R1(X, Z) � + E � φ3 d(X, Z)2R2(X, Z) � + E � φ3 d(X, Z)3R3(X, Z) � + E � φ4 d(X, Z)3R4(X, Z) � , exploiting again the symmetry of the laws of X and Z, using the integrals in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4, the dominated convergence theorem we can then show that lim d→+∞ dE � φd(X, Z)3� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 Bound on Second Moment of Acceptance Ratio for the Laplace Dis- tribution Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let Z be a standard normal random variable and σ = ℓ/dα for α = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Then, there exists a constant C > 0 such that for all a ∈ R and d ∈ N: E � φd(a, Z)2� ≤ C d2α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We consider the case a ≥ 0 and r ≥ σ2(m−1) only, all the other cases follow from identical arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' As in the derivation of the moments of φd in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1, we distinguish four regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We recall that σ = ℓ/dα for α = 1/3 and thus σp+1 ≤ σp for all p ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Take r ≥ σ−2(m−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' for R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 79 we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' using H¨older’s inequality multiple times,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' E � φ1 d(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)21R1(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � = E � φ1 d(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)21B1∪B2(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='≤ Cσ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� σ2m−1r/2−(1−1/σ2(m−1)r)a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−(1−1/σ2(m−1)r)a/σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e−z2/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ2m−1ra − z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2σ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2σ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ2(m−1)r − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ4(m−1)r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='a − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ2(m−1)r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2�2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� dz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ Cσ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� −(1−1/σ2(m−1)r)a/σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−σ2m−1r/2−(1−1/σ2(m−1)r)a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e−z2/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='��2a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ2m−1ra + z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2σ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2σ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ2(m−1)r − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ4(m−1)r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='a − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ2(m−1)r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2�2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� dz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='≤ Cσ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� +∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e−z2/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ2m−1ra ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ 2z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='dz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='+ Cσ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� σ2m−1r/2−(1−1/σ2(m−1)r)a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−σ2m−1r/2−(1−1/σ2(m−1)r)a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e−z2/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2σ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2σ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ2(m−1)r − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ4(m−1)r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='a − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ2(m−1)r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�2�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='≤ Cσ2 + Cσ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� σ2m−1r/2−(1−1/σ2(m−1)r)a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='−σ2m−1r/2−(1−1/σ2(m−1)r)a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='e−z2/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='z4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4σ2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4σ6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ2(m−1)r − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ4(m−1)r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='a4 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ2(m−1)r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='σ4z4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='dz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='≤ Cσ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' where we used the fact that the moments of Z are bounded and a ≤ σ2mr/2 for the first term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' and the fact that z ≤ σ2m−1r/2 for the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proceeding as above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' for R3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' a > 0 and 80 r ≥ σ−2(m−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' we have E � φ3 d(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)21R3(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � = E � φ3 d(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z)21B3∪B4(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Z) � ≤ Cσ2 � +∞ σ2m−1r/2−(1−1/σ2(m−1)r)a/σ e−z2/2 √ 2π �� 1 σ2m−1ra − z �2 + � z2 2σ − 1 2σ3 � 1 σ2(m−1)ra − σz + σ2 2 �2�2� � dz + Cσ2 � −σ2m−1r/2−(1−1/σ2(m−1)r)a/σ −∞ e−z2/2 √ 2π ��2a σ − 1 σ2m−1ra + z �2 + � z2 2σ − 1 2σ3 � 1 σ2(m−1)ra − σz + σ2 2 �2�2� � dz ≤ Cσ2 + Cσ2 � +∞ −∞ e−z2/2 √ 2π � 1 2σ3 � a2 σ4(m−1)r2 + σ4 4 − σ3z + a σ2m−4r − 2az σ2m−3r ��2 dz ≤ Cσ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' where we used again the boundedness of the moments of Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' the fact that a ≤ σ2mr/2 and that σp+1 ≤ σp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For R2 and a > 0, we have E � φ2 d(a, Z)21R2(a, Z) � = E � φ2 d(a, Z)21C1∪C2(a, Z) � ≤ Cσ2 � σ/2+σ2m−1r/2−a/σ σ/2−a/σ e−z2/2 √ 2π ��σ2 2 − σz �2 + � z2 2 − 1 2σ2 � 1 σ2(m−1)ra + � 1 − 1 σ2(m−1)r � �σ2 2 − σz ��2�2� � dz + Cσ2 � σ/2−a/σ σ/2−σ2m−1r/2−a/σ e−z2/2 √ 2π �� 2a − σ2 2 + σz �2 + � z2 2 − 1 2σ2 � 1 σ2(m−1)ra + � 1 − 1 σ2(m−1)r � �σ2 2 − σz ��2�2� � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The first integral is bounded using the moments of Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' while for the third one let us denote 81 χ(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) := a − σ2/2 + σz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' then � σ/2+σ2m−1r/2−a/σ σ/2−σ2m−1r/2−a/σ e−z2/2 √ 2π � z2 2σ − 1 2σ3 � 1 σ2(m−1)ra + � 1 − 1 σ2(m−1)r � �σ2 2 − σz ��2�2 dz = � σ/2+σ2m−1r/2−a/σ σ/2−σ2m−1r/2−a/σ e−z2/2 √ 2π � z2 2σ − 1 2σ3 �χ(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) σ2(m−1)r + σ2 2 − σz �2�2 dz ≤ C � σ/2+σ2m−1r/2−a/σ σ/2−σ2m−1r/2−a/σ e−z2/2 √ 2π � z2 2σ − 1 2σ3 �σ2 2 − σz �2�2 dz + C � σ/2+σ2m−1r/2−a/σ σ/2−σ2m−1r/2−a/σ e−z2/2 √ 2π �χ(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z)2 2σ4m−1r2 �2 + �χ(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z) rσ2m−1 �σ2 2 − σz ��2 dz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' recalling that in R2 we have |χ(a, σ, z)| ≤ σ2mr/2, we obtain that this term is also bounded by Cσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For R4 and a > 0, we have E � φ4 d(a, Z)21R4(a, Z) � = E � φ4 d(a, Z)21C4(a, Z) � = � σ/2−σ2m−1r/2−a/σ −∞ e−z2/2 √ 2π � 2a − σ2 2 + σz �2 dz = σ2 � σ/2−σ2m−1r/2−a/σ −∞ e−z2/2 √ 2π �2a σ − σ 2 + z �2 dz Collecting all the terms together, we obtain E � φd(a, Z)2� = 4 � i=1 E � φi d(a, Z)21Ti(a, Z) � ≤ Cσ2 + Cσ2 � σ/2−a/σ −∞ e−z2/2 √ 2π �2a σ − σ 2 + z �2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Recall that σ = ℓd−1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' To bound the last integral we use H¨older’s inequality � σ/2−a/σ −∞ e−z2/2 √ 2π �2a σ − σ 2 + z �2 dz ≤ C � σ/2−a/σ −∞ e−z2/2 √ 2π ��σ 2 + z �2 + 4 �σ 2 − a σ �2� dz ≤ C � σ/2−a/σ −∞ e−z2/2 √ 2π ��ℓ2 4 + z2 � + 4 �σ 2 − a σ �2� dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The first term is bounded since the moments of Z are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For the second term we use an estimate of the Gaussian cumulative distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let κ(ℓ, d, a) := ℓd−1/3/2 − ad1/3/ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' When z < κ(ℓ, d, a) < 0, we have 1 < z/κ(ℓ, d, a) and therefore (2π)−1/2κ(ℓ, d, a)2 � κ(ℓ,d,a) −∞ e−z2/2dz ≤ (2π)−1/2κ(ℓ, d, a) � κ(ℓ,d,a) −∞ ze−z2/2dz, = (2π)−1/2κ(ℓ, d, a) exp(−κ(ℓ, d, a)2/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 82 However y �→ ye−y2/2 is bounded over R, therefore (a, d) �−→ (2π)−1/2κ(ℓ, d, a) exp(−κ(ℓ, d, a)2/2) is bounded over R∗ + × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' If κ(ℓ, d, a) ≥ 0, then we still have κ(ℓ, d, a) < ℓ and thus have the inequality (2π)−1/2κ(ℓ, d, a)2 � κ(ℓ,d,a) −∞ e−z2/2dz ≤ (2π)−1/2ℓ2 � +∞ −∞ e−z2/2dz = ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The result then follows since σ = ℓd−1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3 Additional Integrals for the Laplace Distribution We collect here two auxiliary Lemmata which are used in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Take X a Laplace random variable and Z a standard normal random variable indepen- dent of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Let ˜X := X − 1 σ2(m−1)rX1|X|≤σ2mr/2 − σ2 2 sgn(X)1|X|>σ2mr/2 + σZ, then, for σ = ℓd−α with α = 1/3, E � 1{sgn(X)̸=sgn( ˜ X)} � → 0 if d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using the same strategy of Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 and the symmetry of the laws of X, Z, we find that E � 1{sgn(X)̸=sgn( ˜ X)} � = 2E [1A1(X)1B2(X, Z)] + 2E [1A3(X)1C2(X, Z)] + 2E [1A1(X)1B4(X, Z)] + 2E [1A3(X)1C4(X, Z)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using the same strategy used to obtain the moments of φd in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1, we find that E [1A1(X)1B2(X, Z)] = o(1), in addition E [1A3(X)1C2(X, Z)] = 1 2 √ 2π � σ/2−σ2m−1r/2 −∞ e−z2/2 � σ2/2−σz+σ2mr/2 σ2/2−σz e−xdx dz + o(1) = 1 2 √ 2π � σ/2−σ2m−1r/2 −∞ e−z2/2 �σ2r 2 δm1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + o(1), where δm1 is a Dirac’s delta, and E [1A1(X)1B4(X, Z)] + E [1A3(X)1C4(X, Z)] = 1 2 √ 2π � σ/2−σ2m−1r −∞ e−z2/2 � σ2/2−σz−σ2mr/2 0 e−xdx dz + o(1) = 1 2 √ 2π � σ/2−σ2m−1r −∞ e−z2/2 [−σz + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='] dz + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 83 Since σ = ℓd−1/3 and the remainder terms of the Taylor expansions are bounded, Lebesque’s dominated convergence theorem gives E [1A3(X)1C2(X, Z)] → 0, E [1A1(X)1B4(X, Z)] + E [1A3(X)1C4(X, Z)] → 0 as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Take X a Laplace random variable and Z a standard normal random variable indepen- dent of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Then, dαE [|Z| |φd (X, Z)|] → 0 for α = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using Cauchy-Schwarz’s inequality we have that E [|Z| |φd (X, Z)|] ≤ E � Z2�1/2 E � φd(X, Z)2�1/2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' the first expectation is equal to one, and the second one converges to zero at rate d1/2 by Proposi- tion 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The result follows straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4 Integrals for Moment Computations We distinguish the case m = 1/2 and m ≥ 1 since the integration bounds significantly differ in these two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For values between 1/2 and 1 the integrals are not finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The expectations below are obtained by integrating w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' x and using a Taylor expansion about σ = 0 to obtain the leading order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Using the Lagrange form of the remainder for the Taylor expansions, we find that the remainder terms are all of the form σ1/α+1f(γ(σ, z))/(1/α + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' where γ(σ, z) is a point between the limits of integration w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' x and f : x �→ p(x)e−x, where p is a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Therefore, using the boundedness of the remainder and Lebesgue’s dominated convergence theorem, the integrals w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' z of the remainder terms all converge to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='1 First Moment For simplicity, we only consider the case for r ≥ σ−2(m−1), the other case follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Region R1 Let us consider φ1 d first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' We have A1 ∩ B1 = � � � � � � � 0 ≤ x ≤ σ2mr 2 if 0 ≤ z ≤ σ 2 0 ≤ x ≤ � σ2mr 2 − σz � � 1 − 1 σ2(m−1)r �−1 if σ 2 ≤ z ≤ σ2m−1r 2 −σz � 1 − 1 σ2(m−1)r �−1 ≤ x ≤ σ2mr 2 if σ 2 − σ2m−1r 2 ≤ z ≤ 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' A1 ∩ B2 = � � � � � � � 0 ≤ x ≤ σ2mr 2 if − σ2m−1r 2 ≤ z ≤ σ 2 − σ2m−1r 2 0 ≤ x ≤ −σz � 1 − 1 σ2(m−1)r �−1 if σ 2 − σ2m−1r 2 ≤ z ≤ 0 − � σ2mr 2 + σz � � 1 − 1 σ2(m−1)r �−1 ≤ x ≤ σ2mr 2 if σ 2 − σ2m−1r ≤ z ≤ − σ2m−1r 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 84 and A1 ∩ (B1 ∪ B2) = � � � � � � � � � � � � � 0 ≤ x ≤ σ2mr 2 if − σ2m−1r 2 ≤ z ≤ σ 2 0 ≤ x ≤ � σ2mr 2 − σz � � 1 − 1 σ2(m−1)r �−1 if σ 2 ≤ z ≤ σ2m−1r 2 − � σ2mr 2 + σz � � 1 − 1 σ2(m−1)r �−1 ≤ x ≤ σ2mr 2 if σ 2 − σ2m−1r ≤ z ≤ − σ2m−1r 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The corresponding expectations are E �� X σ2(m−1)r − σZ � 1A1(X)1B1(X, Z) � = 1 2 √ 2π � σ2m−1r/2 σ/2 e−z2/2 � z4−2mσ4−2mξ(r) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � 0 σ/2−rσ2m−1/2 e−z2/2 � z4−2mσ4−2mξ(r) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + o(1), E �� 2X − X σ2(m−1)r + σZ � 1A1(X)1B2(X, Z) � = 1 2 √ 2π � 0 σ/2−σ2m−1r/2 e−z2/2 � z4−2mσ4−2mξ(r) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � −σ2m−1r/2 σ/2−σ2m−1r e−z2/2 � z4−2mσ4−2mξ(r) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + o(1), where ξ : [0, +∞) → R is a function of r only which might change from one line to the other, and E �� Z2 2 − 1 2σ2 �� 2 σ2(m−1)r − 1 σ4(m−1)r2 � X − � 1 − 1 σ2(m−1)r � σZ �2� 1A1(X)1B1∪B2(X, Z) � = 1 2 √ 2π � σ/2 −σ2m−1r/2 e−z2/2 [+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ] dz + 1 2 √ 2π � σ2m−1r/2 σ/2 e−z2/2 [+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' ] dz + 1 2 √ 2π � −σ2m−1r/2 σ/2−σ2m−1r e−z2/2 � z2σ2ξ(r) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz, where ξ : [0, +∞) → R is a function of r only which might change from one line to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 85 Region R2 For φ2 d, we have A3 ∩ C1 = � σ2/2 − σz ≤ x ≤ −σz + σ2/2 + σ2mr/2 if z < σ/2 − σ2m−1r/2 σ2mr/2 < x ≤ −σz + σ2/2 + σ2mr/2 if σ/2 − σ2m−1r/2 ≤ z ≤ σ/2 , A3 ∩ C2 = � σ2/2 − σz − σ2mr/2 ≤ x ≤ σ2/2 − σz if z < σ/2 − σ2m−1r σ2mr/2 < x ≤ σ2/2 − σz if σ/2 − σ2m−1r < z < σ/2 − σ2m−1r/2 and A3 ∩ (C1 ∪ C2) = � � � � � σ2mr/2 ≤ x ≤ σ2mr/2 + σ2/2 − σz if σ/2 − σ2m−1r ≤ z ≤ σ/2 −σ2mr/2 + σ2/2 − σz < x ≤ σ2mr/2 + σ2/2 − σz if z < σ/2 − σ2m−1r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' The corresponding expectations are E ��σ2 2 − σZ � 1A3(X)1C1(X, Z) � = 1 2 √ 2π � σ/2−σ2m−1r/2 −∞ e−z2/2 � −rz 2 σ2m+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � σ/2 σ/2−σ2m−1r/2 e−z2/2 � z2σ2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz, E �� 2X − σ2 2 + σZ � 1A3(X)1C2(X, Z) � = 1 2 √ 2π � σ/2−σ2m−1r −∞ e−z2/2 � −rz 2 σ2m+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � σ/2−σ2m−1r/2 σ/2−σ2m−1r e−z2/2 � −rz 2 σ2m+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz, and E �� Z2 2 − 1 2σ2 � 1 σ2(m−1)rX + � 1 − 1 σ2(m−1)r � �σ2 2 − σZ ��2� 1A3(X)1C1∪C2(X, Z) � = 1 2 √ 2π � σ/2−σ2m−1r −∞ e−z2/2 �rz 2 σ2m+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � σ/2 σ/2−σ2m−1r e−z2/2 � −z3 2rσ3−2m + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Region R3 For φ3 d, we have, in the case r ≥ σ−2(m−1), A1 ∩ B3 = � 0 ≤ x ≤ σ2mr 2 if z > σ2m−1r 2 � σ2mr 2 − σz � � 1 − 1 σ2(m−1)r �−1 < x ≤ σ2mr 2 if σ 2 ≤ z ≤ σ2m−1r 2 , A1 ∩ B4 = � 0 ≤ x ≤ σ2mr 2 if z < σ 2 − σ2m−1r 0 ≤ x < � σ2mr 2 − σz � � 1 − 1 σ2(m−1)r �−1 if σ 2 − σ2m−1r ≤ z ≤ − σ2m−1r 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 86 The corresponding expectations are E �� 1 σ2(m−1)rX − σZ + Z2 2 − 1 2σ2 � 1 σ2(m−1)rX − σZ + σ2 2 �2� 1A1(X)1B3(X, Z) � = 1 2 √ 2π � +∞ σ2m−1r/2 e−z2/2 � −rz 8 σ2m+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � σ2m−1r/2 σ/2 e−z2/2 � z3−2mσ3−2mξ(r) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz, E �� 2X − 1 σ2(m−1)rX + σZ + Z2 2 − 1 2σ2 � 1 σ2(m−1)rX − σZ − σ2 2 �2� 1A1(X)1B4(X, Z) � = 1 2 √ 2π � σ/2−σ2m−1r/2 −∞ e−z2/2 �3 8σ2m+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � −σ2m−1r/2 σ/2−σ2m−1r/2 e−z2/2 � z3σ3−2mξ(r) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz, where ξ : [0, +∞) → R is a function of r only which might change from one line to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Region R4 Finally, for φ4 d we have A3 ∩ C4 = � z < σ 2 − σ2m−1r, σ2mr 2 < x ≤ σ2 2 − σz − σ2mr 2 � , and E �� 2X − σ2 2 + σZ � 1A3(X)1C4(X, Z) � = 1 2 √ 2π � σ/2−σ2m−1r −∞ e−z2/2 �z3 6 σ3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='2 Second Moment For simplicity, we only consider the case for r ≥ σ−2(m−1), the other case follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Region R1 For φ1 d we have E � φ1 d(X, Z)21A1(X)1B1(X, Z) � = 1 2 √ 2π � σ2m−1r/2 σ/2 e−z2/2 � z3σ3ξ(r) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � 0 σ/2−rσ2m−1/2 e−z2/2 � z3σ3ξ(r) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + o(1), E � φ1 d(X, Z)21A1(X)1B2(X, Z) � = 1 2 √ 2π � 0 σ/2−σ2m−1r/2 e−z2/2 � z3σ3ξ(r) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � −σ2m−1r/2 σ/2−σ2m−1r e−z2/2 � z3σ3ξ(r) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + o(1), 87 where ξ : [0, +∞) → R is a function of r only which might change from one line to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Region R2 For φ2 d we have E � φ2 d(X, Z)21A3(X)1C1(X, Z) � = 1 2 √ 2π � σ/2−σ2m−1r/2 −∞ e−z2/2 �rz2 2 σ2m+2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � σ/2 σ/2−σ2m−1r/2 e−z2/2 � −z3σ3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz, E � φ2 d(X, Z)21A3(X)1C2(X, Z) � = 1 2 √ 2π � σ/2−σ2m−1r −∞ e−z2/2 � z2σ2m+2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � σ/2−σ2m−1r/2 σ/2−σ2m−1r e−z2/2 � −rz2 2 σ2m+2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Region R3 For φ3 d we have E � � � 1 σ2(m−1)rX − σZ + Z2 2 − 1 2σ2 � 1 σ2(m−1)rX − σZ + σ2 2 �2�2 1A1(X)1B3(X, Z) � � = 1 2 √ 2π � +∞ σ2m−1r/2 e−z2/2 �rz2 24 σ2m+2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � σ2m−1r/2 σ/2 e−z2/2 �rz2 24 σ2m+2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz, E � � � 2X − 1 σ2(m−1)rX + σZ + Z2 2 − 1 2σ2 � 1 σ2(m−1)rX − σZ − σ2 2 �2�2 1A1(X)1B4(X, Z) � � = 1 2 √ 2π � σ/2−σ2m−1r/2 −∞ e−z2/2 �7rz2 24 σ2m+2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � −σ2m−1r/2 σ/2−σ2m−1r/2 e−z2/2 �7rz2 24 σ2m+2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz, Region R4 For φ4 d we have E �� 2X − σ2 2 + σZ �2 1A3(X)1C4(X, Z) � = 1 2 √ 2π � σ/2−σ2m−1r −∞ e−z2/2 � −z3 3 σ3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='3 Third Moment Having established that the only possible scaling is given by α = 1/3, β = m/3 with m ≥ 1, we now proceed to bound the third moment of φd in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For simplicity, we only consider the case for r ≥ 1, the other case follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 88 Since m ≥ 1, we find that E � φ1 d(X, Z)31R1(X, Z) � = o(1) as d → ∞ since the limits of integra- tion all converge to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Then, using H¨older’s inequality for φ2 d, we have E � φ2 d(X, Z)3� ≤ CE ��σ2 2 − σZ �3 1A3(X)1C1(X, Z) � + CE �� 2X − σ2 2 + σZ �3 1A3(X)1C2(X, Z) � + CE � � � Z2 2 − 1 2σ2 � 1 σ2(m−1)rX + � 1 − 1 σ2(m−1)r � �σ2 2 − σZ ��2�3 ×1A3(X)1C1∪C2(X, Z)] = C 2 √ 2π � σ/2−σ2m−1r/2 −∞ e−z2/2 � −rz3 2 σ5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + C 2 √ 2π � σ/2−σ2m−1r −∞ e−z2/2 � −rz3 2 σ5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + C 2 √ 2π � σ/2−σ2m−1r −∞ e−z2/2 � z3σ5ξ(r) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + o(1), where ξ : [0, +∞) → R is a function of r only which might change from one line to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' For 89 φ3 d, we have, using again H¨older’s inequality, E � � � 1 σ2(m−1)rX − σZ + Z2 2 − 1 2σ2 � 1 σ2(m−1)rX − σZ + σ2 2 �2�3 1A1(X)1B3(X, Z) � � ≤ CE �� 1 σ2(m−1)rX − σZ �3 1A1(X)1B3(X, Z) � + CE � � � Z2 2 − 1 2σ2 � 1 σ2(m−1)rX − σZ + σ2 2 �2�3 1A1(X)1B3(X, Z) � � = 1 2 √ 2π � +∞ σ2m−1r/2 e−z2/2 � −z3r 2 σ5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � +∞ σ2m−1r/2 e−z2/2 � z3σ5ξ(r) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + o(1) E � � � 2X − 1 σ2(m−1)rX + σZ + Z2 2 − 1 2σ2 � 1 σ2(m−1)rX − σZ − σ2 2 �2�3 1A1(X)1B4(X, Z) � � ≤ CE �� 2X − 1 σ2(m−1)rX + σZ �3 1A1(X)1B4(X, Z) � + CE � � � Z2 2 − 1 2σ2 � 1 σ2(m−1)rX − σZ − σ2 2 �2�3 1A1(X)1B4(X, Z) � � = 1 2 √ 2π � +∞ σ2m−1r/2 e−z2/2 �z3r 2 σ5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + 1 2 √ 2π � +∞ σ2m−1r/2 e−z2/2 � z3σ5ξ(r) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz + o(1), where ξ : [0, +∞) → R is a function of r only which might change from one line to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' Finally, for φ4 d we have E �� 2X − σ2 2 + σZ �3 1A3(X)1C4(X, Z) � = 1 2 √ 2π � σ/2−σ2m−1r −∞ e−z2/2 �z3 10σ5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} +page_content=' 90' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfigFl/content/2301.02446v1.pdf'} diff --git a/LNE2T4oBgHgl3EQfpwjv/content/tmp_files/2301.04033v1.pdf.txt b/LNE2T4oBgHgl3EQfpwjv/content/tmp_files/2301.04033v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bb3ad64f78a813d2663882f81e6175c2134719cb --- /dev/null +++ b/LNE2T4oBgHgl3EQfpwjv/content/tmp_files/2301.04033v1.pdf.txt @@ -0,0 +1,1202 @@ +1 + +Numerical investigation of progressive damage and associated seismicity on a +laboratory fault +Qi Zhao1,2*, Nicola Tisato3, Aly Abdelaziz2, Johnson Ha2, and Giovanni Grasselli2 +1Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, +Hung Hom, Hong Kong SAR, China. +2Department of Civil and Mineral Engineering, The University of Toronto, 35 St. George Street, +Toronto, Ontario M5S 1A4, Canada. +3Department of Geological Sciences, Jackson School of Geosciences, The University of Texas at +Austin, 2305 Speedway Stop C1160, Austin, TX 78712-1692, USA. +*Corresponding author: Qi Zhao (qi.qz.zhao@polyu.edu.hk) + +Abstract +Understanding rock shear failure behavior is crucial to gain insights into slip-related geohazards +such as rock avalanches, landslides, and earthquakes. However, descriptions of the progressive +damage on the shear surface are still incomplete or ambiguous. In this study, we use the hybrid +finite-discrete element method (FDEM) to simulate a shear experiment and obtain a detailed +comprehension of shear induced progressive damage and the associated seismic activity. We built +a laboratory fault model from high resolution surface scans and micro-CT imaging. Our results +show that under quasi-static shear loading, the fault surface experiences local dynamic seismic +activities. We found that the seismic activity is related to the stress concentration on interlocking +asperities. This interlocking behavior (i) causes stress concentration at the region of contact that +could reach the compressive strength, and (ii) produces tensile stress up to the tensile strength in +the region adjacent to the contact area. Thus, different failure mechanisms and damage patterns +including crushing and sub-vertical fracturing are observed on the rough surface. Asperity failure +creates rapid local slips resulting in significant stress perturbations that alter the overall stress +condition and may trigger the slip of adjacent critically stressed asperities. We found that the + +2 + +spatial distribution of the damaged asperities and the seismic activity is highly heterogeneous; +regions with intense asperity interactions formed gouge material, while others exhibit minimal to +no damage. These results emphasize the important role of surface roughness in controlling the +overall shear behavior and the local dynamic seismic activities on faults. +Keywords +Shear behavior; surface roughness; asperity; shear induced damage; seismicity + + +3 + +1 Introduction +Understanding shear behavior along rock discontinuities at various scales, such as joints +and faults, is essential to rock engineering projects and geohazard mitigation. Rock discontinuities +are planes of weakness and are responsible for many geohazards, for example, rock avalanches, +landslides, and earthquakes. Numerous laboratory shear experiments have been conducted on a +large variety of rock types under different conditions (e.g., Bandis et al., 1983; Beeler, 1996; +Marone, 1998; Di Toro et al., 2004; Grasselli, 2001; Reches and Lockner, 2010; Tisato et al., 2012; +Kim and Jeon, 2019; Zhao et al., 2020; Morad et al., 2022). Among these studies, many suggested +the importance of surface roughness and contact condition in controlling the shear behavior. +However, the progressive damaging process on faults is still not well understood because fault +surfaces cannot be observed directly during shear, with the exception of a few studies utilizing +transparent halite samples (e.g., Renard et al., 2012) and in situ and operando testes conducted +under X-ray micro-computed tomography (micro-CT) (e.g., Zhao et al, 2018; Zhao et al., 2020). +Shear processes control coseismic damage and friction on fault, but the constitutive friction +theories are not yet fully understood. +To observe and gain insights into damage processes on rough rock surfaces undergoing +shear deformation, Tatone and Grasselli (2013) used micro-CT to image the joint surfaces after +the shear test; and Crandall et al. (2017) used micro-CT to obtain geometrical information from +fractured shale core that is incrementally sheared. Recently, direct and detailed observations of the +evolution of laboratory fault were achieved by using an in situ rotary shear experimental apparatus +under X-ray micro-CT (Zhao et al., 2017). Such experimental results help draw the connections +between microscopic damage and macroscopic shear behavior, the formation and accumulation of +gouge material, and shear-induced secondary fractures (Zhao, 2017; Zhao et al., 2018; Zhao et al., + +4 + +2020). However, due to technological limitations, time-continuous observations of the shear +surface evolution and the in situ stress condition remain shortfalls. +Numerical simulation methods have been extensively used to study the shear behavior of +rock discontinuities. To simulate the interaction and breakage of asperities and the frictional +sliding behavior, numerical methods that can capture solid fracturing and interaction are typically +used. For example, the particle-based lattice solid model (Mora and Place 1993) uses a numerical +concept similar to the discrete element method (DEM) to simulate frictional behavior and +fracturing in solids. Mora and Place (1998) and Place and Mora (2000) used their model to study +the role fault gouge on the frictional behavior of faults, offering a possible explanation for the heat +flow paradox (Henyey and Wasserburg, 1971; Lachenbruch and Sass, 1992). Bonded particle- +based methods, such as the particle flow code (PFC) (Cundall and Strack, 1979), are commonly +used for simulating rock shear behavior. Park and Song (2009) used PFC3D to simulate direct +shear tests and demonstrated that the method can simulate typical rock joint shear behavior, and +they found that the peak shear strength and peak dilation angle was strongly influenced by the +friction coefficient, roughness, and bond strength, while the residual shear strength and residual +friction angle was influenced by the particle size, friction coefficient, and bond strength. Asadi et +al. (2012) used a similar approach in two dimensions (PFC2D) to simulate direct shear experiments +on synthetic joint profiles of varying roughness and boundary conditions to assess asperity +degradation and intact material damage. They showed that as the joint is sheared, highly localized +asperity interaction on the joint surface and the geometry of the asperities has a significant +influence on how joints fail. Types of failure include asperity sliding, cut-off, separation, and +crushing, typically associated with tensile failure into the intact material in conditions with steep +asperities and/or high normal stress. However, Bahaaddini et al. (2013) identified a significant + +5 + +shortcoming of the particle-based methods due to the unrealistic shear and dilation behavior of +joints as a result of particle interlocking due to the inherent micro-scale roughness of the joint. To +overcome this limitation, they implemented the smooth-joint model (Pierce et al., 2007), where +the blocks associated with either side of the joint are generated separately to appropriately define +and apply the smooth joint model to the interface. Lambert and Coll (2014) created a synthetic +rock joint by importing the real morphology of the joint surface into a bonded particle assembly +and studied the shear behavior using the smooth-joint model. Their results reproduced the +progressive degradation of the asperities upon shearing. +The hybrid finite-discrete element method (FDEM) is increasingly used to investigate rock +shear behavior. FDEM is a micromechanical numerical method first introduced by Munjiza et al., +(1995) combining the finite element method and discrete element method. In doing so, the +numerical method can model the transition of a continuous material to a discontinuous material as +it deforms, yields, and breaks. Karami and Stead (2008) and Tatone (2014) used FDEM to model +direct shear tests and relate progressive asperity degradation mechanisms with the measured shear +stress and dilation during shearing. In addition, Tatone (2014) verified the numerical modelling +observations by coupling their study with X-ray micro-CT imaging on post-mortem specimens. It +was found that tensile fractures develop in asperities at and near the peak shear stress, followed by +a reduction in shear resistance as asperities continue to fail in both tension and shear, and finally, +a residual shear resistance is reached once asperities are completely broken and gouge is formed. +In this study, we used the two-dimensional (2D) FDEM to simulate an experiment on the +gradual evolution of deformation of a laboratory fault, and we improve the understanding of shear +behavior of rough faults through combined interpretation of the simulation and experimental +results. First, we provide a brief review of the FDEM, emphasizing the modeling of damage and + +6 + +seismic activity. Second, we develop a clustering algorithm to improve the comprehension of the +simulated fractures and seismic events. Next, we build a model based on the laboratory experiment +and analyze the simulated results focusing on three aspects that are hardly accessible by +experiments: (1) the time-continuous variation of stress conditions on the shear surface, (2) the +progressive failure of the asperities and accumulation of gouge, and (3) the seismic activity related +to shear-induced damage. +The carefully built and calibrated numerical model is able to simulate the emergent rock +mechanical and frictional behaviors. We observe that shear-induced damage and seismic activities +are heterogeneously distributed along the fault surface due to the surface roughness. Seismic +events occur at the locations of asperity failure due to the interlocking-induced stress concentration. +Such events radiate seismic waves and significantly change the overall stress conditions. Some +areas on the fault were covered by gouge material and free from damage. These results agree with +the laboratory observations and further elaborate on the importance of surface roughness in +controlling shear behavior, which is critical to rock engineering practices and earthquake studies. +2 Material and methods +2.1 In situ shear test under X-ray micro-CT +The numerical simulation in this study is based on the experimental work using in situ +shear tests under micro-CT reported by Zhao et al. (2018) and Zhao et al. (2020), and a brief review +is provided here for completeness. The tested specimen was a cylindrical Flowstone (microfine +calcium sulfate cement mortar) 32 mm in length and 12 mm in diameter. The specimen was divided +into top and bottom parts by a three-point bending test that created two semi-samples divided by +a discontinuity (i.e., laboratory fault) with two matching rough surfaces. An unconfined rotary +shear test was conducted on the two semi-samples by shearing the fault under the initial normal + +7 + +stress of 2.5 MPa. The top semi-sample was forced to slip incrementally against the fixed bottom +semi-sample. Normal force and torque were recorded during rotation and used to calculate the +friction coefficient. After each incremental slip of 6°, a three-dimensional (3D) micro-CT scan was +conducted, which allows for imaging of the gradual morphological evolution of the specimen (Fig. +1a). This experimental work provided detailed information of the shear-induced secondary +fractures (Fig. 1b) and the progressive damage on the slipping surface (Fig. 1c) in the sample +volume; however, the observation of the shear surface damage evolution was only available at +discrete time points coincident with each shear step, while an actual time-continuous observation +of the shear surface evolution and the local stress condition on the rough surface was not available. + +Fig. 1 Summary of the laboratory set-up and results. (a) Schematic of the in situ shear test and a +zoom-in view of the shear surface (i.e., the zone of interest). (b) 3D visualization of the +development of shear induced fractures with increasing shear displacement. (c) 2D unwrapped +micro-CT image slice showing the progressive damage on the slipping surface with increasing +shear displacement, viewed at the radius (R = 5.6 mm) corresponding to the highest asperity. Red + +8 + +dashed boxes in (c) indicate (from left to right) shear-induced aperture opening, fracturing, and +progressive damage and gouge formation (modified from Zhao et al. (2018) and Zhao et al. (2020)). +2.2 The hybrid finite-discrete element method +The hybrid finite-discrete element method (FDEM) combines continuum mechanics +principles with discrete element principles to simulate interaction, deformation, and fracturing of +materials (Munjiza et al., 1995; Munjiza, 2004). FDEM has been used to investigate a wide range +of rock mechanics and geophysics problems including, but not limited to, tunneling and excavation, +micromechanics, rock joint shear behavior, hydraulic fracturing, thermal-mechanical/hydro- +thermal-mechanical coupling, and fault dynamics (e.g., Karami and Stead 2008; Mahabadi et al., +2012; Lisjak et al., 2014; Zhao et al., 2014; Yan et al., 2016; Huang et al., 2017; Lei et al., 2017; +Ma et al., 2017; Fukuda et al., 2019; Okubo et al., 2019; Knight et al., 2020). Simulating the entire +shear behavior and evolution of a rough surface is a challenging task that requires advanced +computational resources encompassing, for example, the 3D FDEM method. However, 3D models +explicitly capturing the surface roughness at sub-millimeter resolution and the entire shear process +is not practical due to the demanded computation power. On the other hand, 2D FDEM simulations +has the merit of reducing the computational demand, and it has been shown to provide insights +into the mechanical behavior of rock joints and faults (e.g., Karami and Stead, 2008; Tatone, 2014; +Okubo et al., 2019). +FDEM models synthesize the macroscopic behavior of materials from the interaction of +the micromechanical constituents. In a 2D FDEM model, the simulated material is first discretized +based on a finite element mesh consisting of nodes and triangular elements. Then, the finite +element mesh is enriched by inserting a four-node cohesive crack element (CCE) between each +adjacent triangular element pair. Motion for the discretized system is calculated by an explicit time + +9 + +integration scheme, and the nodal coordinates of the elements are updated at each simulation step +(Munjiza, 2004). FDEM models the progressive damage and failure of brittle material according +to the principles of non-linear elastic fracture mechanics (Dugdale, 1960; Barenblatt, 1962), and it +captures the fracturing behavior of solids by modeling the entire failure path, including elastic +deformation, yielding, and fracturing (Fig. 2). + +Fig. 2 Schematic diagram showing the FDEM approach of simulating fracturing. (a) Propagation +of a fracture and the creation of fracture process zone (FPZ). (b) Realization of the fracturing +process in FDEM involves the yielded cohesive crack elements and broken cohesive crack +elements (BCCE). +Depending on the local stress and deformation field, the CCE undergoes elastic +deformation, yielding, and breakage, simulating the damage development of the fracture process +zone (FPZ) (Fig. 3) (Labuz et al., 1985). During elastic loading, the relationships between bonding +stresses (normal bonding stress, σ and shear bonding stress, τ) and the corresponding crack +displacement (opening, o and slip, s) are as follows (Munjiza et al., 1999): + +Cracktip10 + +𝜎 = { +2𝑜 +𝑜𝑝 𝑓𝑡 (𝑜 < 0, compression) +[ +2𝑜 +𝑜𝑝 − ( +𝑜 +𝑜𝑝) +2 +] 𝑓𝑡 (0 < 𝑜 < 𝑜𝑝, tension) + + + + (1) +𝜏 = [ +2𝑠 +𝑠𝑝 − ( +𝑠 +𝑠𝑝) +2 +] 𝑓𝑠 (|𝑠| ≤ |𝑠𝑝|, shear) + + + + (2) +where ft and fs are the peak tensile and shear bonding strength of a CCE, respectively. The peak +shear bonding strength is calculated based on the Mohr-Coulomb failure criterion using the +cohesion (c) and internal friction angle (ϕ): 𝑓𝑠 = 𝑐 + 𝜎 tan 𝜙. op and sp are the peak opening and +slip values at the peak bonding stresses calculated as 𝑜𝑝 = 2ℎ𝑓𝑡 𝑝𝑓 +⁄ + and 𝑠𝑝 = 2ℎ𝑓𝑠 𝑝𝑓 +⁄ +, where h is +the nominal element edge length, and pf is the fracture penalty value. A CCE yields once the stress +reaches the peak, then it experiences a post-peak softening behavior with the bonding stresses +gradually decreased (Munjiza et al., 1999): +𝜎 = 𝐹(𝐷)𝑓𝑡 + + + + + + (3) +𝜏 = 𝐹(𝐷)𝑓𝑠 + + + + + (4) +F(D) is an empirical function that approximates the shape of the experimental stress-displacement +failure curve according to Evans and Marathe (1968): +F(𝐷) = [1 − +𝑎+𝑏−1 +𝑎+𝑏 exp (𝐷 +𝑎+𝑐𝑏 +(𝑎+𝑏)(1−𝑎−𝑏))] ∙ [𝑎(1 − 𝐷) + 𝑏(1 − 𝐷)𝑐] + +(5) +where a, b, c are empirical curve fitting parameters equal to 0.63, 1.8, and 6.0, respectively. The +damage coefficient (D) is calculated for Mode I, II, and I-II as +𝐷I = +𝑜−𝑜𝑝 +𝑜𝑟−𝑜𝑝 + + + + + + (6) +𝐷II = +𝑠−𝑠𝑝 +𝑠𝑟−𝑠𝑝 + + + + + + (7) + +11 + +𝐷I−II = √𝐷I +2 + 𝐷II +2 + + + + + (8) +with the subscripts indicating the mode of failure. The CCE breaks when D = 1, which corresponds +to a residual opening (or) or a residual slip (sr), for pure Mode I or II failure, respectively. For +Mode I-II failure, DI-II = 1 corresponds to a mixed failure opening and slip (of and sf). The values +of or and sr are calculated using the predefined numerical fracture energy GfI and GfII, for opening +failure and shear failure, respectively. The failure mode of the CCE (κ) is computed as +𝜅 = { +1 (pure tensile, Mode I) +1 + 𝐷II (mixed mode, Mode I − II) +2 (pure shear, Mode II) + + + + (9) + + +a) +b +Mode I +Mode II +Gf +0 +Sp +Opening +Slip +O, t +Normal/tangential bonding stress +0, s +Opening/slip +c) +f,fs +Tensile/shear strength +Internal friction angle +Pf +Fracture penalty +h +Nominal elementedgelength +Mode I-HI +G/G Energy consumed by Mode I/II fracture +f×f(D) +0 +Failure path +Broken +Mode I-II +012 + +Fig. 3 Deformation and failure criteria of the cohesive crack element (CCE). (a) Mode I, tensile +mode, (b) Mode II, shear mode, and (c) Mode I-II, mixed-mode. Shaded areas highlight the total +fracture energy consumed during the failure process of a CCE. The blue curve (failure path) +indicates the stress condition during the yielding and failure processes of the CCE. +When both DI and DII are satisfied at the same time, the failure is also considered as Mode +I-II, and a value of 1.5 is assigned to these events during post-processing. The broken cohesive +crack element (BCCE) is then considered as a new crack with no cohesion, and its behavior is +handled by the interaction algorithms, which are discussed in detail in the literature (Munjiza 2004; +Mahabadi et al., 2012). +2.3 Simulation of fracture propagation and seismicity in FDEM +Modeling seismic activity in rocks can provide quantitative information of the rock failure +process, and a validated model can improve the understanding of laboratory and field seismic +observations. In FDEM, upon breakage of the CCE, the accumulated strain energy is released, +resembling seismic activity. The coordinates, failure time, kinetic energy at failure, and failure +mode of the related BCCE can be recorded (Lisjak et al., 2013). However, a limitation of this +approach is that it considers each BCCE as one single seismic event. Consequently, the properties +of the fracture and the associated seismic events are highly dependent on the mesh size and mesh +orientation (Munjiza and John, 2002). In nature, the breakage of CCEs can be regarded as acoustic +emissions associated with the breakage of several mineral grains and grain boundaries (Zhao et al., +2015; Abdelaziz et al., 2018). In most cases, such a mesh dependency needs to be addressed to +obtain a better physical meaning of the failure process of CCEs. Zhao et al. (2014) attempted to +mitigate the problem with a clustering algorithm considering the temporal and spatial distribution +of BCCEs. In their method, each BCCE is viewed as an advancing crack tip, and BCCEs + +13 + +connecting to the crack tip are clustered together as a continuous fracture. However, this +implementation did not consider the physical meaning of fracture propagation. The propagating +fracture can arrest and then continue to propagate according to the stress conditions and material +heterogeneities (Van der Pluijm and Marshak, 2004), and from an energy dissipation point of view, +choosing the yielding point of a BCCE as the fracture tip is more consistent with the cohesive +crack model (Shet and Chandra, 2002). +Stemming from Zhao et al. (2014) and Zhao (2017), we implemented a new clustering +algorithm to mimic fracture propagation process during a seismic event. Note that we consider +only seismic activities related to the formation of new fractures, and seismic events created by +slipping on existing fracture surfaces are not considered. The algorithm proceeds as follows: +(1) The first BCCE that yields at time ty and fails at time tf is considered the initial crack of a +cluster. The search algorithm is then executed to include BCCEs connecting to either side of this +BCCE (i.e., fracture tips). +(2) BCCEs that are connected to the fracture tips and yield within the time window between ty and +tf are included in the same cluster and then treated as new fracture tips. At each output frame, the +same searching criterion is applied to such new fracture tips until no new BCCEs are found. Then, +this cluster of BCCEs is considered to be one continuous fracture, whose growth has produced one +seismic event. +(3) Repeat steps 1-2, until all recorded BCCEs are processed. +(4) Calculate the source parameters of the clustered seismic events as follows (for a cluster of n +BCCEs): +(a) event time is the breakage time tf of the initial BCCE in this cluster; + +14 + +(b) the hypocentre location is the centre coordinates of the initial BCCE in this cluster; +(c) the kinetic energy, Ee, is calculated as the sum of the kinetic energy of all BCCEs in +this cluster, 𝐸e = ∑ +𝐸k +𝑖 +𝑛 +𝑖=1 +, where 𝐸k +𝑖 = +1 +2 ∑ +𝑚𝑗𝑣𝑗 +2 +4 +𝑗=1 + is the kinetic energy of a BCCE, and +mj and vj are the nodal mass and velocity of the BCCE at the time of breakage. We adopt +the empirical relation between radiated energy and magnitude to calculate the magnitude +of the seismic events: 𝑀𝑒 = +2 +3 (log𝐸𝑒 − 4.8) (Gutenberg, 1956; Lisjak et al., 2013). +(d) the dominant source mechanism (ζ) of each cluster is calculated as a weighted average +of the failure modes of all BCCEs in this cluster: +𝜁 = +∑ +𝐸k +𝑖 +𝑛 +𝑖=1 +𝜅𝑖 +∑ +𝐸k +𝑖 +𝑛 +𝑖=1 + + + + + + +(10) +Where κi is the failure mode of the ith BCCE, and its associated kinetic energy, Eki is taken +as its weight. ζ = 1 and 2 represent pure tensile (Mode I) and shear events (Mode II), +respectively, while events having 1 < ζ < 2 have tensile and shear failure components +(Mode I-II). +This algorithm considers multiple BCCEs created by a single fracturing event, resulting in a more +realistic representation of the source mechanism and event energy than previous studies. Note that +if a series of connected CCEs break simultaneously due to mechanisms such as crushing or +pulverization, they will also be clustered as one event under this algorithm. +2.4 Numerical model setup + +15 + + +Fig. 4 Preparation of 2D surface profiles for the FDEM model. (a) The top (left) and bottom (right) +parts of the sample used in the rotary shear experiment. (b)–(c) 3D surface scan of the shear +surfaces. Red dashed lines indicate the extracted profiles. (d) The initial condition by micro-CT +imaging. (e) Comparison of the profiles (red dashed curves) with the micro-CT image showing the +initial condition of the shear simulation. Note that profiles are vertically offset for clearer +illustration. +3D shear simulations would mimic at best the deformation processes, but this is currently +impossible due to computational limitations. Instead, we built the 2D FDEM model that considers +not only the geometry of the experimental specimen but also the initial contact condition on the +rough surface. A 2D circular profile at the radius of 5.6 mm, which corresponds to the roughest +region (i.e., highest asperities) on the surface, was extracted (Fig. 4a-c). We chose such a profile +because the work by Zhao et al. (2018) suggested that this region with the largest roughness plays +an important role in controlling the shear strength and fracture development during the experiment. +To capture the geometry of the slipping surface, we digitized the top and bottom surfaces before +the experiment using a 3D surface scanner (ATOS II by GOM) at a horizontal grid interval of +44 μm. The relative location of the two profiles were adjusted to recreate the initial contact +conditions according to the micro-CT image (Fig. 4d-e). We subsampled the profiles to a 0.1 mm + +a) +b) +C) +1 mm +0.30.20.10.0-0.1-0.2-0.3 +d) +Elevation(mm) +e) +Sheardirection +0 +5 +10 +15 +20 +25 +30 +(mm)16 + +nominal grid interval, which was chosen as an acceptable compromise between computation time +and accuracy in representing the surface geometry. In addition, to mimic the rotary shear behavior, +the two ends of the profiles were extended by 3 mm (i.e., the desired total shear displacement) +using the same geometry as their opposite ends to create an effective periodic boundary. These +profiles formed the initial shear surfaces of the numerical model (Fig. 4e). + +Fig. 5 (a) Mesh topology and boundary conditions of the shear test simulation. The blue dotted +line indicates the location of the virtual measurement line. (b) Zoom in view of the refined mesh +at the shear surfaces, and the arrows indicate the smallest gap between top and bottom surfaces. +The bodies of the top and bottom model were 15 mm in thickness, resulting in a total +vertical height of 30 mm, similar to the sample used in the laboratory experiment (Fig. 5a). The +corners at the ends of the shear surfaces were filleted with a radius of 0.2 mm to avoid stress +concentrations that may result in unrealistic damage. To reduce computational time in applying +the normal stress during the simulation, the initial vertical distance between the top and bottom +semi-sample was adjusted to 2×10−6 mm (Fig. 5b). Moreover, two rigid boxes were added to +simulate the sample holders encasing the two semi-samples. The region of interest (i.e., within +1 mm distance from the shear surface) was discretized with a constant nominal element size of +0.1 mm. The remaining parts of the model were meshed with linearly increasing mesh size as a + +b) +5 mm +V +0.2 mm +x +V17 + +function of the distance from the shear surface, with the coarsest element size being 3 mm. As a +result, the model was meshed into 20,240 triangular elements. These elements were assigned with +the calibrated numerical properties (Table 1&2), while the shear boxes had properties of stainless +steel (Young’s modulus at 200 GPa, density at 8100 kg/m3, and Poisson’s ratio at 0.25). +Table 1 Laboratory measured macromechanical properties (i.e., calibration targets) and emergent +properties of the calibrated FDEM model (after Tatone and Grasselli, 2015; Zhao, 2017). +Properties (unit) +Laboratory +measurement +Calibrated +FDEM model + Density (kg·m−3) +1704 +1704 + Young’s modulus (GPa) +15.0 +15.0 + Poisson’s ratio (-) +0.24 +0.24 + Internal friction angle (Degrees) +23 +23 + Internal cohesion (MPa) +16.4 +16.4 + Tensile strength (MPa) +2.6 +2.7 + Uniaxial compressive strength (MPa) +50.3 +49.9 + +FDEM models synthesize the macroscopic behavior of materials from the interaction of +the micromechanical constituents. The overall deformation and failure behavior of the simulated +material are controlled by the combined effect of the input parameters defining the elastic +triangular elements and CCEs. As a result, the macroscopic mechanical properties (as listed in +Table 1, except for the density that needs no calibration) measured by standard laboratory tests +cannot be used directly. Rather, an iterative calibration approach is carried out to obtain input +parameters representative of the material, and the laboratory measured properties were used as the +calibration targets. In this approach, numerical compressive and tensile strength test models are +created and simulated using an initial set of input parameters. The macroscopic mechanical +properties and failure patterns are obtained from the simuation and compared against laboratory + +18 + +measurements. In a successful calibration, the numerical model will replicate both the macroscopic +mechanical properties measured from the experiments and the overall failure mode of the material. +If the simulation result is inadequate, the input parameters are iteratively fine-tuned until the +calibration targets are met (Tatone and Grasselli, 2015). The laboratory-measured properties and +the emergent macromechanical properties of the calibrated FDEM model are listed in Table 1, and +the calibrated FDEM model parameters are listed in Table 2. +Table 2 Calibrated FDEM model input parameters (after Zhao, 2017). +Parameter (unit) +Value +Continuum triangular elements + + Density, ρ (kg·m−3) +1704 + Young’s modulus, E (GPa) +15.6 + Poisson’s ratio, υ (-) +0.22 + Viscous damping factor, α +1 +Cohesive crack elements + + Internal cohesion, c (MPa) +17.5 + Tensile strength, σt (MPa) +2.55 + Friction angle, ϕ (Degree) +24.5 + Mode I fracture energy, GIc (J·m−2) +3.8 + Mode II fracture energy, GIIc (J·m−2) +90 + Fracture penalty, Pf (GPa) +156 + Normal contact penalty, Pn (GPa) +156 + Tangential contact penalty, Pt (GPa) +156 + +2.5 Simulation procedure and boundary conditions +The simulation was computed using the Irazu FDEM software (Geomechanica Inc., 2021) +with GPU (graphics processing unit) parallelization. The shear test simulation was conducted in +three phases (Table 3). In phase 1, the initial normal stress was applied by compressing the sample +at a constant vertical velocity of 0.2 m/s until the vertical stress reaches 2.5 MPa, which + +19 + +corresponds to the initial normal stress condition of the laboratory experiment. In phase 2, the top +and bottom boxes were constrained to their vertical position, and the horizontal shear velocity was +increased gradually to 0.3 m/s. This transition phase allows the oscillation induced by the +instantaneous stop of normal loading to dampen oscillations due to the shear acceleration. In +phase 3, the top and bottom boxes were fixed in their vertical positions (i.e., this is a constant +normal stiffness shear test) and moved in the horizontal direction at a constant velocity of 0.3 m/s +until the desired shear displacement of 3 mm was reached. Note that the loading velocities used in +the study are significantly higher (1000 times) than those used in laboratory experiments; however, +such a speed has been verified to provide a quasi-static loading condition while allowing a +reasonable computation time (Mahabadi, 2012). The model has 26 million simulation time steps, +and each step represents a simulation time of 4×10−10 s. +Table 3 Simulation phases and boundary conditions applied to the model. The applied velocities +in the x (vx) and y (vy) directions, and the resultant shear displacement (u) are listed. +Phase Simulation steps +vx (m/s)[1] vy (m/s)[2] u (mm) +1 +1–66,400 +0 +0.1 +0 +2 +66,401–964,000 +0–0.15[3] +0 +0–0.02 +3 +964,000–26,000,000 0.15 +0 +0.02–3.02 +[1] Positive (→) on the top box and negative (←) on the bottom box. +[2] Negative (↓) on the top box and positive (↑) on the bottom box. +[3] Linearly interpolated every time step to ramp up the shear velocity gradually. + +Normal and shear stresses were measured along a line parallel to the fault and placed 5 mm +above the rigid box in the bottom sample (Fig. 5a). This measurement line monitored the stress +conditions every 13,000 simulation steps, equivalent to a 200 kHz monitoring rate. The recorded +stress values in all elements along the measurement line were averaged to obtain the overall normal + +20 + +stress (σn) and shear stress (τ), which were used to calculate the friction coefficient μ = τ/σn, similar +to the laboratory-measured apparent friction coefficient. +3 Results and data analysis +3.1 Shear behavior + +Fig. 6 Calculated friction coefficient of (a) the laboratory test results plotted as a function of the +equivalent slip distance at a radius of 5.6 mm and (b) the numerical simulation. The first ~0.3 mm +are detailed in Fig. 7. Red arrows indicate significant drops of frictional resistance associated to +seismic events 1, 2, and 3, which are investigated in Section 3.2. +The simulated μ showed a similar trend with the experimental data. It reached the peak +value of 0.22 at a shear displacement of 0.32 mm, followed by a significant drop (Fig. 6b). The +simulated μ experienced many abrupt drops during the slipping process and then stabilized at +approximately 0.04 after approximately 1.7 mm of shear displacement. The simulated μ was +significantly lower than the value reported in the laboratory experiment, with many more +oscillations. + +a) +Experiment +1.5 +1.0 +0.5 +0 +0 +0.5 +1.2 +1.8 +2.4 +3.0 +Equivalent slip distance (mm) +b) +Simulation +0.2 +0.1 +0 +Event 2 & 3 +-0.1 +-Event 1 +Fig. 7b +-0.2 +0 +0.5 +1 +1.5 +2 +2.5 +3 +Shear displacement (mm)21 + +The simulated stress conditions of the first 0.3 mm showed intriguing similarities to the +laboratory experimental data (Fig. 7a&b). In this interval, the shear behavior observed in the +experiment can be divided into four stages (Fig. 7a): (I) τ, σn, and the resultant μ ramped up +gradually; (II) τ experienced a relatively stable stage with minor change, and σn decreased +continuously, causing minor change of μ; (III) τ and σn gradually increased to a peak shear stress, +and μ increased to the peak value; and (IV) τ, σn, and μ dropped rapidly. + +Fig. 7 Comparison of the overall normal and shear stresses and the friction coefficient of stages I- +IV between (a) the experimental data and (b) the simulated data. (c) and (d) are the zoom in views +of the local shear and normal stresses, respectively, at the asperity responsible for the stress drop +at stage IV. Orange circles numbered 1-6 indicate the horizontal shear displacements (u). (e) and +(f) are the micro-CT image of the laboratory specimen corresponding to frame 1 and 6 in (c) and +(d). The initial surface profiles from the surface scan data (red curves) are placed next to the +laboratory fault for comparison. + +a)Experiment +b) Simulation +II +III +IV +II +III +IV +70.6 +T(MPa) +2 +0.4 +(MPa) +0.2 +0 +0 +3 +On (MPa) +2.2 +6 +tttttt! +1.8 +1 +0.2 +0.1 +≥0.5 +、. +0 +e) +0 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.05 +0.15 +0.2 +0.25 +0.3 +Shear displacement (mm) +Shear displacement, u (mm) +1 +u=0mm +②u=0.092mm +③ u= 0.136 mm +4u=0.183mm +5 u=0.320mm +6u=0.321mm +5mm +Localshear/normalstress(MPa) +0 +5 +10 +2022 + +The numerical simulation qualitatively captured the general trend of these stages (Fig. 7b); +however, simulated σn in stage I decreased gradually, and more oscillations are observed in the +curves in the simulated data. To further investigate the mechanisms behind the shear behavior +during these stages, we examined the simulated local stress conditions around the asperity whose +breakage was responsible for the large and sudden drop of frictional resistance at stage IV (Fig. +7c&d). During stage I, the simulation shows that the shear surface is at the initial contact condition. +As the shear displacement increases, the shear stress increases gradually due to frictional resistance +of the initial contact area. Note that the numerical model did not capture the minor normal stress +increase measured in the experiment at this stage. Such an increase may be related to the interaction +of the asperities in the direction perpendicular to shear (i.e., out-of-plane motion) that does not +exist in the 2D simulation. During stage II, the top and bottom surfaces adjusted to a more +conforming contact, which resulted in the decrease of the shear and normal stress. During stage III, +new contact points were established, and asperities engage and interlock, causing the shear stress +to increase rapidly reaching the peak shear stress at the end of this stage. Asperities survived and +climbed onto each other, causing dilation that increased the normal stress. At stage IV, the highly +stressed asperity underwent high-stress concentration and failure, releasing the accumulated strain +energy that resulted in the sudden and significant drop of stresses and frictional resistance. The +simulated failure pattern, in terms of location and mechanism, resembled the laboratory +observation (Fig. 7e&f). More importantly, the numerical model can provide the evolution of +surface contacts and stress conditions throughout the shear process. +3.2 Progressive damage, gouge formation, and seismic activity +Progressive damage on the shear surface and fault gouge formation was simulated by +BCCEs (Fig. 8). The first several BCCEs occurred when the top and bottom semi-sample were + +23 + +loaded with the initial normal stress. Before ⁓1 mm of shear displacement, the damage was +concentrated in the vicinity of the shear surface. After ⁓1 mm of shear displacement, a number of +sub-vertical fractures penetrated the sample body, resembling the fracturing observed in the +laboratory (Fig. 9). The distribution of the shear-induced damage was mostly concentrated close +to the fault surface and heterogeneously distributed along the fault. Broken asperities formed the +gouge layer that accumulated between the semi-samples. As a result, some portions of the bare +fracture surface were protected from wearing (Fig. 9a), and this phenomenon is also observed in +the laboratory micro-CT image (Fig. 9b). + +24 + + +Fig. 8 Damage of the shear surface and the accumulation of gouge material with increasing shear +displacement. Damage is represented by broken cohesive crack elements. + +25 + + +Fig. 9 (a) Zoom-in view of a portion of the simulated fault surface at 3 mm of slip. The dashed red +lines highlight intact fault walls that were not damaged. (b) Zoom-in view of the micro-CT image +of a portion of the laboratory fault at a similar location to (a) (adopted from Zhao et al., (2018)). + +Fig. 10 Simulated seismic activities. (a) Magnitude, location, and failure mode of the clustered +seismic activity. (b) Event count in each bin. +A total of 7,557 CCEs were broken throughout the simulation, and they were clustered into +1,561 seismic events. Most of the BCCEs near the shear surface failed in shear mode (Mode II), +and almost all sub-vertical fractures propagated in tensile mode (Mode I). The magnitude of these +seismic events ranged between −11.1 and −4.4, with an average magnitude of −7.3. In general, + +a) +1mm +-Intact surface +Intact surface +-Gouge +Gouge +Shear + Shear induced fracture +induced +fractures +mmMode I-IH +20 +30 +35 +40 +2026 + +large magnitude events were mostly produced by shear-mode failures, while small magnitude +events mostly arose from tensile-mode failures (Fig. 10a). Along the vertical direction (y direction), +the spatial distribution of seismic activity coincides with the damage pattern: events were +concentrated within ±4 mm of the fault. We divided the horizontal length of the fault, i.e., the x +direction, into 100 bins and examined the spatial distribution of the seismic events along such a +path (Fig. 10b). Seismic events were distributed heterogeneously along the fault: bins at x = +29.8 mm and 34.9 mm had the largest number of events at 44; bins at x ranging 10.5 to 12.2 mm +and 20.5 to 21.0 mm had no seismic events. +Each asperity failure resulted in the sudden and significant drop of frictional resistance and +the release of accumulated strain energy. These large magnitude events caused stick-slip-like +responses and released high amplitude stress waves propagating across the model. Prior to these +dynamic seismic events, their corresponding locations experienced low shear velocity due to the +interlocking of asperities and are referred to as interlocking zones (ILZs) in the following +discussion. Stress concentrated at ILZs and eventually broke the asperities, releasing the +accumulated strain energy (see animated figures Fig. S1 in Supplementary Material for the velocity +fields). Three seismic events (Events 1-3, as indicated in Fig. 6) with distinct wave radiation +patterns are chosen as examples for further examination. Event 1 at u⁓0.3 mm was related to the +most significant friction drop. Events 2 and 3 were two consecutive events that occurred on the +slipping surface 12.5 mm apart from each other with a 0.005 ms time delay. We examined the +stress field (Fig. 11) and observed that the magnitude of seismic events was directly correlated to +the magnitude of the stress concentration at the asperities that failed. We observed that stress +concentration at the ILZs reached values as high as the compressive strength of the material, +causing compressive failure. Due to interlocking, the non-interlocking regions slightly ahead (with + +27 + +respect to the shear direction) of the ILZs were subjected to significant tensile stress that reached +the tensile strength of the material, thus, causing tensile fracturing. By examining the particle +velocity field (Fig. 12), we found that prior to the seismic events, the locations of the ILZ were +experiencing particle velocities lower than the loading velocity (i.e., < 0.1 m/s). As the seismic +events occurred, the source region had particle velocities that were two orders of magnitude higher +than that of the ILZs (i.e., > 10 m/s). Interestingly, considering that P- and S-wave velocities are +2967 m/s and 1884 m/s, respectively, Event 3 occurred right after the arrival of the P-wave induced +by Event 2, but prior to the arrival of the S-wave. Therefore, Event 3 may have been triggered by +the stress perturbation from Event 2. +Fig. 11 Output frames of the numerical model showing the horizontal stress σxx at (a-c) Event 1 +and (d-g) Events 2 and 3. Interlocking zones (ILZs) that are related to the selected seismic events +are highlighted by the yellow arrows. Note that the simulation time interval between two frames +is 0.052 ms. + + +evenl +erem28 + + +Fig. 12 Output frames of the numerical model showing the particle velocity at (a-c) Event 1 and +(d-g) Events 2 and 3. ILZs are highlighted by the yellow arrows and the P-wave wavefront of +Event 2 is labelled. Note the color map is in log scale. +4 Discussion +The FDEM numerical model qualitatively captured the mechanical behavior observed in +the laboratory experiments, highlighting the dominant role of surface roughness on the shear +behavior of rocks at low-stress conditions. Both the laboratory experiment and the numerical +simulation show a slip weakening behavior where the friction coefficient ramps up to the peak +value and then decreases to a residual value. In the numerical simulation, the shear stress and +friction reached a steady-state and residual value around ~1.7 mm of total displacement, in +agreement with the laboratory value (Zhao et al., 2018). +During the first ~0.3 mm of shear displacement, the discrepancies between the +experimental and simulation results in stage I and the additional stress oscillations in the simulation +results may be because the 2D model was not able to capture 3D asperity interactions. However, +the overall variation trend of stresses and the damage pattern on the shear surface showed close + +Event! +front +ILZ +Eyent2 +Event329 + +similarities, suggesting that the 2D profile that we used is a proxy for the laboratory specimen. +This also supports our previous interpretation that the highest asperity was responsible for the +formation of the large secondary sub-vertical fractures and the associated sudden drop in shear +resistance (Zhao et al., 2018). These results suggest that our numerical technique, which uses a +combination of surface scanning, X-ray micro-CT imaging, and FDEM modelling, represents a +promising approach to simulate realistic fault behavior. Our simulation provides the continuous +evolution of contacts on the shear surface and the stress conditions that complement the laboratory +observations in achieving a better comprehension of how the interaction between asperities +controls the stress conditions and damage patterns in faults. +During the shear process, asperities interact in various modes including climbing onto each +other, interlocking, and breaking (Scholz, 1990). Our experimental and numerical results show +that such interactions directly influenced the stress conditions and damage patterns. When the slip +displacement is small (u < 1.5 mm), weak asperities (i.e., millimetric scale unevenness) controlled +the frictional behavior, creating gouge material. These observations agree with the laboratory +observations on the post-mortem sample and suggest the importance of surface roughness in +controlling the formation of the gouge layer. As the slip displacement increases (u > 1.5 mm), the +large-scale roughness of the shear surface (i.e., centimetric scale waviness) becomes important to +the shear behavior and damage pattern. Large scale waviness causes high stress concentration +through interlocking and climbing and may cause sub-vertical secondary fractures. +The damage and seismic event distributions are closely related to the stress heterogeneity +on the shear surface caused by the surface roughness. Depending on the geometry of the asperity, +the stress concentration at the ILZs could reach the compressive strength of the material, causing +compressive failure. This mechanism creates gouge material in the vicinity of the shear surface. + +30 + +On the other hand, the areas ahead of the ILZs experience tensile stress up to the tensile strength +of the material, thus, creating tensile fractures. This mechanism creates large sub-vertical +secondary fractures. Breakage of strong asperities release the accumulated strain energy in the +whole model, causing an overall shear stress drop, giving a stick-slip-like shear behavior. Such a +lock-and-fail mechanism is recently found to be the key process of stick-slip behavior of bare +surfaces (Chen et al., 2020; Morad et al., 2022). Note that the overall shear loading in our model +is considered quasi-static, but the local seismic events are dynamic activities with particle velocity +more than 100 times the quasi-static loading velocity. This suggests that on a rough shear surface, +quasi-static shear consists of numerous heterogeneously distributed local dynamic seismic +activities, and this process may complicate the slip process on rough faults and the estimation of +the energy budget (Tinti et al., 2005). +Observations on Events 2 and 3 suggest that the stress perturbation from asperities +breakage may trigger events on adjacent interlocking zones. From an earthquake perspective, there +are two possible mechanisms that may trigger seismic events in the near field: (1) static stress +redistribution (e.g., King et al., 1994; Toda et al., 1998) and (2) dynamic stress wave perturbation +(e.g., Kilb et al., 2000; Gomberg et al., 2001). In our simulation, the modeled body did not slip as +a rigid body, rather, the slipping consisted of pulses of local movements, accompanied by +numerous continuously changing of contacts and asperities breakages. When the asperity +associated to Event 1 breaks, the dynamic stress perturbation was damped out, and the static stress +concentration is transferred to nearby asperities, which eventually caused failure of other asperities. +On the other hand, Events 2 and 3 showed a more interesting correlation. Event 3 occurred +between the arrival times of P- and S-waves from Event 2. Within this time window, stress +redistribution had not reached a steady state, suggesting that the perturbation of the dynamic stress + +31 + +wave radiated from Event 2 may have triggered Event 3. These results imply that static stress +transfer and dynamic stress perturbation triggering may occur on the same fault and contribute to +the movement of fault slip. However, due to the limitation of the model output frequency and post- +processing method, the triggering is not conclusive, Event 2 and 3 may have been independent +seismic events occurred in a narrow time window, and more investigation is needed in future +research. +The numerical simulation has the advantage of continuously modeling the fault shear +process, fault surface damage, and associated stress conditions. However, the simulated sample +experienced more damage than the laboratory sample, which is probably related to the limitation +of 2D simulations not accounting for the motion in the third dimension. For the same reason, the +simulated stresses suffered significant fluctuations, and the friction coefficient was much lower +than the experimental measurement, which is a common limitation of 2D simulations. The +laboratory experiment by Frye and Marone (2002) and the numerical simulation by Hazzard and +Mair (2003) demonstrated that 2D numerical models exhibit friction values notably lower than 3D +models and suffer from greater stress fluctuations due to the lack of particle motion in the third +dimension. In addition, we meshed the shear surface at a relatively high resolution (0.1 mm), +resulting in a large number of asperities at various sizes. Hence, the interlocking and breakage of +these asperities caused stress oscillations (i.e., microseismic events). Even though we qualitatively +captured the shear behavior that matches the laboratory measurements, to fully capture the shear +behavior of the rotary shear experiment, a 3D model capturing the surface geometry and asperity +interaction on the entire shear surface will be required. +5 Conclusion + +32 + +In this study, we used a carefully built and calibrated FDEM numerical model to simulate a +laboratory shear experiment. We introduced a new clustering algorithm to improve the +understanding of the simulated fracturing and associated seismic events. The model was able to +qualitatively capture the frictional behavior observed in the laboratory experiment, providing the +missing information in the experimental observation regarding the continuous variation of stresses +and the progressive evolution on the shear surfaces. +Our numerical model matches the experimental results particularly well at the beginning +of the shear deformation (~0.3 mm). We were able to identify similar stress variation trends and +damage patterns. The simulation results provided detailed evolution processes of the contacts on +the shear surface and the local stress conditions, which are not available in experimental +observations. Combining the numerical and experimental results, we conclude that interlocking of +asperities can cause compressive stress concentration on the front side (i.e., facing the shear +direction) of the asperity, which could induce compressive failure (e.g., crushing) near the shear +surface; on the other hand, tensile stress concentration is generated on the leeward side of the +asperity, which could cause sub-vertical tensile fractures that could propagate into the host rock. +Progressive surface damage and the associated microseismic events occur at the locations of +asperity interactions and is highly heterogeneous. Several locations experienced no damage even +after large shear displacement, these locations are either not in contact or were protected by gouge +materials. +As a result of the interlocking and breakdown of asperities, local dynamic failure events +occur, even though the overall loading is quasistatic. These events are considered microseismic +events, and their magnitudes range between −11.1 and −4.4. Strain energy stored in the medium +was released during these events, causing dynamic perturbation to the overall stress condition, and + +33 + +the particle velocity in the source reached > 10 m/s, two orders of magnitude larger than the +surrounding regions. This high amplitude stress perturbation could even trigger the failure of +adjacent critically stressed asperities. +Both the numerical model and the experiment suggested the importance of shear surface +roughness in controlling slip behavior, and we were able to explain the laboratory observations +with the help of numerical results. Shear surface evolution is a complicated process that involves +frictional sliding, fracturing, gouge comminution, and seismicity. The high degree of agreement +between simulation and experiment data leads to a promising future of predicting fault behavior +through, laboratory testing, surface characterization, and numerical simulations. These results +improved the understanding of shear behavior and demonstrated that micromechanical based +numerical simulation is a capable approach to study fault mechanics. +Declaration of competing interest +The authors declare that they have no known competing financial interests or personal +relationships that could have appeared to influence the work reported in this paper. +Acknowledgements +Q. Zhao is supported by the FCE Start-up Fund for New Recruits at the Hong Kong Polytechnic +University (Project ID P0034042) and the Early Career Scheme of the Research Grants Council of +the Hong Kong Special Administrative Region, China (Project No. PolyU 25220021). This work +has also been supported through the NSERC Discovery Grants 341275, CFILOF Grant 18285, +Carbon Management Canada (CMC), and NSERC/Energi Simulation Industrial Research Chair +Program. The authors would like to thank Geomechanica Inc. for providing the Irazu FDEM +simulation software. Q. Zhao would like to thank Dr. Andrea Lisjak and Dr. Bin Chen for + +34 + +discussions and suggestions. The authors appreciate the constructive suggestions and comments +from the editor and the reviewers. +References +Abdelaziz, A., Zhao, Q., & Grasselli, G. (2018). Grain based modelling of rocks using the +combined finite-discrete element method. Computers and Geotechnics, 103, 73-81. +Asadi, M. S., Rasouli, V., & Barla, G. (2012). A bonded particle model simulation of shear strength +and asperity degradation for rough rock fractures. Rock Mechanics and Rock Engineering, 45, +649–675. doi:10.1007/s00603-012-0231-4. +Bahaaddini, M., Sharrock, G., & Hebblewhite, B. (2013). 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K., Liu, Q., & Grasselli, G. (2014). Numerical simulation of +hydraulic fracturing and associated microseismicity using finite-discrete element method. Journal +of Rock Mechanics and Geotechnical Engineering, 6, 574–581. +Zhao, Q., Tisato, N., Grasselli, G., Mahabadi, O. K., Lisjak, A., & Liu, Q. (2015). Influence of in +situ stress variations on acoustic emissions: a numerical study. Geophysical Journal International, +203, 1246–1252. +Zhao, Q., Tisato, N., & Grasselli, G. (2017). Rotary shear experiments under X-ray micro- +computed tomography. Review of Scientific Instruments, 88(1), 015110. + +38 + +Zhao, Q.; Tisato, N.; Kovaleva, O.; and Grasselli, G. (2018). Direct Observation of Faulting by +Means of Rotary Shear Tests Under X-Ray Micro-Computed Tomography. Journal of +Geophysical Research, 123(9). +Zhao, Q.; Glaser, S.; Tisato, N.; and Grasselli, G. (2020). Assessing Energy Budget of +Laboratory Fault Slip Using Rotary Shear Experiments and Micro-Computed Tomography. +Geophysical Research Letters, 47(1). + diff --git a/LNE2T4oBgHgl3EQfpwjv/content/tmp_files/load_file.txt b/LNE2T4oBgHgl3EQfpwjv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2b545f7bee4eeace393453afa5f162438a0318e --- /dev/null +++ b/LNE2T4oBgHgl3EQfpwjv/content/tmp_files/load_file.txt @@ -0,0 +1,1114 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf,len=1113 +page_content='1 Numerical investigation of progressive damage and associated seismicity on a laboratory fault Qi Zhao1,2*, Nicola Tisato3, Aly Abdelaziz2, Johnson Ha2, and Giovanni Grasselli2 1Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong SAR, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 2Department of Civil and Mineral Engineering, The University of Toronto, 35 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' George Street, Toronto, Ontario M5S 1A4, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 3Department of Geological Sciences, Jackson School of Geosciences, The University of Texas at Austin, 2305 Speedway Stop C1160, Austin, TX 78712-1692, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' *Corresponding author: Qi Zhao (qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='qz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='zhao@polyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='hk) Abstract Understanding rock shear failure behavior is crucial to gain insights into slip-related geohazards such as rock avalanches, landslides, and earthquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' However, descriptions of the progressive damage on the shear surface are still incomplete or ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In this study, we use the hybrid finite-discrete element method (FDEM) to simulate a shear experiment and obtain a detailed comprehension of shear induced progressive damage and the associated seismic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' We built a laboratory fault model from high resolution surface scans and micro-CT imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Our results show that under quasi-static shear loading, the fault surface experiences local dynamic seismic activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' We found that the seismic activity is related to the stress concentration on interlocking asperities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' This interlocking behavior (i) causes stress concentration at the region of contact that could reach the compressive strength, and (ii) produces tensile stress up to the tensile strength in the region adjacent to the contact area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Thus, different failure mechanisms and damage patterns including crushing and sub-vertical fracturing are observed on the rough surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Asperity failure creates rapid local slips resulting in significant stress perturbations that alter the overall stress condition and may trigger the slip of adjacent critically stressed asperities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' We found that the 2 spatial distribution of the damaged asperities and the seismic activity is highly heterogeneous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' regions with intense asperity interactions formed gouge material, while others exhibit minimal to no damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' These results emphasize the important role of surface roughness in controlling the overall shear behavior and the local dynamic seismic activities on faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Keywords Shear behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' surface roughness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' asperity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' shear induced damage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' seismicity 3 1 Introduction Understanding shear behavior along rock discontinuities at various scales, such as joints and faults, is essential to rock engineering projects and geohazard mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Rock discontinuities are planes of weakness and are responsible for many geohazards, for example, rock avalanches, landslides, and earthquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Numerous laboratory shear experiments have been conducted on a large variety of rock types under different conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', Bandis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Beeler, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Marone, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Di Toro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Grasselli, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Reches and Lockner, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Tisato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Kim and Jeon, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Morad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Among these studies, many suggested the importance of surface roughness and contact condition in controlling the shear behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' However, the progressive damaging process on faults is still not well understood because fault surfaces cannot be observed directly during shear, with the exception of a few studies utilizing transparent halite samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', Renard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2012) and in situ and operando testes conducted under X-ray micro-computed tomography (micro-CT) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', Zhao et al, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Shear processes control coseismic damage and friction on fault, but the constitutive friction theories are not yet fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' To observe and gain insights into damage processes on rough rock surfaces undergoing shear deformation, Tatone and Grasselli (2013) used micro-CT to image the joint surfaces after the shear test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' and Crandall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2017) used micro-CT to obtain geometrical information from fractured shale core that is incrementally sheared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Recently, direct and detailed observations of the evolution of laboratory fault were achieved by using an in situ rotary shear experimental apparatus under X-ray micro-CT (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Such experimental results help draw the connections between microscopic damage and macroscopic shear behavior, the formation and accumulation of gouge material, and shear-induced secondary fractures (Zhao, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 4 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' However, due to technological limitations, time-continuous observations of the shear surface evolution and the in situ stress condition remain shortfalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Numerical simulation methods have been extensively used to study the shear behavior of rock discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' To simulate the interaction and breakage of asperities and the frictional sliding behavior, numerical methods that can capture solid fracturing and interaction are typically used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' For example, the particle-based lattice solid model (Mora and Place 1993) uses a numerical concept similar to the discrete element method (DEM) to simulate frictional behavior and fracturing in solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Mora and Place (1998) and Place and Mora (2000) used their model to study the role fault gouge on the frictional behavior of faults, offering a possible explanation for the heat flow paradox (Henyey and Wasserburg, 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Lachenbruch and Sass, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Bonded particle- based methods, such as the particle flow code (PFC) (Cundall and Strack, 1979), are commonly used for simulating rock shear behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Park and Song (2009) used PFC3D to simulate direct shear tests and demonstrated that the method can simulate typical rock joint shear behavior, and they found that the peak shear strength and peak dilation angle was strongly influenced by the friction coefficient, roughness, and bond strength, while the residual shear strength and residual friction angle was influenced by the particle size, friction coefficient, and bond strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Asadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2012) used a similar approach in two dimensions (PFC2D) to simulate direct shear experiments on synthetic joint profiles of varying roughness and boundary conditions to assess asperity degradation and intact material damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' They showed that as the joint is sheared, highly localized asperity interaction on the joint surface and the geometry of the asperities has a significant influence on how joints fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Types of failure include asperity sliding, cut-off, separation, and crushing, typically associated with tensile failure into the intact material in conditions with steep asperities and/or high normal stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' However, Bahaaddini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2013) identified a significant 5 shortcoming of the particle-based methods due to the unrealistic shear and dilation behavior of joints as a result of particle interlocking due to the inherent micro-scale roughness of the joint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' To overcome this limitation, they implemented the smooth-joint model (Pierce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2007), where the blocks associated with either side of the joint are generated separately to appropriately define and apply the smooth joint model to the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Lambert and Coll (2014) created a synthetic rock joint by importing the real morphology of the joint surface into a bonded particle assembly and studied the shear behavior using the smooth-joint model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Their results reproduced the progressive degradation of the asperities upon shearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The hybrid finite-discrete element method (FDEM) is increasingly used to investigate rock shear behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' FDEM is a micromechanical numerical method first introduced by Munjiza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', (1995) combining the finite element method and discrete element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In doing so, the numerical method can model the transition of a continuous material to a discontinuous material as it deforms, yields, and breaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Karami and Stead (2008) and Tatone (2014) used FDEM to model direct shear tests and relate progressive asperity degradation mechanisms with the measured shear stress and dilation during shearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In addition, Tatone (2014) verified the numerical modelling observations by coupling their study with X-ray micro-CT imaging on post-mortem specimens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' It was found that tensile fractures develop in asperities at and near the peak shear stress, followed by a reduction in shear resistance as asperities continue to fail in both tension and shear, and finally, a residual shear resistance is reached once asperities are completely broken and gouge is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In this study, we used the two-dimensional (2D) FDEM to simulate an experiment on the gradual evolution of deformation of a laboratory fault, and we improve the understanding of shear behavior of rough faults through combined interpretation of the simulation and experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' First, we provide a brief review of the FDEM, emphasizing the modeling of damage and 6 seismic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Second, we develop a clustering algorithm to improve the comprehension of the simulated fractures and seismic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Next, we build a model based on the laboratory experiment and analyze the simulated results focusing on three aspects that are hardly accessible by experiments: (1) the time-continuous variation of stress conditions on the shear surface, (2) the progressive failure of the asperities and accumulation of gouge, and (3) the seismic activity related to shear-induced damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The carefully built and calibrated numerical model is able to simulate the emergent rock mechanical and frictional behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' We observe that shear-induced damage and seismic activities are heterogeneously distributed along the fault surface due to the surface roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Seismic events occur at the locations of asperity failure due to the interlocking-induced stress concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Such events radiate seismic waves and significantly change the overall stress conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Some areas on the fault were covered by gouge material and free from damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' These results agree with the laboratory observations and further elaborate on the importance of surface roughness in controlling shear behavior, which is critical to rock engineering practices and earthquake studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 2 Material and methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='1 In situ shear test under X-ray micro-CT The numerical simulation in this study is based on the experimental work using in situ shear tests under micro-CT reported by Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2018) and Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2020), and a brief review is provided here for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The tested specimen was a cylindrical Flowstone (microfine calcium sulfate cement mortar) 32 mm in length and 12 mm in diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The specimen was divided into top and bottom parts by a three-point bending test that created two semi-samples divided by a discontinuity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', laboratory fault) with two matching rough surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' An unconfined rotary shear test was conducted on the two semi-samples by shearing the fault under the initial normal 7 stress of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The top semi-sample was forced to slip incrementally against the fixed bottom semi-sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Normal force and torque were recorded during rotation and used to calculate the friction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' After each incremental slip of 6°, a three-dimensional (3D) micro-CT scan was conducted, which allows for imaging of the gradual morphological evolution of the specimen (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' This experimental work provided detailed information of the shear-induced secondary fractures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 1b) and the progressive damage on the slipping surface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 1c) in the sample volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' however, the observation of the shear surface damage evolution was only available at discrete time points coincident with each shear step, while an actual time-continuous observation of the shear surface evolution and the local stress condition on the rough surface was not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 1 Summary of the laboratory set-up and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (a) Schematic of the in situ shear test and a zoom-in view of the shear surface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', the zone of interest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (b) 3D visualization of the development of shear induced fractures with increasing shear displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (c) 2D unwrapped micro-CT image slice showing the progressive damage on the slipping surface with increasing shear displacement, viewed at the radius (R = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='6 mm) corresponding to the highest asperity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Red 8 dashed boxes in (c) indicate (from left to right) shear-induced aperture opening, fracturing, and progressive damage and gouge formation (modified from Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2018) and Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 The hybrid finite-discrete element method The hybrid finite-discrete element method (FDEM) combines continuum mechanics principles with discrete element principles to simulate interaction, deformation, and fracturing of materials (Munjiza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Munjiza, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' FDEM has been used to investigate a wide range of rock mechanics and geophysics problems including, but not limited to, tunneling and excavation, micromechanics, rock joint shear behavior, hydraulic fracturing, thermal-mechanical/hydro- thermal-mechanical coupling, and fault dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', Karami and Stead 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Mahabadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Lisjak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Fukuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Okubo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Knight et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Simulating the entire shear behavior and evolution of a rough surface is a challenging task that requires advanced computational resources encompassing, for example, the 3D FDEM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' However, 3D models explicitly capturing the surface roughness at sub-millimeter resolution and the entire shear process is not practical due to the demanded computation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' On the other hand, 2D FDEM simulations has the merit of reducing the computational demand, and it has been shown to provide insights into the mechanical behavior of rock joints and faults (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', Karami and Stead, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Tatone, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Okubo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' FDEM models synthesize the macroscopic behavior of materials from the interaction of the micromechanical constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In a 2D FDEM model, the simulated material is first discretized based on a finite element mesh consisting of nodes and triangular elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Then, the finite element mesh is enriched by inserting a four-node cohesive crack element (CCE) between each adjacent triangular element pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Motion for the discretized system is calculated by an explicit time 9 integration scheme, and the nodal coordinates of the elements are updated at each simulation step (Munjiza, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' FDEM models the progressive damage and failure of brittle material according to the principles of non-linear elastic fracture mechanics (Dugdale, 1960;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Barenblatt, 1962), and it captures the fracturing behavior of solids by modeling the entire failure path, including elastic deformation, yielding, and fracturing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 2 Schematic diagram showing the FDEM approach of simulating fracturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (a) Propagation of a fracture and the creation of fracture process zone (FPZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (b) Realization of the fracturing process in FDEM involves the yielded cohesive crack elements and broken cohesive crack elements (BCCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Depending on the local stress and deformation field, the CCE undergoes elastic deformation, yielding, and breakage, simulating the damage development of the fracture process zone (FPZ) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 3) (Labuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' During elastic loading, the relationships between bonding stresses (normal bonding stress, σ and shear bonding stress, τ) and the corresponding crack displacement (opening, o and slip, s) are as follows (Munjiza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 1999): Cracktip10 𝜎 = { 2𝑜 𝑜𝑝 𝑓𝑡 (𝑜 < 0, compression) [ 2𝑜 𝑜𝑝 − ( 𝑜 𝑜𝑝) 2 ] 𝑓𝑡 (0 < 𝑜 < 𝑜𝑝, tension) (1) 𝜏 = [ 2𝑠 𝑠𝑝 − ( 𝑠 𝑠𝑝) 2 ] 𝑓𝑠 (|𝑠| ≤ |𝑠𝑝|, shear) (2) where ft and fs are the peak tensile and shear bonding strength of a CCE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The peak shear bonding strength is calculated based on the Mohr-Coulomb failure criterion using the cohesion (c) and internal friction angle (ϕ): 𝑓𝑠 = 𝑐 + 𝜎 tan 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' op and sp are the peak opening and slip values at the peak bonding stresses calculated as 𝑜𝑝 = 2ℎ𝑓𝑡 𝑝𝑓 ⁄ and 𝑠𝑝 = 2ℎ𝑓𝑠 𝑝𝑓 ⁄ , where h is the nominal element edge length, and pf is the fracture penalty value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' A CCE yields once the stress reaches the peak, then it experiences a post-peak softening behavior with the bonding stresses gradually decreased (Munjiza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 1999): 𝜎 = 𝐹(𝐷)𝑓𝑡 (3) 𝜏 = 𝐹(𝐷)𝑓𝑠 (4) F(D) is an empirical function that approximates the shape of the experimental stress-displacement failure curve according to Evans and Marathe (1968): F(𝐷) = [1 − 𝑎+𝑏−1 𝑎+𝑏 exp (𝐷 𝑎+𝑐𝑏 (𝑎+𝑏)(1−𝑎−𝑏))] ∙ [𝑎(1 − 𝐷) + 𝑏(1 − 𝐷)𝑐] (5) where a, b, c are empirical curve fitting parameters equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='63, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='8, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The damage coefficient (D) is calculated for Mode I, II, and I-II as 𝐷I = 𝑜−𝑜𝑝 𝑜𝑟−𝑜𝑝 (6) 𝐷II = 𝑠−𝑠𝑝 𝑠𝑟−𝑠𝑝 (7) 11 𝐷I−II = √𝐷I 2 + 𝐷II 2 (8) with the subscripts indicating the mode of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The CCE breaks when D = 1, which corresponds to a residual opening (or) or a residual slip (sr), for pure Mode I or II failure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' For Mode I-II failure, DI-II = 1 corresponds to a mixed failure opening and slip (of and sf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The values of or and sr are calculated using the predefined numerical fracture energy GfI and GfII, for opening failure and shear failure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The failure mode of the CCE (κ) is computed as 𝜅 = { 1 (pure tensile, Mode I) 1 + 𝐷II (mixed mode, Mode I − II) 2 (pure shear, Mode II) (9) a) b Mode I Mode II Gf 0 Sp Opening Slip O, t Normal/tangential bonding stress 0, s Opening/slip c) f,fs Tensile/shear strength Internal friction angle Pf Fracture penalty h Nominal elementedgelength Mode I-HI G/G Energy consumed by Mode I/II fracture f×f(D) 0 Failure path Broken Mode I-II 012 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 3 Deformation and failure criteria of the cohesive crack element (CCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (a) Mode I, tensile mode, (b) Mode II, shear mode, and (c) Mode I-II, mixed-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Shaded areas highlight the total fracture energy consumed during the failure process of a CCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The blue curve (failure path) indicates the stress condition during the yielding and failure processes of the CCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' When both DI and DII are satisfied at the same time, the failure is also considered as Mode I-II, and a value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 is assigned to these events during post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The broken cohesive crack element (BCCE) is then considered as a new crack with no cohesion, and its behavior is handled by the interaction algorithms, which are discussed in detail in the literature (Munjiza 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Mahabadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='3 Simulation of fracture propagation and seismicity in FDEM Modeling seismic activity in rocks can provide quantitative information of the rock failure process, and a validated model can improve the understanding of laboratory and field seismic observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In FDEM, upon breakage of the CCE, the accumulated strain energy is released, resembling seismic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The coordinates, failure time, kinetic energy at failure, and failure mode of the related BCCE can be recorded (Lisjak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' However, a limitation of this approach is that it considers each BCCE as one single seismic event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Consequently, the properties of the fracture and the associated seismic events are highly dependent on the mesh size and mesh orientation (Munjiza and John, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In nature, the breakage of CCEs can be regarded as acoustic emissions associated with the breakage of several mineral grains and grain boundaries (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Abdelaziz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In most cases, such a mesh dependency needs to be addressed to obtain a better physical meaning of the failure process of CCEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2014) attempted to mitigate the problem with a clustering algorithm considering the temporal and spatial distribution of BCCEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In their method, each BCCE is viewed as an advancing crack tip, and BCCEs 13 connecting to the crack tip are clustered together as a continuous fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' However, this implementation did not consider the physical meaning of fracture propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The propagating fracture can arrest and then continue to propagate according to the stress conditions and material heterogeneities (Van der Pluijm and Marshak, 2004), and from an energy dissipation point of view, choosing the yielding point of a BCCE as the fracture tip is more consistent with the cohesive crack model (Shet and Chandra, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Stemming from Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2014) and Zhao (2017), we implemented a new clustering algorithm to mimic fracture propagation process during a seismic event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Note that we consider only seismic activities related to the formation of new fractures, and seismic events created by slipping on existing fracture surfaces are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The algorithm proceeds as follows: (1) The first BCCE that yields at time ty and fails at time tf is considered the initial crack of a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The search algorithm is then executed to include BCCEs connecting to either side of this BCCE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', fracture tips).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2) BCCEs that are connected to the fracture tips and yield within the time window between ty and tf are included in the same cluster and then treated as new fracture tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' At each output frame, the same searching criterion is applied to such new fracture tips until no new BCCEs are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Then, this cluster of BCCEs is considered to be one continuous fracture, whose growth has produced one seismic event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (3) Repeat steps 1-2, until all recorded BCCEs are processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (4) Calculate the source parameters of the clustered seismic events as follows (for a cluster of n BCCEs): (a) event time is the breakage time tf of the initial BCCE in this cluster;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 14 (b) the hypocentre location is the centre coordinates of the initial BCCE in this cluster;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (c) the kinetic energy, Ee, is calculated as the sum of the kinetic energy of all BCCEs in this cluster, 𝐸e = ∑ 𝐸k 𝑖 𝑛 𝑖=1 , where 𝐸k 𝑖 = 1 2 ∑ 𝑚𝑗𝑣𝑗 2 4 𝑗=1 is the kinetic energy of a BCCE, and mj and vj are the nodal mass and velocity of the BCCE at the time of breakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' We adopt the empirical relation between radiated energy and magnitude to calculate the magnitude of the seismic events: 𝑀𝑒 = 2 3 (log𝐸𝑒 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='8) (Gutenberg, 1956;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Lisjak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (d) the dominant source mechanism (ζ) of each cluster is calculated as a weighted average of the failure modes of all BCCEs in this cluster: 𝜁 = ∑ 𝐸k 𝑖 𝑛 𝑖=1 𝜅𝑖 ∑ 𝐸k 𝑖 𝑛 𝑖=1 (10) Where κi is the failure mode of the ith BCCE, and its associated kinetic energy, Eki is taken as its weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' ζ = 1 and 2 represent pure tensile (Mode I) and shear events (Mode II), respectively, while events having 1 < ζ < 2 have tensile and shear failure components (Mode I-II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' This algorithm considers multiple BCCEs created by a single fracturing event, resulting in a more realistic representation of the source mechanism and event energy than previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Note that if a series of connected CCEs break simultaneously due to mechanisms such as crushing or pulverization, they will also be clustered as one event under this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='4 Numerical model setup 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 4 Preparation of 2D surface profiles for the FDEM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (a) The top (left) and bottom (right) parts of the sample used in the rotary shear experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (b)–(c) 3D surface scan of the shear surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Red dashed lines indicate the extracted profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (d) The initial condition by micro-CT imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (e) Comparison of the profiles (red dashed curves) with the micro-CT image showing the initial condition of the shear simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Note that profiles are vertically offset for clearer illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 3D shear simulations would mimic at best the deformation processes, but this is currently impossible due to computational limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Instead, we built the 2D FDEM model that considers not only the geometry of the experimental specimen but also the initial contact condition on the rough surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' A 2D circular profile at the radius of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='6 mm, which corresponds to the roughest region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', highest asperities) on the surface, was extracted (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 4a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' We chose such a profile because the work by Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2018) suggested that this region with the largest roughness plays an important role in controlling the shear strength and fracture development during the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' To capture the geometry of the slipping surface, we digitized the top and bottom surfaces before the experiment using a 3D surface scanner (ATOS II by GOM) at a horizontal grid interval of 44 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The relative location of the two profiles were adjusted to recreate the initial contact conditions according to the micro-CT image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 4d-e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' We subsampled the profiles to a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='1 mm a) b) C) 1 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='3 d) Elevation(mm) e) Sheardirection 0 5 10 15 20 25 30 (mm)16 nominal grid interval, which was chosen as an acceptable compromise between computation time and accuracy in representing the surface geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In addition, to mimic the rotary shear behavior, the two ends of the profiles were extended by 3 mm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', the desired total shear displacement) using the same geometry as their opposite ends to create an effective periodic boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' These profiles formed the initial shear surfaces of the numerical model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 4e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 5 (a) Mesh topology and boundary conditions of the shear test simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The blue dotted line indicates the location of the virtual measurement line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (b) Zoom in view of the refined mesh at the shear surfaces, and the arrows indicate the smallest gap between top and bottom surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The bodies of the top and bottom model were 15 mm in thickness, resulting in a total vertical height of 30 mm, similar to the sample used in the laboratory experiment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The corners at the ends of the shear surfaces were filleted with a radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 mm to avoid stress concentrations that may result in unrealistic damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' To reduce computational time in applying the normal stress during the simulation, the initial vertical distance between the top and bottom semi-sample was adjusted to 2×10−6 mm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Moreover, two rigid boxes were added to simulate the sample holders encasing the two semi-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The region of interest (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', within 1 mm distance from the shear surface) was discretized with a constant nominal element size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The remaining parts of the model were meshed with linearly increasing mesh size as a b) 5 mm V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 mm x V17 function of the distance from the shear surface, with the coarsest element size being 3 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' As a result, the model was meshed into 20,240 triangular elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' These elements were assigned with the calibrated numerical properties (Table 1&2), while the shear boxes had properties of stainless steel (Young’s modulus at 200 GPa, density at 8100 kg/m3, and Poisson’s ratio at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Table 1 Laboratory measured macromechanical properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', calibration targets) and emergent properties of the calibrated FDEM model (after Tatone and Grasselli, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Zhao, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Properties (unit) Laboratory measurement Calibrated FDEM model Density (kg·m−3) 1704 1704 Young’s modulus (GPa) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='0 Poisson’s ratio (-) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='24 Internal friction angle (Degrees) 23 23 Internal cohesion (MPa) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='4 Tensile strength (MPa) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='7 Uniaxial compressive strength (MPa) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='9 FDEM models synthesize the macroscopic behavior of materials from the interaction of the micromechanical constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The overall deformation and failure behavior of the simulated material are controlled by the combined effect of the input parameters defining the elastic triangular elements and CCEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' As a result, the macroscopic mechanical properties (as listed in Table 1, except for the density that needs no calibration) measured by standard laboratory tests cannot be used directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Rather, an iterative calibration approach is carried out to obtain input parameters representative of the material, and the laboratory measured properties were used as the calibration targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In this approach, numerical compressive and tensile strength test models are created and simulated using an initial set of input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The macroscopic mechanical properties and failure patterns are obtained from the simuation and compared against laboratory 18 measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In a successful calibration, the numerical model will replicate both the macroscopic mechanical properties measured from the experiments and the overall failure mode of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' If the simulation result is inadequate, the input parameters are iteratively fine-tuned until the calibration targets are met (Tatone and Grasselli, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The laboratory-measured properties and the emergent macromechanical properties of the calibrated FDEM model are listed in Table 1, and the calibrated FDEM model parameters are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Table 2 Calibrated FDEM model input parameters (after Zhao, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Parameter (unit) Value Continuum triangular elements Density, ρ (kg m−3) 1704 Young’s modulus, E (GPa) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='6 Poisson’s ratio, υ ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='22 Viscous damping factor, α 1 Cohesive crack elements Internal cohesion, c (MPa) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 Tensile strength, σt (MPa) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='55 Friction angle, ϕ (Degree) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 Mode I fracture energy, GIc (J·m−2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='8 Mode II fracture energy, GIIc (J·m−2) 90 Fracture penalty, Pf (GPa) 156 Normal contact penalty, Pn (GPa) 156 Tangential contact penalty, Pt (GPa) 156 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 Simulation procedure and boundary conditions The simulation was computed using the Irazu FDEM software (Geomechanica Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2021) with GPU (graphics processing unit) parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The shear test simulation was conducted in three phases (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In phase 1, the initial normal stress was applied by compressing the sample at a constant vertical velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 m/s until the vertical stress reaches 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 MPa, which 19 corresponds to the initial normal stress condition of the laboratory experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In phase 2, the top and bottom boxes were constrained to their vertical position, and the horizontal shear velocity was increased gradually to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='3 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' This transition phase allows the oscillation induced by the instantaneous stop of normal loading to dampen oscillations due to the shear acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In phase 3, the top and bottom boxes were fixed in their vertical positions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', this is a constant normal stiffness shear test) and moved in the horizontal direction at a constant velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='3 m/s until the desired shear displacement of 3 mm was reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Note that the loading velocities used in the study are significantly higher (1000 times) than those used in laboratory experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' however, such a speed has been verified to provide a quasi-static loading condition while allowing a reasonable computation time (Mahabadi, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The model has 26 million simulation time steps, and each step represents a simulation time of 4×10−10 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Table 3 Simulation phases and boundary conditions applied to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The applied velocities in the x (vx) and y (vy) directions, and the resultant shear displacement (u) are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Phase Simulation steps vx (m/s)[1] vy (m/s)[2] u (mm) 1 1–66,400 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='1 0 2 66,401–964,000 0–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='15[3] 0 0–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='02 3 964,000–26,000,000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='02–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='02 [1] Positive (→) on the top box and negative (←) on the bottom box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' [2] Negative (↓) on the top box and positive (↑) on the bottom box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' [3] Linearly interpolated every time step to ramp up the shear velocity gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Normal and shear stresses were measured along a line parallel to the fault and placed 5 mm above the rigid box in the bottom sample (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' This measurement line monitored the stress conditions every 13,000 simulation steps, equivalent to a 200 kHz monitoring rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The recorded stress values in all elements along the measurement line were averaged to obtain the overall normal 20 stress (σn) and shear stress (τ), which were used to calculate the friction coefficient μ = τ/σn, similar to the laboratory-measured apparent friction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 3 Results and data analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='1 Shear behavior Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 6 Calculated friction coefficient of (a) the laboratory test results plotted as a function of the equivalent slip distance at a radius of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='6 mm and (b) the numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The first ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='3 mm are detailed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Red arrows indicate significant drops of frictional resistance associated to seismic events 1, 2, and 3, which are investigated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The simulated μ showed a similar trend with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' It reached the peak value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='22 at a shear displacement of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='32 mm, followed by a significant drop (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The simulated μ experienced many abrupt drops during the slipping process and then stabilized at approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='04 after approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='7 mm of shear displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The simulated μ was significantly lower than the value reported in the laboratory experiment, with many more oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' a) Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='0 Equivalent slip distance (mm) b) Simulation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='1 0 Event 2 & 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='1 Event 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 7b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 3 Shear displacement (mm)21 The simulated stress conditions of the first 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='3 mm showed intriguing similarities to the laboratory experimental data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 7a&b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In this interval, the shear behavior observed in the experiment can be divided into four stages (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 7a): (I) τ, σn, and the resultant μ ramped up gradually;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (II) τ experienced a relatively stable stage with minor change, and σn decreased continuously, causing minor change of μ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (III) τ and σn gradually increased to a peak shear stress, and μ increased to the peak value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' and (IV) τ, σn, and μ dropped rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 7 Comparison of the overall normal and shear stresses and the friction coefficient of stages I- IV between (a) the experimental data and (b) the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (c) and (d) are the zoom in views of the local shear and normal stresses, respectively, at the asperity responsible for the stress drop at stage IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Orange circles numbered 1-6 indicate the horizontal shear displacements (u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (e) and (f) are the micro-CT image of the laboratory specimen corresponding to frame 1 and 6 in (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The initial surface profiles from the surface scan data (red curves) are placed next to the laboratory fault for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' a)Experiment b) Simulation II III IV II III IV 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='6 T(MPa) 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='4 (MPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 0 0 3 On (MPa) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 6 tttttt!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='1 ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 、.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 0 e) 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='3 Shear displacement (mm) Shear displacement, u (mm) 1 u=0mm ②u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='092mm ③ u= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='136 mm 4u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='183mm 5 u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='320mm 6u=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='321mm 5mm Localshear/normalstress(MPa) 0 5 10 2022 The numerical simulation qualitatively captured the general trend of these stages (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 7b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' however, simulated σn in stage I decreased gradually, and more oscillations are observed in the curves in the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' To further investigate the mechanisms behind the shear behavior during these stages, we examined the simulated local stress conditions around the asperity whose breakage was responsible for the large and sudden drop of frictional resistance at stage IV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 7c&d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' During stage I, the simulation shows that the shear surface is at the initial contact condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' As the shear displacement increases, the shear stress increases gradually due to frictional resistance of the initial contact area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Note that the numerical model did not capture the minor normal stress increase measured in the experiment at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Such an increase may be related to the interaction of the asperities in the direction perpendicular to shear (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', out-of-plane motion) that does not exist in the 2D simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' During stage II, the top and bottom surfaces adjusted to a more conforming contact, which resulted in the decrease of the shear and normal stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' During stage III, new contact points were established, and asperities engage and interlock, causing the shear stress to increase rapidly reaching the peak shear stress at the end of this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Asperities survived and climbed onto each other, causing dilation that increased the normal stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' At stage IV, the highly stressed asperity underwent high-stress concentration and failure, releasing the accumulated strain energy that resulted in the sudden and significant drop of stresses and frictional resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The simulated failure pattern, in terms of location and mechanism, resembled the laboratory observation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 7e&f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' More importantly, the numerical model can provide the evolution of surface contacts and stress conditions throughout the shear process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 Progressive damage, gouge formation, and seismic activity Progressive damage on the shear surface and fault gouge formation was simulated by BCCEs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The first several BCCEs occurred when the top and bottom semi-sample were 23 loaded with the initial normal stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Before ⁓1 mm of shear displacement, the damage was concentrated in the vicinity of the shear surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' After ⁓1 mm of shear displacement, a number of sub-vertical fractures penetrated the sample body, resembling the fracturing observed in the laboratory (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The distribution of the shear-induced damage was mostly concentrated close to the fault surface and heterogeneously distributed along the fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Broken asperities formed the gouge layer that accumulated between the semi-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' As a result, some portions of the bare fracture surface were protected from wearing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 9a), and this phenomenon is also observed in the laboratory micro-CT image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 24 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 8 Damage of the shear surface and the accumulation of gouge material with increasing shear displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Damage is represented by broken cohesive crack elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 9 (a) Zoom-in view of a portion of the simulated fault surface at 3 mm of slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The dashed red lines highlight intact fault walls that were not damaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (b) Zoom-in view of the micro-CT image of a portion of the laboratory fault at a similar location to (a) (adopted from Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 10 Simulated seismic activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (a) Magnitude, location, and failure mode of the clustered seismic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (b) Event count in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' A total of 7,557 CCEs were broken throughout the simulation, and they were clustered into 1,561 seismic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Most of the BCCEs near the shear surface failed in shear mode (Mode II), and almost all sub-vertical fractures propagated in tensile mode (Mode I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The magnitude of these seismic events ranged between −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='1 and −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='4, with an average magnitude of −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In general, a) 1mm Intact surface Intact surface Gouge Gouge Shear Shear induced fracture induced fractures mmMode I IH 20 30 35 40 2026 large magnitude events were mostly produced by shear-mode failures, while small magnitude events mostly arose from tensile-mode failures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 10a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Along the vertical direction (y direction), the spatial distribution of seismic activity coincides with the damage pattern: events were concentrated within ±4 mm of the fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' We divided the horizontal length of the fault, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', the x direction, into 100 bins and examined the spatial distribution of the seismic events along such a path (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 10b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Seismic events were distributed heterogeneously along the fault: bins at x = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='8 mm and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='9 mm had the largest number of events at 44;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' bins at x ranging 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='2 mm and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 to 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='0 mm had no seismic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Each asperity failure resulted in the sudden and significant drop of frictional resistance and the release of accumulated strain energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' These large magnitude events caused stick-slip-like responses and released high amplitude stress waves propagating across the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Prior to these dynamic seismic events, their corresponding locations experienced low shear velocity due to the interlocking of asperities and are referred to as interlocking zones (ILZs) in the following discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Stress concentrated at ILZs and eventually broke the asperities, releasing the accumulated strain energy (see animated figures Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' S1 in Supplementary Material for the velocity fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Three seismic events (Events 1-3, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 6) with distinct wave radiation patterns are chosen as examples for further examination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Event 1 at u⁓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='3 mm was related to the most significant friction drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Events 2 and 3 were two consecutive events that occurred on the slipping surface 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 mm apart from each other with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='005 ms time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' We examined the stress field (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 11) and observed that the magnitude of seismic events was directly correlated to the magnitude of the stress concentration at the asperities that failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' We observed that stress concentration at the ILZs reached values as high as the compressive strength of the material, causing compressive failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Due to interlocking, the non-interlocking regions slightly ahead (with 27 respect to the shear direction) of the ILZs were subjected to significant tensile stress that reached the tensile strength of the material, thus, causing tensile fracturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' By examining the particle velocity field (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 12), we found that prior to the seismic events, the locations of the ILZ were experiencing particle velocities lower than the loading velocity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='1 m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' As the seismic events occurred, the source region had particle velocities that were two orders of magnitude higher than that of the ILZs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', > 10 m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Interestingly, considering that P- and S-wave velocities are 2967 m/s and 1884 m/s, respectively, Event 3 occurred right after the arrival of the P-wave induced by Event 2, but prior to the arrival of the S-wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Therefore, Event 3 may have been triggered by the stress perturbation from Event 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 11 Output frames of the numerical model showing the horizontal stress σxx at (a-c) Event 1 and (d-g) Events 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Interlocking zones (ILZs) that are related to the selected seismic events are highlighted by the yellow arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Note that the simulation time interval between two frames is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='052 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' evenl erem28 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 12 Output frames of the numerical model showing the particle velocity at (a-c) Event 1 and (d-g) Events 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' ILZs are highlighted by the yellow arrows and the P-wave wavefront of Event 2 is labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Note the color map is in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 4 Discussion The FDEM numerical model qualitatively captured the mechanical behavior observed in the laboratory experiments, highlighting the dominant role of surface roughness on the shear behavior of rocks at low-stress conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Both the laboratory experiment and the numerical simulation show a slip weakening behavior where the friction coefficient ramps up to the peak value and then decreases to a residual value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In the numerical simulation, the shear stress and friction reached a steady-state and residual value around ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='7 mm of total displacement, in agreement with the laboratory value (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' During the first ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='3 mm of shear displacement, the discrepancies between the experimental and simulation results in stage I and the additional stress oscillations in the simulation results may be because the 2D model was not able to capture 3D asperity interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' However, the overall variation trend of stresses and the damage pattern on the shear surface showed close Event!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' front ILZ Eyent2 Event329 similarities, suggesting that the 2D profile that we used is a proxy for the laboratory specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' This also supports our previous interpretation that the highest asperity was responsible for the formation of the large secondary sub-vertical fractures and the associated sudden drop in shear resistance (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' These results suggest that our numerical technique, which uses a combination of surface scanning, X-ray micro-CT imaging, and FDEM modelling, represents a promising approach to simulate realistic fault behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Our simulation provides the continuous evolution of contacts on the shear surface and the stress conditions that complement the laboratory observations in achieving a better comprehension of how the interaction between asperities controls the stress conditions and damage patterns in faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' During the shear process, asperities interact in various modes including climbing onto each other, interlocking, and breaking (Scholz, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Our experimental and numerical results show that such interactions directly influenced the stress conditions and damage patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' When the slip displacement is small (u < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 mm), weak asperities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', millimetric scale unevenness) controlled the frictional behavior, creating gouge material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' These observations agree with the laboratory observations on the post-mortem sample and suggest the importance of surface roughness in controlling the formation of the gouge layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' As the slip displacement increases (u > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='5 mm), the large-scale roughness of the shear surface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', centimetric scale waviness) becomes important to the shear behavior and damage pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Large scale waviness causes high stress concentration through interlocking and climbing and may cause sub-vertical secondary fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The damage and seismic event distributions are closely related to the stress heterogeneity on the shear surface caused by the surface roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Depending on the geometry of the asperity, the stress concentration at the ILZs could reach the compressive strength of the material, causing compressive failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' This mechanism creates gouge material in the vicinity of the shear surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 30 On the other hand, the areas ahead of the ILZs experience tensile stress up to the tensile strength of the material, thus, creating tensile fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' This mechanism creates large sub-vertical secondary fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Breakage of strong asperities release the accumulated strain energy in the whole model, causing an overall shear stress drop, giving a stick-slip-like shear behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Such a lock-and-fail mechanism is recently found to be the key process of stick-slip behavior of bare surfaces (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Morad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Note that the overall shear loading in our model is considered quasi-static, but the local seismic events are dynamic activities with particle velocity more than 100 times the quasi-static loading velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' This suggests that on a rough shear surface, quasi-static shear consists of numerous heterogeneously distributed local dynamic seismic activities, and this process may complicate the slip process on rough faults and the estimation of the energy budget (Tinti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Observations on Events 2 and 3 suggest that the stress perturbation from asperities breakage may trigger events on adjacent interlocking zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' From an earthquake perspective, there are two possible mechanisms that may trigger seismic events in the near field: (1) static stress redistribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Toda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 1998) and (2) dynamic stress wave perturbation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', Kilb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Gomberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In our simulation, the modeled body did not slip as a rigid body, rather, the slipping consisted of pulses of local movements, accompanied by numerous continuously changing of contacts and asperities breakages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' When the asperity associated to Event 1 breaks, the dynamic stress perturbation was damped out, and the static stress concentration is transferred to nearby asperities, which eventually caused failure of other asperities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' On the other hand, Events 2 and 3 showed a more interesting correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Event 3 occurred between the arrival times of P- and S-waves from Event 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Within this time window, stress redistribution had not reached a steady state, suggesting that the perturbation of the dynamic stress 31 wave radiated from Event 2 may have triggered Event 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' These results imply that static stress transfer and dynamic stress perturbation triggering may occur on the same fault and contribute to the movement of fault slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' However, due to the limitation of the model output frequency and post- processing method, the triggering is not conclusive, Event 2 and 3 may have been independent seismic events occurred in a narrow time window, and more investigation is needed in future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The numerical simulation has the advantage of continuously modeling the fault shear process, fault surface damage, and associated stress conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' However, the simulated sample experienced more damage than the laboratory sample, which is probably related to the limitation of 2D simulations not accounting for the motion in the third dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' For the same reason, the simulated stresses suffered significant fluctuations, and the friction coefficient was much lower than the experimental measurement, which is a common limitation of 2D simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The laboratory experiment by Frye and Marone (2002) and the numerical simulation by Hazzard and Mair (2003) demonstrated that 2D numerical models exhibit friction values notably lower than 3D models and suffer from greater stress fluctuations due to the lack of particle motion in the third dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' In addition, we meshed the shear surface at a relatively high resolution (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='1 mm), resulting in a large number of asperities at various sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Hence, the interlocking and breakage of these asperities caused stress oscillations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', microseismic events).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Even though we qualitatively captured the shear behavior that matches the laboratory measurements, to fully capture the shear behavior of the rotary shear experiment, a 3D model capturing the surface geometry and asperity interaction on the entire shear surface will be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 5 Conclusion 32 In this study, we used a carefully built and calibrated FDEM numerical model to simulate a laboratory shear experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' We introduced a new clustering algorithm to improve the understanding of the simulated fracturing and associated seismic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The model was able to qualitatively capture the frictional behavior observed in the laboratory experiment, providing the missing information in the experimental observation regarding the continuous variation of stresses and the progressive evolution on the shear surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Our numerical model matches the experimental results particularly well at the beginning of the shear deformation (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='3 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' We were able to identify similar stress variation trends and damage patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The simulation results provided detailed evolution processes of the contacts on the shear surface and the local stress conditions, which are not available in experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Combining the numerical and experimental results, we conclude that interlocking of asperities can cause compressive stress concentration on the front side (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', facing the shear direction) of the asperity, which could induce compressive failure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', crushing) near the shear surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' on the other hand, tensile stress concentration is generated on the leeward side of the asperity, which could cause sub-vertical tensile fractures that could propagate into the host rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Progressive surface damage and the associated microseismic events occur at the locations of asperity interactions and is highly heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Several locations experienced no damage even after large shear displacement, these locations are either not in contact or were protected by gouge materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' As a result of the interlocking and breakdown of asperities, local dynamic failure events occur, even though the overall loading is quasistatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' These events are considered microseismic events, and their magnitudes range between −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='1 and −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Strain energy stored in the medium was released during these events, causing dynamic perturbation to the overall stress condition, and 33 the particle velocity in the source reached > 10 m/s, two orders of magnitude larger than the surrounding regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' This high amplitude stress perturbation could even trigger the failure of adjacent critically stressed asperities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Both the numerical model and the experiment suggested the importance of shear surface roughness in controlling slip behavior, and we were able to explain the laboratory observations with the help of numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Shear surface evolution is a complicated process that involves frictional sliding, fracturing, gouge comminution, and seismicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The high degree of agreement between simulation and experiment data leads to a promising future of predicting fault behavior through, laboratory testing, surface characterization, and numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' These results improved the understanding of shear behavior and demonstrated that micromechanical based numerical simulation is a capable approach to study fault mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Acknowledgements Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Zhao is supported by the FCE Start-up Fund for New Recruits at the Hong Kong Polytechnic University (Project ID P0034042) and the Early Career Scheme of the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' PolyU 25220021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' This work has also been supported through the NSERC Discovery Grants 341275, CFILOF Grant 18285, Carbon Management Canada (CMC), and NSERC/Energi Simulation Industrial Research Chair Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The authors would like to thank Geomechanica Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' for providing the Irazu FDEM simulation software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Zhao would like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Andrea Lisjak and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Bin Chen for 34 discussions and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' The authors appreciate the constructive suggestions and comments from the editor and the reviewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' References Abdelaziz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', & Grasselli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Grain based modelling 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of hydraulic fracturing and associated microseismicity using finite-discrete element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Journal of Rock Mechanics and Geotechnical Engineering, 6, 574–581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', Tisato, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', Grasselli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', Mahabadi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', Lisjak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', & Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Influence of in situ stress variations on acoustic emissions: a numerical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Geophysical Journal International, 203, 1246–1252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', Tisato, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=', & Grasselli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Rotary shear experiments under X-ray micro- computed tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Review of Scientific Instruments, 88(1), 015110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' 38 Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Tisato, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Kovaleva, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' and Grasselli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Direct Observation of Faulting by Means of Rotary Shear Tests Under X-Ray Micro-Computed Tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Journal of Geophysical Research, 123(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Glaser, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Tisato, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' and Grasselli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Assessing Energy Budget of Laboratory Fault Slip Using Rotary Shear Experiments and Micro-Computed Tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} +page_content=' Geophysical Research Letters, 47(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE2T4oBgHgl3EQfpwjv/content/2301.04033v1.pdf'} diff --git a/LdE0T4oBgHgl3EQfzwKd/content/tmp_files/2301.02677v1.pdf.txt b/LdE0T4oBgHgl3EQfzwKd/content/tmp_files/2301.02677v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a7ddc2de888933dca8643c998424168e0b4a47f6 --- /dev/null +++ b/LdE0T4oBgHgl3EQfzwKd/content/tmp_files/2301.02677v1.pdf.txt @@ -0,0 +1,1135 @@ +Resonant inelastic X-ray scattering in topological semimetal FeSi +Yao Shen,1 Anirudh Chandrasekaran,2, 3 Jennifer Sears,1 Tiantian Zhang,4, 5 Xin Han,6 Youguo Shi,6 +Jiemin Li,7 Jonathan Pelliciari,7 Valentina Bisogni,7 Mark P. M. Dean,1, ∗ and Stefanos Kourtis8, 2, † +1Condensed Matter Physics and Materials Science Department, +Brookhaven National Laboratory, Upton, New York 11973, USA +2Department of Physics, Boston University, Boston, MA, 02215, USA +3Department of Physics and Centre for the Science of Materials, +Loughborough University, Loughborough LE11 3TU, UK +4Department of Physics, Tokyo Institute of Technology, Okayama, Meguro-ku, Tokyo, Japan +5Tokodai Institute for Element Strategy, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama, Kanagawa, Japan +6Beijing National Laboratory for Condensed Matter Physics, +Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China +7National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA +8Institut quantique & D´epartement de physique, Universit´e de Sherbrooke, J1K 2R1, Qu´ebec, Canada +(Dated: January 10, 2023) +The energy spectrum of topological semimetals contains protected degeneracies in reciprocal space that corre- +spond to Weyl, Dirac, or multifold fermionic states. To exploit the unconventional properties of these states, one +has to access the electronic structure of the three-dimensional bulk. In this work, we resolve the bulk electronic +states of candidate topological semimetal FeSi using momentum-dependent resonant inelastic X-ray scattering +(RIXS) at the Fe L3 edge. We observe a broad excitation continuum devoid of sharp features, consistent with +particle-hole scattering in an underlying electronic band structure. Using density functional theory, we calculate +the electronic structure of FeSi and derive a band theory formulation of RIXS in the fast collision approximation +to model the scattering process. We find that band theory qualitatively captures the number and position of the +main spectral features, as well as the overall momentum dependence of the RIXS intensity. Our work paves the +way for targeted studies of band touchings in topological semimetals with RIXS. +I. +INTRODUCTION +In the last decade and a half, topological matter has become +a cornerstone of quantum materials science [1]. The discovery +of three-dimensional topological insulators [2], in particular, +sparked a flurry of activity in the then nascent field. Elec- +trons in these crystalline materials are effectively noninteract- +ing, giving rise to electronic bands in the bulk that are indis- +tinguishable from those of a trivial band insulator. The elec- +tronic wavefunction, however, is characterized by topologi- +cal indices that dictate the presence of symmetry-protected +Dirac states at the surface of the material, as well as nontrivial +(magneto-)transport responses. +More recently, topological semimetals have been added +to the catalogue of three-dimensional topological materi- +als [3, 4]. These systems also feature topologically protected +boundary states and nontrivial (magneto-)transport, but addi- +tionally have distinct geometric characteristics in their bulk +band structure. In the simplest case of Weyl semimetals, these +geometric characteristics are singly degenerate energy sur- +faces in reciprocal space that contain a band touching point—a +Weyl node—around which electronic bands disperse linearly +in all three directions in reciprocal space [5–7]. Such band +touchings are Berry curvature singularities characterized by +topological indices. The value of the topological index of a +nodal point determines the geometry of the band dispersion in +∗ mdean@bnl.gov +† Stefanos.Kourtis@usherbrooke.ca +the vicinity of nodal points [8–10]. Conversely, the geometry +of the bulk bands becomes a proxy of topology in these ma- +terials. Measuring the electronic density of states in the bulk +can therefore reveal the topological nature of a semimetal. +Resonant inelastic X-ray scattering (RIXS) is a spectro- +scopic technique that yields momentum- and energy-resolved +spectra of charge-neutral electronic excitations. While RIXS +has been extensively used in studying magnetic excitations in +gapped materials like insulators and superconductors [11, 12], +it is increasingly applied in studies of compounds that host +itinerant carriers with small or no charge gaps [13–15]. That +RIXS can be used to map electronic bands of materials, in- +cluding semimetals, has long been established [16–23]. Im- +provements in resolution in recent years have renewed in- +terest in using RIXS to detect band structure effects at meV +energy scales in materials of technological interest, such as +unconventional superconductors [24]. There are even theo- +retical proposals to use RIXS to measure topological indices +of nodal points in topological semimetals [25, 26]. +These +prospects are particularly appealing for probing the bulk of +three-dimensional materials, since alternative techniques such +as angle-resolved photoemission (ARPES) and scanning tun- +neling spectroscopy, probe predominantly the surface rather +than the bulk. Furthermore, topological nodal points may only +appear above the Fermi level or as a result of an applied mag- +netic field, settings in which the resolving power of ARPES is +limited. As these settings may be relevant to the technologi- +cal exploitation of topological materials, alternative methods +to visualize the bulk band structure and to identify topologi- +cal features are sought after. Before honing in on properties +of topological origin, however, one has to determine whether +arXiv:2301.02677v1 [cond-mat.str-el] 6 Jan 2023 + +2 +bulk band structure effects at large are detectable in RIXS of +topological semimetals. +In this work, we use RIXS to probe the bulk of FeSi aim- +ing to quantitatively test how bulk band structure manifests +in RIXS spectra of a putative topological semimetal [27, 28]. +We observe broad continua in the RIXS spectra, consistent +with particle-hole scattering in an underlying band structure. +We model the RIXS process in the fast-collision approxima- +tion using the band structure of FeSi as determined by den- +sity functional theory (DFT) calculations. We find reasonable +agreement between experiment and theory in the number and +hierarchy of dominant spectral features. We interpret this find- +ing as evidence of a bulk band structure in FeSi, with many- +body effects playing only a secondary role in determining the +RIXS spectrum. Our results indicate that higher resolution +experiments — feasible with existing instruments — could vi- +sualize topological nodal points and thus identify and classify +topological semimetals. +II. +THEORY +A. +RIXS cross section and fast collision approximation +We briefly introduce the theoretical description of RIXS at +zero temperature. More comprehensive presentations of RIXS +can be found in Refs. 29 and 30. +In a RIXS experiment, core electrons of an ion are pro- +moted to a state above the Fermi level εF by an intense x- +ray beam, thereby locally exciting the irradiated material into +a highly energetic and short-lived intermediate state. Subse- +quently, the core hole recombines with a valence electron. The +process imparts both energy and momentum to particle-hole +excitations in the material. In what follows, we will consider +excitation of core electrons directly into orbital(s) close to εF , +which give rise to the low-energy physics in the material. This +process, which is often referred to as direct RIXS, is illus- +trated in Fig. 1. +The double differential cross section is a measure of the +total RIXS intensity. Up to a constant prefactor, it is given by +I(kin, kout, ωin, ωout, ϵin, ϵout) += +� +fg +|Ffg(kin, kout, ωin, ϵin, ϵout)|2δ(Eg − Ef + ℏ∆ω) , +(1) +where ℏ∆ω = ℏ(ωin − ωout) is the energy transferred to the +material, kin and kout (ϵin and ϵout) the incoming and outgoing +photon wavevectors (polarizations) and Eg and Ef the ener- +gies corresponding to initial and final many-body states |g⟩ +and |f⟩ of the valence electrons. The scattering amplitude +Ffg in the dipole approximation is +Ffg(kin, kout, ωin, ϵin, ϵout) += ⟨f| �D†(ϵout, kout) �G(ωin) �D(ϵin, kin)|g⟩ , +(2) +where �G is the intermediate-state propagator +�G(ωin) = (Eg + ℏωin + iΓ − � +H)−1 , +(3) +with � +H the Hamiltonian describing the system in the interme- +diate excited state and Γ the intermediate-state inverse life- +time. The dipole operators �D and �D † represent the x-ray ab- +sorption and emission, respectively. For a crystalline material, +they can be written as +�D(ϵ, k) = ϵ · �Dk , +(4) +�Dk = +� +µ,ν +⟨µ|�r − rµ|ν⟩ +� +R +eik·R �d † +Rµ �pRν , +(5) +where R is the lattice position. States |µ⟩ and |ν⟩ express +single-electron valence and core states respectively. The com- +bined valence (core) index µ (ν) encodes spin, orbital, and +sublattice degrees of freedom. +Core states |ν⟩ are conve- +niently expressed as atomic orbitals, whereas valence states +|µ⟩ can be appropriately chosen Wannier functions, both lo- +calized in space around the same position rµ of each atomic +site within the unit cell. The position operator �r, defined with +respect to each lattice position R, is the same for all ions. +For the L2/3 and M2/3 resonant edges, the operators �d † +Rµ and +�pRν create a d-orbital electron and a p-orbital core hole, re- +spectively. +Physical arguments allow us to simplify the RIXS scatter- +ing amplitude. First, core holes do not hop appreciably; they +are created and annihilated at the same site. Taking this into +account, Ffg becomes [30] +Ffg(q, ωin, ϵin, ϵout) += +� +µ,ν,µ′,ν′ +Tµνµ′ν′(ϵin, ϵout)Fµνµ′ν′(q, ωin) , +(6) +where q = kin − kout. The scattering amplitude has been +factored in the atomic scattering tensor +Tµνµ′ν′(ϵin, ϵout) = ⟨µ|ϵout · �r|ν⟩∗⟨µ′|ϵin · �r|ν′⟩ +(7) +and the fundamental scattering amplitude +Fµνµ′ν′(q, ωin) += ⟨f| +� +R +e−iq·R �p † +Rν �dRµ �G(ωin) �d † +Rµ′ �pRν′|g⟩ . +(8) +The intrinsic spectral characteristics of a material are carried +by the tensor F, which is typically the main quantity of inter- +est in theoretical studies. The tensor T modulates the scatter- +ing amplitude according to the geometry of the localized core +and valence states. The entries of T can be calculated given +knowledge of the valency of the targeted ion and the symme- +try group of the crystal [31–34]. +Then, within the fast-collision approximation, one assumes +that �G(ωin) ≈ 1/Γ, where Γ is the inverse core-hole lifetime. +In this approximation, the RIXS process reduces to the intro- +duction of a particle-hole excitation with fixed momentum and +energy in the material — see Fig 1 for an example. +Before proceeding to derive the theory of RIXS in band +structures, we evaluate the geometric modulation of the RIXS +spectrum owing purely to the orbital content of the quantum + +3 +εk +k +εF +energy +core level +εF +εk +k +INITIAL +energy +core level +εF +εk +k +FINAL +fast +collision +approximation +FIG. 1. Illustration of the direct RIXS process and reduction to effective particle-hole scattering via the fast-collision approximation. +states involved in the RIXS process. This is obtained by set- +ting the fundamental scattering amplitude Fµνµ′ν′ to unity: +T (ϵin, ϵout) = +������ +� +µ,ν,µ′,ν′ +Tµνµ′ν′(ϵin, ϵout) +������ +2 +. +(9) +We shall use T as a diagnostic to disentangle contributions to +the modulation of the RIXS intensity as a function of scatter- +ing angles. The first contribution comes through the polariza- +tion vectors, which are angle-dependent — see Fig. 2. The +second contribution is the intrinsic momentum dependence +coming from electronic dispersion in the material. In Sec. V +we calculate the geometric modulation in Eq. (9) for FeSi and +compare it to the scattering angle dependence of RIXS inten- +sity. +B. +RIXS process in a band structure +We wish to describe the RIXS response of crystalline mate- +rials in which electrons are, to a good approximation, non- +interacting. +Valence electrons in these materials are well- +described by band theory. The states |g⟩ and |f⟩ in Eq. (2) +are then collections of Bloch modes. +For a given RIXS edge, one then sums over core states +|ν⟩ and valence Wannier states |µ⟩ and +��µ′� +connected by the +dipole operators �D, �D†. Here we study the Fe L3 edge, hence +we consider 2p3/2 orbitals for core electrons and the 3d shell +for valence electrons. +In k-space, the band eigenbasis is given by a unitary rota- +tion of a basis of Wannier states |µ⟩ per lattice position R to a +basis of Bloch states |kµ⟩. The Wannier states have wavefunc- +tions ϕµ(x) = ⟨x|µ⟩ that are centered about different points +in the unit cell, possibly atomic sites. Let ϕkµ(x) = ⟨x|kµ⟩ +be the spatial wavefunction of |kµ⟩, which could be a spinor. +We then have +ϕkµ(x) = +1 +√ +N +� +R +eik·Rϕµ(x − R) . +(10) +The raising and lowering operators of the Bloch wavefunc- +tions are �d † +kµ and �dkµ. They are defined by �d † +kµ|Ω⟩ = |kµ⟩, +{�dkµ, �dk′µ′} = 0 and {�dkµ, �d † +k′µ′} = δk,k′δµ,µ′, where |Ω⟩ is +the vacuum of valence excitations and core holes. A general +Hamiltonian describing noninteracting valence electrons is +� +Hband = +� +k∈BZ +� +µ,µ′ +�d † +kµ Hµµ′(k) �dkµ′ , +(11) +where Hµµ′(k) are the elements of the matrix H(k). +Let U(k) be a matrix that diagonalizes H(k), such that +U †(k)H(k)U(k) is a diagonal matrix containing the eigen- +values εl(k), which constitute the dispersing bands. We can +then write +�dkµ = +� +l +Uµl(k) �ψkl, +(12) +where l denotes an energy band index and �ψkl annihilates the +corresponding eigenstate. The ground state at zero tempera- +ture is obtained by populating all states below the Fermi level: +|g⟩ = +� +� � +l,k∈BZ +Θ(εF − εl(k)) �ψ† +kl +� +� |Ω⟩ , +(13) +with Θ the Heaviside step function. The Wannier lowering +operators at any lattice site R can be expressed in terms of the +band operators as +�dRµ = 1 +√ +N +� +k∈BZ +e−ik·R �dkµ +(14a) += 1 +√ +N +� +l,k∈BZ +e−ik·R Uµl(k) �ψkl. +(14b) +We now assume, as per the fast collision approximation, +that the intermediate-state Hamiltonian is well approximated +by the band Hamiltonian, along with a core hole inverse life- +time Γ in the intermediate-state propagator. Due to this as- +sumption, core electron operators cancel out and the interme- +diate state propagator becomes simply +�G(ωin) =(Eg + ℏωin + iΓ − � +Hband)−1 , +(15a) += +� +k +� +l +|k, l⟩⟨k, l| +Eg + ℏωin + iΓ − εl(k) . +(15b) + +4 +where |k, l⟩ are band eigenstates, and we treat Eg and Γ as +free parameters to be determined by fitting the x-ray absorp- +tion spectrum (see App. B). +Substituting the expression for �dRµ in Eq. (14b) in Eq. (8), +we obtain the fundamental RIXS scattering amplitude in a +band structure +Fµµ′(q, ωin) = ⟨f| +� +k,k′∈BZ +� +l,l′ +1 +N +� +R +e−i(k+q−k′)·R +× Uµl(k) U ∗ +µ′l′(k′) �ψkl �G(ωin) �ψ † +k′l′|g⟩ . +(16) +Notice that F is independent of the core orbitals at this level +of description. The sum over R evaluates to Nδk′,k+q, which +enforces k′ = k + q. When εl′(k + q) > εF, we have that +�ψ † +k+ql′|g⟩ is an eigenstate of the band Hamiltonian with en- +ergy Eg + εl′(k + q) (otherwise the single particle state is +already occupied and this term evaluates to zero). The action +of �G(ωin) on �ψ † +k+ql′|g⟩ is +�G(ωin) �ψ † +k+ql′|g⟩ = +Θ(εl′(k + q) − εF) +ℏωin − εl′(k + q) + iΓ +�ψ † +k+ql′|g⟩. +(17) +Furthermore, the action of �ψkl on �ψ † +k+ql′|g⟩ is non-zero only +if εl(k) < εF (we need this single particle level to be occupied +for the term to be non-zero). Using this we obtain +Fµµ′(q, ωin) = +� +l,l′ +� +k∈BZ +� +⟨f| �ψkl �ψ † +k+ql′|g⟩ +× Θ(εl′(k + q) − εF) +× Θ(εF − εl(k)) +× +Uµl(k) U ∗ +µ′l′(k + q) +ℏωin − εl′(k + q) + iΓ +� +. +(18) +The sum over final states |f⟩ can be taken over the eigenstates +of � +Hband. The pair of step functions in the fundamental scat- +tering amplitude given above in Eq. (18) ensures that there +is a unique |f⟩ that makes the inner product ⟨f| �ψkl �ψ † +k+ql′|g⟩ +non-zero, since the role of the operator pair is to simply cre- +ate particle-hole excitations across the Fermi level. Thus, the +final sum over |f⟩ can be replaced as +� +f +→ +� +l,l′ +� +k∈BZ +Θ(εl′(k + q) − εF) Θ(εF − εl(k)). +(19) +This corresponds to summing over final states with one +particle-hole excitation in the valence bands. The inner prod- +uct is then redundant and can be removed. +The final form of the RIXS intensity for systems well- +described by band theory is +I(q, ωin, ∆ω, ϵin, ϵout) = +� +l,l′ +� +k∈BZ +Θ(εl′(k + q) − εF) Θ(εF − εl(k)) +× +������ +� +µ,ν,µ′ +⟨µ|ϵout · �r|ν⟩∗⟨µ′|ϵin · �r|ν⟩ +Uµl(k) U ∗ +µ′l′(k + q) +ℏωin − εl′(k + q) + iΓ +������ +2 +× +η +[εl(k) − εl′(k + q) + ℏ∆ω]2 + η2 , +(20) +where we have replaced the Dirac δ-function with a +Lorentzian of peak broadening η to represent finite experi- +mental resolution. +With respect to a local set of coordinate axes, the incoming +X-ray polarization ϵ has components (ϵx, ϵy, ϵz). The outgo- +ing polarization is usually not measured, and hence one sums +over either polarizations parallel to and perpendicular to the +scattering plane or over left, linear, and right polarizations. +We list the polarization matrix elements for the specific case +of the L3 edge of a 3d transition-metal in Table I. +III. +EXPERIMENTAL METHODS +A. +Sample preparation +Single crystals of FeSi were prepared using a Ga flux +method. +We mixed the starting materials in a molar ratio +of 1:1:15 in a glove box filled with argon. +This mixture +was placed in an alumina crucible and sealed in an evacuated +quartz tube. The crucible was heated to 1150◦C and held for +10 h, before cooling to 950◦C at 2 K/h, after which the flux +was centrifuged. The crystals were washed with diluted hy- +drochloric acid in order to remove Ga flux from the surface of +the samples. + +5 +B. +RIXS and experimental geometry +FIG. 2. Schematic of the RIXS setup. kin and kout respectively +denote the ingoing and outgoing scattering vectors. The components +of the ingoing and outgoing photon polarization within the scattering +plane are denoted by πin and πout while the σ polarization direction +is the same for both. The incident angle θi is measured with respect +to the sample surface, that is the a direction in the sample coordinates +while the c direction is the normal to the sample surface. +RIXS measurements were performed at the Soft Inelas- +tic X-Ray (SIX) beamline at the National Syncrotron Light +Source-II (NSLS-II). The energy resolution was 23 meV. The +experimental setup is depicted in Fig. 2. The lab coordinates +are denoted by x, y, z while the crystallographic axes are la- +belled a, b, c. We define the incident and outgoing beam an- +gles with respect to the sample coordinate system, wherein +the c direction is the normal to the sample surface and the ac +plane is the scattering plane. Experimental data are corrected +to account for self-absorption effects. +With θi denoting the incident angle measured with respect +to the sample surface and 2θ denoting the angle between the +incident and outgoing beam, we can easily verify the follow- +ing in the sample frame: +kin = kin(− cos θi, 0, − sin θi) , +(21a) +kout = kout (− cos(2θ − θi), 0, sin(2θ − θi)) , +(21b) +ϵπ,in = (− sin θi, 0, cos θi) , +(21c) +ϵπ,out = (sin(2θ − θi), 0, cos(2θ − θi)) , +(21d) +ϵσ,in = ϵσ,out = (0, 1, 0) . +(21e) +Although in reality the ingoing and outgoing photon mo- +mentum magnitudes kin and kout is different owing to non- +zero energy transfer ∆ω, the difference is negligible since +∆ω ≪ ωin, and hence kin ≈ kout = k. The momentum +transfer to the material q is then +q = k (cos(2θ − θi) − cos θi, 0, − sin(2θ − θi) − sin θi) . +(22) +0 +1 +2 +3 +4 +5 +705 706 707 708 709 710 705 706 707 708 709 710 +Energy Loss (eV) +(a) Experiment +(b) Theory +Incident energy (eV) +705 706 707 708 709 710 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +RIXS (arb. units) +FIG. 3. Color maps of RIXS intensity with 2θ = 150◦ and θi = 68◦ +for π-polarized incident beam as a function of incident photon energy +ℏωin and energy loss ∆ω at the L3 edge of Fe in FeSi as obtained (a) +in experiment and (b) in the band theory formalism of Sec. II B. +In practice the ingoing and outgoing beam directions are set +to specific values, which defines a specific 2θ. By rotating +the sample about the y axis (or equivalently b axis), we can +change q by changing θi . +IV. +DENSITY FUNCTIONAL THEORY AND +TIGHT-BINDING MODEL +The band structure of FeSi was simulated in a similar way +to prior studies of FeSi [35]. We performed first-principles +calculations based on DFT [36] within the Perdew-Burke- +Ernzerhof exchange-correlation [37] implemented in the Vi- +enna ab initio simulation package (VASP) [38]. The plane- +wave cutoff energy was 450 eV with a 9×9×9 k-mesh in the +BZ for self-consistent calculation without considering spin- +orbit coupling. Maximally localized Wannier functions [39] +were used to obtain the tight-binding model of bulk FeSi with +the lattice constants a = b = c = 4.48 ˚A. +V. +RIXS SPECTRUM OF FESI +The RIXS intensity at the Fe L3 edge with π-polarized in- +cident beam is shown in Fig. 3 in the incident energy-energy +loss plane. The absence of prominent sharp inelastic features +suggests a particle-hole continuum, consistent with particle- +hole excitations in a partially filled band structure. +Momentum-resolved RIXS spectra are shown in Fig. 4 for +two values of 2θ. Spectral weight from inelastic processes +lies predominantly in a window of width ∼ 5 eV. Within that +window, all spectra have similar lineshape, featuring a peak +around 2 eV that disperses to higher energies with increasing +θi and a dispersionless shoulder above 3 eV. Overall inelastic +intensity increases with increasing θi for both values of 2θ. +We use the band theory formulation of Sec. II B and +Eq. (20) to theoretically model the RIXS process in FeSi. A +fit of the absorption spectrum (see App. B) yields Γ = 0.8 + +6 +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +Exp +(a) 2� D 150ı +(b) 2� D 70ı +RIXS (arb. units) +�i +�i +Thy +Energy loss (eV) +FIG. 4. Momentum-resolved RIXS spectra at ℏωin = 708.7 eV +with π x-ray polarization for (a) 2θ = 150◦ and (b) 2θ = 70◦ +and comparison to simulations within band theory (bottom pan- +els) using Eq. (20) with Γ = 0.8 eV and ε0 += 707.67 eV. +For scattering angle 2θ = 150◦, the incident angle values are +θi = 10◦, 30◦, 45◦, 68◦, 120◦, while for 2θ = 70◦ we have θi = +10◦, 30◦, 60◦. +eV and Eg = 707.67 eV, and we choose a peak broadening +η = 100 meV. We use a 48 × 48 × 48 grid of k points in the +Brillouin zone for the 32-band tight binding model detailed +above. +The simulated spectra show a structure similar to that ob- +served experimentally, with inelastic weight in a ∼ 5 eV win- +dow containing a peak at ∆ω ∼ 2.5 eV and shoulder at +∆ω ∼ 3.5 eV. As in experiment, overall inelastic intensity +increases with increasing θi for both values of 2θ, though to +a lesser extent. Compared to experiment, features are shifted +to higher energies in simulated spectra, while for 2θ = 70◦ +and θi = 60◦ the main peak subsides, leading to theory and +experimental spectra that look qualitatively different. Finally, +experimental spectra also contain subdominant features close +to the elastic line (∆ω < 1 eV) that, as discussed later, deviate +from the band theory predictions. +To investigate what causes the overall increase in RIXS in- +tensity with increasing θi, we calculate the atomic scattering +tensor (7). From this we obtain the modulation of the RIXS +spectrum (9) coming purely from the orbital content of core +and valence states. After summing over outgoing polariza- +tions, we obtain the behavior shown in Fig. 5. Comparing to +the θi dependence of RIXS spectra in Fig. 4, we see that ge- +ometric considerations are insufficient to explain the momen- +tum dependence of RIXS intensity in experiment: the mo- +mentum dependence of the atomic scattering tensor is differ- +ent from that observed, even showing a reverse trend in the +0 +0.5 +1 +1.5 +2 +2.5 +0ı +30ı +60ı +90ı +120ı 150ı 0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +0ı +30ı +60ı +90ı +120ı 150ı +Orbital RIXS +Incident angle �i +(a) 2� D 150ı +(b) 2� D 70ı +T .�; �/ C T .�; �/ +FIG. 5. The contribution of the dipole matrix elements to the RIXS +spectrum of FeSi for π ingoing polarisation as given by Eq. (9) after +summing over σ and π polarisations of the outgoing beam. +case of 2θ = 70◦. Reinstating the band structure fundamen- +tal scattering amplitudes in Eq. (20) yields the experimentally +observed trend of overall momentum dependence, albeit only +qualitatively, as shown in Fig. 4. +VI. +DISCUSSION & CONCLUSION +We have seen that the theoretical formulation of RIXS +based on band theory captures the overall momentum depen- +dence of the experimental Fe L3-edge RIXS spectra of FeSi +better than a calculation based on just the atomic multiplet. +Band theory also reproduces the right bandwidth for the in- +elastic part of the spectrum, as well as the right number of +features therein, at roughly the right energy. +Discrepancies between theory and experiment exist. While +the overall momentum dependence of the RIXS spectrum is +reproduced by band theory, experimental spectra depend more +sensitively on θi. +Spectral features also do not align per- +fectly between experiment and theory. This includes a fea- +ture around 0.3 eV in the experiment, which is only present +as a weak shoulder in the theory. We discuss potential rea- +sons for these discrepancies. First, the band theory of RIXS +ignores electronic correlations. The extent to which correla- +tions play a role in the 3d shell of FeSi is not clear [40, 41]. A +more detailed theoretical study of the RIXS spectrum would +require identifying the precise type, range, and magnitude of +interactions present in FeSi. Fully incorporating the effects +of interactions in theoretical studies of RIXS is nevertheless a +challenge, since we are dealing with a three-dimensional ma- +terial with 32 relevant orbitals per unit cell. Numerical sim- +ulation of the RIXS spectrum based on dynamical mean field +theory [42] may eventually be up to this task. Second, in the +fast collision approximation we ignore the effects of a finite +core-hole lifetime, which may be appreciable in 3d transition +metal compounds [43]. Future simulations could be improved +by incorporating dynamics and interactions with the core hole +in the intermediate state. +In conclusion, we have reported RIXS spectra of FeSi at + +7 +the Fe L3 edge. We observe an excitation continuum without +sharp features. Through a band theory formulation of RIXS +in the fast collision approximation, we model the RIXS pro- +cess using the ab initio band structure of FeSi. We obtain +reasonable agreement for the spectrum bandwidth, as well as +the number and position of main features. Theory also repro- +duces the dispersion trend of the RIXS spectrum, albeit only +qualitatively. This work paves the path to ever finer resolution +of distinctive band structure features in topological materials +with RIXS. +ACKNOWLEDGMENTS +A.C. was supported by DOE Grant No. +DE-FG02- +06ER46316 and EPSRC grant EP/T034351/1. +S.K. ac- +knowledges support from the Minist`ere de l’´Economie et de +l’Innovation du Qu´ebec and the Canada First Research Excel- +lence Fund. Work at Brookhaven National Laboratory (x-ray +scattering and analysis) was supported by the U.S. Depart- +ment of Energy, Office of Science, Office of Basic Energy +Sciences. This research used resources at the SIX beamline +of the National Synchrotron Light Source II, a U.S. DOE Of- +fice of Science User Facility operated for the DOE Office of +Science by Brookhaven National Laboratory under Contract +No. DE-SC0012704. We acknowledge National Natural Sci- +ence Foundation of China (U2032204), the Strategic Prior- +ity Research Program of the Chinese Academy of Sciences +(XDB33010000) for funding sample synthesis. We thank Yue +Cao, Siddhant Das, Claudio Chamon, Michael El-Batanouny, +Jungho Kim, and Karl Ludwig for useful discussions. +Appendix A: Polarization matrix elements and the atomic +scattering tensor +The DFT derived tight-binding model used for the calcula- +tions presented in the paper involves thirty two basis orbitals +per unit cell of the crystal lattice. Due to the assumption of +zero spin orbit coupling for the valence bands, this gives rise +to thirty-two, two-fold spin-degenerate bands. The orbitals +used are the five 3d orbitals of each of the four Fe atoms and +the three 3p orbitals of each of the four silicon atoms within a +unit cell, giving a total of 32 orbitals per unit cell. +Since the tight binding model is expressed in terms of 3d or- +bitals whose local axes are perfectly aligned with crystal axis +for each of the four Fe atoms in the unit cell, we need to com- +pute the matrix elements of the 2p3/2 → 3d transitions for +just one of the atoms. The 2p orbitals all have the same radial +part of the wavefunction, φ2p(r) and, likewise, the 3d orbitals +have same radial wavefunction φ3d(r). The radial integral of +the various matrix elements in the atomic scattering tensor is +simply the radial integral of the product φ2p(r) · φ3d(r) and +the radial part of the dipole transition operator. Since this is a +common term that just provides an overall multiplicative fac- +tor for the RIXS cross section, we ignore it and compute only +the azimuthal and polar integrals of the matrix elements. We +document the relevant matrix elements of the dipole operator +in Table I, which were verified by comparing to open source +RIXS code EDRIXS [44]. +Appendix B: X-ray absorption spectrum and theoretical fit +To align the experimental RIXS spectra with theoretical re- +sults obtained through ab initio calculations, we need to de- +termine the absolute energy scale Eg of the initial state. We +determine Eg through a fit of the experimental X-ray absorp- +tion intensity with the calculated absorption spectrum +Iabs(q, ωin, ϵin) = +� +ϵout +� +l,l′ +� +k∈BZ +Θ(εl′(k + q) − εF) Θ(εF − εl(k)) +������ +� +µ,ν,µ′ +⟨µ|ϵout · �r|ν⟩∗⟨µ′|ϵin · �r|ν⟩ Uµl(k) U ∗ +µ′l′(k + q) +Eg + ℏωin − εl′(k + q) + iΓ +������ +2 +, +(B1) +which is obtained by integrating over ∆ω the RIXS spectrum +in Eq. (20). In addition to Eg, we consider the core hole in- +verse lifetime Γ as a tunable parameter in the fit. We sum over +outgoing polarizations since the measured spectrum is not po- +larization resolved. Fig. 6 shows the fit that minimizes the +average absolute deviation and yields the values Γ = 0.8 eV +and Eg = 707.67 eV. +[1] M. Z. Hasan and C. L. Kane, Reviews of modern physics 82, +3045 (2010). +[2] M. Z. Hasan and J. E. Moore, Annu. Rev. Condens. Matter +Phys. 2, 55 (2011). +[3] A. Burkov, Nature materials 15, 1145 (2016). +[4] N. P. Armitage, E. J. Mele, +and A. Vishwanath, Rev. Mod. +Phys. 90, 015001 (2018). +[5] J. von Neumann and E. P. Wigner, in Collect. Work. 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Kotliar, Computer +Physics Communications 243, 151 (2019). + diff --git a/LdE0T4oBgHgl3EQfzwKd/content/tmp_files/load_file.txt b/LdE0T4oBgHgl3EQfzwKd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6ebf1e8c0af01c4ae541b07629121b3667946822 --- /dev/null +++ b/LdE0T4oBgHgl3EQfzwKd/content/tmp_files/load_file.txt @@ -0,0 +1,820 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf,len=819 +page_content='Resonant inelastic X-ray scattering in topological semimetal FeSi Yao Shen,1 Anirudh Chandrasekaran,2, 3 Jennifer Sears,1 Tiantian Zhang,4, 5 Xin Han,6 Youguo Shi,6 Jiemin Li,7 Jonathan Pelliciari,7 Valentina Bisogni,7 Mark P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Dean,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' ∗ and Stefanos Kourtis8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' † 1Condensed Matter Physics and Materials Science Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' New York 11973,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' USA 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Boston University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 02215,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' USA 3Department of Physics and Centre for the Science of Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Loughborough University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Loughborough LE11 3TU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' UK 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Tokyo Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Okayama,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Meguro-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Japan 5Tokodai Institute for Element Strategy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Tokyo Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Nagatsuta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Midori-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Yokohama,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Kanagawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Japan 6Beijing National Laboratory for Condensed Matter Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' China 7National Synchrotron Light Source II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' New York 11973,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' USA 8Institut quantique & D´epartement de physique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Universit´e de Sherbrooke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' J1K 2R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Qu´ebec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Canada (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 2023) The energy spectrum of topological semimetals contains protected degeneracies in reciprocal space that corre- spond to Weyl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Dirac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' or multifold fermionic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' To exploit the unconventional properties of these states, one has to access the electronic structure of the three-dimensional bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' In this work, we resolve the bulk electronic states of candidate topological semimetal FeSi using momentum-dependent resonant inelastic X-ray scattering (RIXS) at the Fe L3 edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We observe a broad excitation continuum devoid of sharp features, consistent with particle-hole scattering in an underlying electronic band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Using density functional theory, we calculate the electronic structure of FeSi and derive a band theory formulation of RIXS in the fast collision approximation to model the scattering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We find that band theory qualitatively captures the number and position of the main spectral features, as well as the overall momentum dependence of the RIXS intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Our work paves the way for targeted studies of band touchings in topological semimetals with RIXS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' INTRODUCTION In the last decade and a half, topological matter has become a cornerstone of quantum materials science [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The discovery of three-dimensional topological insulators [2], in particular, sparked a flurry of activity in the then nascent field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Elec- trons in these crystalline materials are effectively noninteract- ing, giving rise to electronic bands in the bulk that are indis- tinguishable from those of a trivial band insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The elec- tronic wavefunction, however, is characterized by topologi- cal indices that dictate the presence of symmetry-protected Dirac states at the surface of the material, as well as nontrivial (magneto-)transport responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' More recently, topological semimetals have been added to the catalogue of three-dimensional topological materi- als [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' These systems also feature topologically protected boundary states and nontrivial (magneto-)transport, but addi- tionally have distinct geometric characteristics in their bulk band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' In the simplest case of Weyl semimetals, these geometric characteristics are singly degenerate energy sur- faces in reciprocal space that contain a band touching point—a Weyl node—around which electronic bands disperse linearly in all three directions in reciprocal space [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Such band touchings are Berry curvature singularities characterized by topological indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The value of the topological index of a nodal point determines the geometry of the band dispersion in ∗ mdean@bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='gov † Stefanos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='Kourtis@usherbrooke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='ca the vicinity of nodal points [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Conversely, the geometry of the bulk bands becomes a proxy of topology in these ma- terials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Measuring the electronic density of states in the bulk can therefore reveal the topological nature of a semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Resonant inelastic X-ray scattering (RIXS) is a spectro- scopic technique that yields momentum- and energy-resolved spectra of charge-neutral electronic excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' While RIXS has been extensively used in studying magnetic excitations in gapped materials like insulators and superconductors [11, 12], it is increasingly applied in studies of compounds that host itinerant carriers with small or no charge gaps [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' That RIXS can be used to map electronic bands of materials, in- cluding semimetals, has long been established [16–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Im- provements in resolution in recent years have renewed in- terest in using RIXS to detect band structure effects at meV energy scales in materials of technological interest, such as unconventional superconductors [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' There are even theo- retical proposals to use RIXS to measure topological indices of nodal points in topological semimetals [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' These prospects are particularly appealing for probing the bulk of three-dimensional materials, since alternative techniques such as angle-resolved photoemission (ARPES) and scanning tun- neling spectroscopy, probe predominantly the surface rather than the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Furthermore, topological nodal points may only appear above the Fermi level or as a result of an applied mag- netic field, settings in which the resolving power of ARPES is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' As these settings may be relevant to the technologi- cal exploitation of topological materials, alternative methods to visualize the bulk band structure and to identify topologi- cal features are sought after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Before honing in on properties of topological origin, however, one has to determine whether arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='02677v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='str-el] 6 Jan 2023 2 bulk band structure effects at large are detectable in RIXS of topological semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' In this work, we use RIXS to probe the bulk of FeSi aim- ing to quantitatively test how bulk band structure manifests in RIXS spectra of a putative topological semimetal [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We observe broad continua in the RIXS spectra, consistent with particle-hole scattering in an underlying band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We model the RIXS process in the fast-collision approxima- tion using the band structure of FeSi as determined by den- sity functional theory (DFT) calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We find reasonable agreement between experiment and theory in the number and hierarchy of dominant spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We interpret this find- ing as evidence of a bulk band structure in FeSi, with many- body effects playing only a secondary role in determining the RIXS spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Our results indicate that higher resolution experiments — feasible with existing instruments — could vi- sualize topological nodal points and thus identify and classify topological semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' THEORY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' RIXS cross section and fast collision approximation We briefly introduce the theoretical description of RIXS at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' More comprehensive presentations of RIXS can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 29 and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' In a RIXS experiment, core electrons of an ion are pro- moted to a state above the Fermi level εF by an intense x- ray beam, thereby locally exciting the irradiated material into a highly energetic and short-lived intermediate state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Subse- quently, the core hole recombines with a valence electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The process imparts both energy and momentum to particle-hole excitations in the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' In what follows, we will consider excitation of core electrons directly into orbital(s) close to εF , which give rise to the low-energy physics in the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' This process, which is often referred to as direct RIXS, is illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The double differential cross section is a measure of the total RIXS intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Up to a constant prefactor, it is given by I(kin, kout, ωin, ωout, ϵin, ϵout) = � fg |Ffg(kin, kout, ωin, ϵin, ϵout)|2δ(Eg − Ef + ℏ∆ω) , (1) where ℏ∆ω = ℏ(ωin − ωout) is the energy transferred to the material, kin and kout (ϵin and ϵout) the incoming and outgoing photon wavevectors (polarizations) and Eg and Ef the ener- gies corresponding to initial and final many-body states |g⟩ and |f⟩ of the valence electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The scattering amplitude Ffg in the dipole approximation is Ffg(kin, kout, ωin, ϵin, ϵout) = ⟨f| �D†(ϵout, kout) �G(ωin) �D(ϵin, kin)|g⟩ , (2) where �G is the intermediate-state propagator �G(ωin) = (Eg + ℏωin + iΓ − � H)−1 , (3) with � H the Hamiltonian describing the system in the interme- diate excited state and Γ the intermediate-state inverse life- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The dipole operators �D and �D † represent the x-ray ab- sorption and emission, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' For a crystalline material, they can be written as �D(ϵ, k) = ϵ · �Dk , (4) �Dk = � µ,ν ⟨µ|�r − rµ|ν⟩ � R eik·R �d † Rµ �pRν , (5) where R is the lattice position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' States |µ⟩ and |ν⟩ express single-electron valence and core states respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The com- bined valence (core) index µ (ν) encodes spin, orbital, and sublattice degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Core states |ν⟩ are conve- niently expressed as atomic orbitals, whereas valence states |µ⟩ can be appropriately chosen Wannier functions, both lo- calized in space around the same position rµ of each atomic site within the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The position operator �r, defined with respect to each lattice position R, is the same for all ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' For the L2/3 and M2/3 resonant edges, the operators �d † Rµ and �pRν create a d-orbital electron and a p-orbital core hole, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Physical arguments allow us to simplify the RIXS scatter- ing amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' First, core holes do not hop appreciably;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' they are created and annihilated at the same site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Taking this into account, Ffg becomes [30] Ffg(q, ωin, ϵin, ϵout) = � µ,ν,µ′,ν′ Tµνµ′ν′(ϵin, ϵout)Fµνµ′ν′(q, ωin) , (6) where q = kin − kout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The scattering amplitude has been factored in the atomic scattering tensor Tµνµ′ν′(ϵin, ϵout) = ⟨µ|ϵout · �r|ν⟩∗⟨µ′|ϵin · �r|ν′⟩ (7) and the fundamental scattering amplitude Fµνµ′ν′(q, ωin) = ⟨f| � R e−iq·R �p † Rν �dRµ �G(ωin) �d † Rµ′ �pRν′|g⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (8) The intrinsic spectral characteristics of a material are carried by the tensor F, which is typically the main quantity of inter- est in theoretical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The tensor T modulates the scatter- ing amplitude according to the geometry of the localized core and valence states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The entries of T can be calculated given knowledge of the valency of the targeted ion and the symme- try group of the crystal [31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Then, within the fast-collision approximation, one assumes that �G(ωin) ≈ 1/Γ, where Γ is the inverse core-hole lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' In this approximation, the RIXS process reduces to the intro- duction of a particle-hole excitation with fixed momentum and energy in the material — see Fig 1 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Before proceeding to derive the theory of RIXS in band structures, we evaluate the geometric modulation of the RIXS spectrum owing purely to the orbital content of the quantum 3 εk k εF energy core level εF εk k INITIAL energy core level εF εk k FINAL fast collision approximation FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Illustration of the direct RIXS process and reduction to effective particle-hole scattering via the fast-collision approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' states involved in the RIXS process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' This is obtained by set- ting the fundamental scattering amplitude Fµνµ′ν′ to unity: T (ϵin, ϵout) = ������ � µ,ν,µ′,ν′ Tµνµ′ν′(ϵin, ϵout) ������ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (9) We shall use T as a diagnostic to disentangle contributions to the modulation of the RIXS intensity as a function of scatter- ing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The first contribution comes through the polariza- tion vectors, which are angle-dependent — see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The second contribution is the intrinsic momentum dependence coming from electronic dispersion in the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' V we calculate the geometric modulation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (9) for FeSi and compare it to the scattering angle dependence of RIXS inten- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' RIXS process in a band structure We wish to describe the RIXS response of crystalline mate- rials in which electrons are, to a good approximation, non- interacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Valence electrons in these materials are well- described by band theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The states |g⟩ and |f⟩ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (2) are then collections of Bloch modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' For a given RIXS edge, one then sums over core states |ν⟩ and valence Wannier states |µ⟩ and ��µ′� connected by the dipole operators �D, �D†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Here we study the Fe L3 edge, hence we consider 2p3/2 orbitals for core electrons and the 3d shell for valence electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' In k-space, the band eigenbasis is given by a unitary rota- tion of a basis of Wannier states |µ⟩ per lattice position R to a basis of Bloch states |kµ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The Wannier states have wavefunc- tions ϕµ(x) = ⟨x|µ⟩ that are centered about different points in the unit cell, possibly atomic sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Let ϕkµ(x) = ⟨x|kµ⟩ be the spatial wavefunction of |kµ⟩, which could be a spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We then have ϕkµ(x) = 1 √ N � R eik·Rϕµ(x − R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (10) The raising and lowering operators of the Bloch wavefunc- tions are �d † kµ and �dkµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' They are defined by �d † kµ|Ω⟩ = |kµ⟩, {�dkµ, �dk′µ′} = 0 and {�dkµ, �d † k′µ′} = δk,k′δµ,µ′, where |Ω⟩ is the vacuum of valence excitations and core holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' A general Hamiltonian describing noninteracting valence electrons is � Hband = � k∈BZ � µ,µ′ �d † kµ Hµµ′(k) �dkµ′ , (11) where Hµµ′(k) are the elements of the matrix H(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Let U(k) be a matrix that diagonalizes H(k), such that U †(k)H(k)U(k) is a diagonal matrix containing the eigen- values εl(k), which constitute the dispersing bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We can then write �dkµ = � l Uµl(k) �ψkl, (12) where l denotes an energy band index and �ψkl annihilates the corresponding eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The ground state at zero tempera- ture is obtained by populating all states below the Fermi level: |g⟩ = � � � l,k∈BZ Θ(εF − εl(k)) �ψ† kl � � |Ω⟩ , (13) with Θ the Heaviside step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The Wannier lowering operators at any lattice site R can be expressed in terms of the band operators as �dRµ = 1 √ N � k∈BZ e−ik·R �dkµ (14a) = 1 √ N � l,k∈BZ e−ik·R Uµl(k) �ψkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (14b) We now assume, as per the fast collision approximation, that the intermediate-state Hamiltonian is well approximated by the band Hamiltonian, along with a core hole inverse life- time Γ in the intermediate-state propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Due to this as- sumption, core electron operators cancel out and the interme- diate state propagator becomes simply �G(ωin) =(Eg + ℏωin + iΓ − � Hband)−1 , (15a) = � k � l |k, l⟩⟨k, l| Eg + ℏωin + iΓ − εl(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (15b) 4 where |k, l⟩ are band eigenstates, and we treat Eg and Γ as free parameters to be determined by fitting the x-ray absorp- tion spectrum (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Substituting the expression for �dRµ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (14b) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (8), we obtain the fundamental RIXS scattering amplitude in a band structure Fµµ′(q, ωin) = ⟨f| � k,k′∈BZ � l,l′ 1 N � R e−i(k+q−k′)·R × Uµl(k) U ∗ µ′l′(k′) �ψkl �G(ωin) �ψ † k′l′|g⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (16) Notice that F is independent of the core orbitals at this level of description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The sum over R evaluates to Nδk′,k+q, which enforces k′ = k + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' When εl′(k + q) > εF, we have that �ψ † k+ql′|g⟩ is an eigenstate of the band Hamiltonian with en- ergy Eg + εl′(k + q) (otherwise the single particle state is already occupied and this term evaluates to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The action of �G(ωin) on �ψ † k+ql′|g⟩ is �G(ωin) �ψ † k+ql′|g⟩ = Θ(εl′(k + q) − εF) ℏωin − εl′(k + q) + iΓ �ψ † k+ql′|g⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (17) Furthermore, the action of �ψkl on �ψ † k+ql′|g⟩ is non-zero only if εl(k) < εF (we need this single particle level to be occupied for the term to be non-zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Using this we obtain Fµµ′(q, ωin) = � l,l′ � k∈BZ � ⟨f| �ψkl �ψ † k+ql′|g⟩ × Θ(εl′(k + q) − εF) × Θ(εF − εl(k)) × Uµl(k) U ∗ µ′l′(k + q) ℏωin − εl′(k + q) + iΓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (18) The sum over final states |f⟩ can be taken over the eigenstates of � Hband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The pair of step functions in the fundamental scat- tering amplitude given above in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (18) ensures that there is a unique |f⟩ that makes the inner product ⟨f| �ψkl �ψ † k+ql′|g⟩ non-zero, since the role of the operator pair is to simply cre- ate particle-hole excitations across the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Thus, the final sum over |f⟩ can be replaced as � f → � l,l′ � k∈BZ Θ(εl′(k + q) − εF) Θ(εF − εl(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (19) This corresponds to summing over final states with one particle-hole excitation in the valence bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The inner prod- uct is then redundant and can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The final form of the RIXS intensity for systems well- described by band theory is I(q, ωin, ∆ω, ϵin, ϵout) = � l,l′ � k∈BZ Θ(εl′(k + q) − εF) Θ(εF − εl(k)) × ������ � µ,ν,µ′ ⟨µ|ϵout · �r|ν⟩∗⟨µ′|ϵin · �r|ν⟩ Uµl(k) U ∗ µ′l′(k + q) ℏωin − εl′(k + q) + iΓ ������ 2 × η [εl(k) − εl′(k + q) + ℏ∆ω]2 + η2 , (20) where we have replaced the Dirac δ-function with a Lorentzian of peak broadening η to represent finite experi- mental resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' With respect to a local set of coordinate axes, the incoming X-ray polarization ϵ has components (ϵx, ϵy, ϵz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The outgo- ing polarization is usually not measured, and hence one sums over either polarizations parallel to and perpendicular to the scattering plane or over left, linear, and right polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We list the polarization matrix elements for the specific case of the L3 edge of a 3d transition-metal in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' EXPERIMENTAL METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Sample preparation Single crystals of FeSi were prepared using a Ga flux method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We mixed the starting materials in a molar ratio of 1:1:15 in a glove box filled with argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' This mixture was placed in an alumina crucible and sealed in an evacuated quartz tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The crucible was heated to 1150◦C and held for 10 h, before cooling to 950◦C at 2 K/h, after which the flux was centrifuged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The crystals were washed with diluted hy- drochloric acid in order to remove Ga flux from the surface of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' RIXS and experimental geometry FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Schematic of the RIXS setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' kin and kout respectively denote the ingoing and outgoing scattering vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The components of the ingoing and outgoing photon polarization within the scattering plane are denoted by πin and πout while the σ polarization direction is the same for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The incident angle θi is measured with respect to the sample surface, that is the a direction in the sample coordinates while the c direction is the normal to the sample surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' RIXS measurements were performed at the Soft Inelas- tic X-Ray (SIX) beamline at the National Syncrotron Light Source-II (NSLS-II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The energy resolution was 23 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The experimental setup is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The lab coordinates are denoted by x, y, z while the crystallographic axes are la- belled a, b, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We define the incident and outgoing beam an- gles with respect to the sample coordinate system, wherein the c direction is the normal to the sample surface and the ac plane is the scattering plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Experimental data are corrected to account for self-absorption effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' With θi denoting the incident angle measured with respect to the sample surface and 2θ denoting the angle between the incident and outgoing beam, we can easily verify the follow- ing in the sample frame: kin = kin(− cos θi, 0, − sin θi) , (21a) kout = kout (− cos(2θ − θi), 0, sin(2θ − θi)) , (21b) ϵπ,in = (− sin θi, 0, cos θi) , (21c) ϵπ,out = (sin(2θ − θi), 0, cos(2θ − θi)) , (21d) ϵσ,in = ϵσ,out = (0, 1, 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (21e) Although in reality the ingoing and outgoing photon mo- mentum magnitudes kin and kout is different owing to non- zero energy transfer ∆ω, the difference is negligible since ∆ω ≪ ωin, and hence kin ≈ kout = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The momentum transfer to the material q is then q = k (cos(2θ − θi) − cos θi, 0, − sin(2θ − θi) − sin θi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (22) 0 1 2 3 4 5 705 706 707 708 709 710 705 706 707 708 709 710 Energy Loss (eV) (a) Experiment (b) Theory Incident energy (eV) 705 706 707 708 709 710 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='9 1 RIXS (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' units) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Color maps of RIXS intensity with 2θ = 150◦ and θi = 68◦ for π-polarized incident beam as a function of incident photon energy ℏωin and energy loss ∆ω at the L3 edge of Fe in FeSi as obtained (a) in experiment and (b) in the band theory formalism of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' In practice the ingoing and outgoing beam directions are set to specific values, which defines a specific 2θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' By rotating the sample about the y axis (or equivalently b axis), we can change q by changing θi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' DENSITY FUNCTIONAL THEORY AND TIGHT-BINDING MODEL The band structure of FeSi was simulated in a similar way to prior studies of FeSi [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We performed first-principles calculations based on DFT [36] within the Perdew-Burke- Ernzerhof exchange-correlation [37] implemented in the Vi- enna ab initio simulation package (VASP) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The plane- wave cutoff energy was 450 eV with a 9×9×9 k-mesh in the BZ for self-consistent calculation without considering spin- orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Maximally localized Wannier functions [39] were used to obtain the tight-binding model of bulk FeSi with the lattice constants a = b = c = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='48 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' RIXS SPECTRUM OF FESI The RIXS intensity at the Fe L3 edge with π-polarized in- cident beam is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 3 in the incident energy-energy loss plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The absence of prominent sharp inelastic features suggests a particle-hole continuum, consistent with particle- hole excitations in a partially filled band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Momentum-resolved RIXS spectra are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 4 for two values of 2θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Spectral weight from inelastic processes lies predominantly in a window of width ∼ 5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Within that window, all spectra have similar lineshape, featuring a peak around 2 eV that disperses to higher energies with increasing θi and a dispersionless shoulder above 3 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Overall inelastic intensity increases with increasing θi for both values of 2θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We use the band theory formulation of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' II B and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (20) to theoretically model the RIXS process in FeSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' A fit of the absorption spectrum (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' B) yields Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='8 6 0 1 2 3 4 5 0 1 2 3 4 5 Exp (a) 2� D 150ı (b) 2� D 70ı RIXS (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' units) �i �i Thy Energy loss (eV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Momentum-resolved RIXS spectra at ℏωin = 708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='7 eV with π x-ray polarization for (a) 2θ = 150◦ and (b) 2θ = 70◦ and comparison to simulations within band theory (bottom pan- els) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (20) with Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='8 eV and ε0 = 707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='67 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' For scattering angle 2θ = 150◦, the incident angle values are θi = 10◦, 30◦, 45◦, 68◦, 120◦, while for 2θ = 70◦ we have θi = 10◦, 30◦, 60◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' eV and Eg = 707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='67 eV, and we choose a peak broadening η = 100 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We use a 48 × 48 × 48 grid of k points in the Brillouin zone for the 32-band tight binding model detailed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The simulated spectra show a structure similar to that ob- served experimentally, with inelastic weight in a ∼ 5 eV win- dow containing a peak at ∆ω ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='5 eV and shoulder at ∆ω ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' As in experiment, overall inelastic intensity increases with increasing θi for both values of 2θ, though to a lesser extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Compared to experiment, features are shifted to higher energies in simulated spectra, while for 2θ = 70◦ and θi = 60◦ the main peak subsides, leading to theory and experimental spectra that look qualitatively different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Finally, experimental spectra also contain subdominant features close to the elastic line (∆ω < 1 eV) that, as discussed later, deviate from the band theory predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' To investigate what causes the overall increase in RIXS in- tensity with increasing θi, we calculate the atomic scattering tensor (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' From this we obtain the modulation of the RIXS spectrum (9) coming purely from the orbital content of core and valence states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' After summing over outgoing polariza- tions, we obtain the behavior shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Comparing to the θi dependence of RIXS spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 4, we see that ge- ometric considerations are insufficient to explain the momen- tum dependence of RIXS intensity in experiment: the mo- mentum dependence of the atomic scattering tensor is differ- ent from that observed, even showing a reverse trend in the 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='5 0ı 30ı 60ı 90ı 120ı 150ı 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='4 0ı 30ı 60ı 90ı 120ı 150ı Orbital RIXS Incident angle �i (a) 2� D 150ı (b) 2� D 70ı T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' �/ C T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' �/ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The contribution of the dipole matrix elements to the RIXS spectrum of FeSi for π ingoing polarisation as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (9) after summing over σ and π polarisations of the outgoing beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' case of 2θ = 70◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Reinstating the band structure fundamen- tal scattering amplitudes in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (20) yields the experimentally observed trend of overall momentum dependence, albeit only qualitatively, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' DISCUSSION & CONCLUSION We have seen that the theoretical formulation of RIXS based on band theory captures the overall momentum depen- dence of the experimental Fe L3-edge RIXS spectra of FeSi better than a calculation based on just the atomic multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Band theory also reproduces the right bandwidth for the in- elastic part of the spectrum, as well as the right number of features therein, at roughly the right energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Discrepancies between theory and experiment exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' While the overall momentum dependence of the RIXS spectrum is reproduced by band theory, experimental spectra depend more sensitively on θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Spectral features also do not align per- fectly between experiment and theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' This includes a fea- ture around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='3 eV in the experiment, which is only present as a weak shoulder in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We discuss potential rea- sons for these discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' First, the band theory of RIXS ignores electronic correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The extent to which correla- tions play a role in the 3d shell of FeSi is not clear [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' A more detailed theoretical study of the RIXS spectrum would require identifying the precise type, range, and magnitude of interactions present in FeSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Fully incorporating the effects of interactions in theoretical studies of RIXS is nevertheless a challenge, since we are dealing with a three-dimensional ma- terial with 32 relevant orbitals per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Numerical sim- ulation of the RIXS spectrum based on dynamical mean field theory [42] may eventually be up to this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Second, in the fast collision approximation we ignore the effects of a finite core-hole lifetime, which may be appreciable in 3d transition metal compounds [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Future simulations could be improved by incorporating dynamics and interactions with the core hole in the intermediate state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' In conclusion, we have reported RIXS spectra of FeSi at 7 the Fe L3 edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We observe an excitation continuum without sharp features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Through a band theory formulation of RIXS in the fast collision approximation, we model the RIXS pro- cess using the ab initio band structure of FeSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We obtain reasonable agreement for the spectrum bandwidth, as well as the number and position of main features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Theory also repro- duces the dispersion trend of the RIXS spectrum, albeit only qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' This work paves the path to ever finer resolution of distinctive band structure features in topological materials with RIXS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' ACKNOWLEDGMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' was supported by DOE Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' DE-FG02- 06ER46316 and EPSRC grant EP/T034351/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' ac- knowledges support from the Minist`ere de l’´Economie et de l’Innovation du Qu´ebec and the Canada First Research Excel- lence Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Work at Brookhaven National Laboratory (x-ray scattering and analysis) was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Depart- ment of Energy, Office of Science, Office of Basic Energy Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' This research used resources at the SIX beamline of the National Synchrotron Light Source II, a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' DOE Of- fice of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' DE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We acknowledge National Natural Sci- ence Foundation of China (U2032204), the Strategic Prior- ity Research Program of the Chinese Academy of Sciences (XDB33010000) for funding sample synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We thank Yue Cao, Siddhant Das, Claudio Chamon, Michael El-Batanouny, Jungho Kim, and Karl Ludwig for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Appendix A: Polarization matrix elements and the atomic scattering tensor The DFT derived tight-binding model used for the calcula- tions presented in the paper involves thirty two basis orbitals per unit cell of the crystal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Due to the assumption of zero spin orbit coupling for the valence bands, this gives rise to thirty-two, two-fold spin-degenerate bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The orbitals used are the five 3d orbitals of each of the four Fe atoms and the three 3p orbitals of each of the four silicon atoms within a unit cell, giving a total of 32 orbitals per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Since the tight binding model is expressed in terms of 3d or- bitals whose local axes are perfectly aligned with crystal axis for each of the four Fe atoms in the unit cell, we need to com- pute the matrix elements of the 2p3/2 → 3d transitions for just one of the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The 2p orbitals all have the same radial part of the wavefunction, φ2p(r) and, likewise, the 3d orbitals have same radial wavefunction φ3d(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The radial integral of the various matrix elements in the atomic scattering tensor is simply the radial integral of the product φ2p(r) · φ3d(r) and the radial part of the dipole transition operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Since this is a common term that just provides an overall multiplicative fac- tor for the RIXS cross section, we ignore it and compute only the azimuthal and polar integrals of the matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We document the relevant matrix elements of the dipole operator in Table I, which were verified by comparing to open source RIXS code EDRIXS [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Appendix B: X-ray absorption spectrum and theoretical fit To align the experimental RIXS spectra with theoretical re- sults obtained through ab initio calculations, we need to de- termine the absolute energy scale Eg of the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We determine Eg through a fit of the experimental X-ray absorp- tion intensity with the calculated absorption spectrum Iabs(q, ωin, ϵin) = � ϵout � l,l′ � k∈BZ Θ(εl′(k + q) − εF) Θ(εF − εl(k)) ������ � µ,ν,µ′ ⟨µ|ϵout · �r|ν⟩∗⟨µ′|ϵin · �r|ν⟩ Uµl(k) U ∗ µ′l′(k + q) Eg + ℏωin − εl′(k + q) + iΓ ������ 2 , (B1) which is obtained by integrating over ∆ω the RIXS spectrum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' In addition to Eg, we consider the core hole in- verse lifetime Γ as a tunable parameter in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' We sum over outgoing polarizations since the measured spectrum is not po- larization resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 6 shows the fit that minimizes the average absolute deviation and yields the values Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='8 eV and Eg = 707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='67 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Hasan and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Kane, Reviews of modern physics 82, 3045 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Hasan and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Moore, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Matter Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 2, 55 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Burkov, Nature materials 15, 1145 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' [4] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Armitage, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Mele, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Vishwanath, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 90, 015001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' von Neumann and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Wigner, in Collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Eugene 8 J = 3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Jz = − 3 2 J = 3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Jz = − 1 2 J = 3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Jz = 1 2 J = 3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Jz = 3 2 d3z2−r2↑ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0) � − 1 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' i 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 2 3 � � 1 2 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' i 2 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � d3z2−r2↓ � − 1 2 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' i 2 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 2 3 � � 1 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' i 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0) dxz↑ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0) � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 1 2 √ 3 � � 1 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' − 1 2 � dxz↓ � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 1 2 � � 1 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' − 1 2 √ 3 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0) dyz↑ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0) � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' − i 2 √ 3 � � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 1 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' − i 2 � dyz↓ � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' − i 2 � � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 1 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' − i 2 √ 3 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0) dx2−y2↑ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0) � 1 2 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' i 2 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0) � − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' i 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � dx2−y2↓ � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' i 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0) � − 1 2 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' i 2 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0) dxy↑ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0) � − i 2 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 1 2 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0) � − i 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � dxy↓ � − i 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0) � − i 2 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' − 1 2 √ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 0) TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Dipole matrix elements relevant for the L3 edge of FeSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Only the polar and azimuthal integrals are evaluated since the radial integral is the same for all the core-valence pairs, and provides only an overall prefactor to the theoretical RIXS spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 704 705 706 707 708 709 710 0 500 1000 1500 2000 Incident Energy (eV) XAS (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' units) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Optimal average absolute deviation fit to the L3-edge x- ray absorption spectrum (XAS) of FeSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The black dots represent the experimental absorption spectrum while the continuous blue line represents the theoretical spectrum calculated using the tight-binding model described in Sec IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' The fit yields Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='8 eV and Eg = 707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content='67 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Paul Wigner (Springer Berlin Heidelberg, Berlin, Heidelberg, 1993) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' 294–297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Herring, Phys.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Dean, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} +page_content=' Kotliar, Computer Physics Communications 243, 151 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE0T4oBgHgl3EQfzwKd/content/2301.02677v1.pdf'} diff --git a/MNAyT4oBgHgl3EQfgfgN/content/tmp_files/2301.00358v1.pdf.txt b/MNAyT4oBgHgl3EQfgfgN/content/tmp_files/2301.00358v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..99c5b60144ec119fff9b210e871febe5dc942bad --- /dev/null +++ b/MNAyT4oBgHgl3EQfgfgN/content/tmp_files/2301.00358v1.pdf.txt @@ -0,0 +1,1317 @@ +arXiv:2301.00358v1 [gr-qc] 1 Jan 2023 +Noname manuscript No. +(will be inserted by the editor) +Nonsingular Black Holes in Higher dimensions +Bikash Chandra Paul +Received: date / Accepted: date +Abstract We present a class of new nonsingular black holes in higher dimensional +theories of gravity. Assuming a specific form of the stress energy tensor exact an- +alytic solutions of the field equation are generated in general theory of relativity +(GR) and Rastall theory. The non-singular black hole solutions are obtained with +a finite pressure at the centre in D = 4 dimensions. For D > 4 the transverse pres- +sure is found finite at the centre for a set of model parameters. In the later case the +transverse pressure is more than that in the usual four dimensions. The exact ana- +lytic solution of the field equations in higher dimensions for large r coincides with +the Schwarzschild black hole solution in the usual four and in higher dimensions +which is singularity free. The different features of the generalized non-singular +black hole in GR and modified GR are explored. A new vacuum nonsingular black +hole is found in Rastall gravity. We also study the motion of massive and massless +particles around the black holes. +1 Introduction +The idea that spacetime dimensions should be extended from four to higher dimen- +sions came from the seminal work of Kaluza and Klein [1,2] who first tried to unify +gravity with electromagnetism. The Kaluza-Klein approach has been revived and +considerably generalized after realizing that many interesting theories of particle +interactions need spacetime dimensions more than four for their formulation. Dur- +ing the last few decades considerable research activities in progress to understand +the quantum properties of gravity. The investigation seems to lead some people to +believe that a consistent theory of quantum gravity cannot be obtained within the +framework of point-field theories. For example, superstring theory is considered +B. C. Paul +E-mail: bcpaul@nbu.ac.in +Department of Physics, University of North Bengal, Siliguri, Dist. : Darjeeling 734 013, West +Bengal, India +and +IUCAA Centre for Astronomy Research and Development, North Bengal + +2 +Bikash Chandra Paul +to be the promising candidate which may unify gravity with the other fundamen- +tal forces in nature which requires ten dimensions for consistent formulation. The +advent of string theory has opened up new and interesting possibilities in this con- +text. The discovery that a supergravity theory coupled to Yang-Mills fields with a +gauge group SO(32) or E8 ×E8 is anomaly-free in ten dimensions had inspired con- +siderable activities in this area. Although the expected breakthrough has not yet +come, worldwide hectic activities have served to focus on a number of issues which +need further investigations in higher dimensions. The present ideas in dimensional +reduction suggest that our cosmos may be a 3-brane evolving in a D-dimensional +spacetime. Cadeau and Woolgar [3] addressed this issue in the context of black +holes which led to homogeneous but non- FRW-braneworld cosmologies. Classical +General Relativity (GR) in more than the usual four dimensions is thus a sub- +ject of increasing attention in recent years. A successful development of counting +of the five dimensional black hole entropy [4] and the AdS/CFT correspondence +relates the properties of a D-dimensional black hole with those of a quantum field +theory in (D − 1) dimensions [5]. There has been a growing interest to investigate +the physics of higher-dimensional black holes [6] which is markedly different, and +much richer in structure compared to four dimensions. +In connection with localized sources, higher dimensional generalization of the +spherically symmetric Schwarzschild, Reisner-Nordstr¨om black holes, Kerr black +holes can be found in the literature [7,8,9,10,11,12]. The generalization of the +rotating Kerr black hole [8,9,13,14] and black holes in compactified spacetime [7, +12] are also found in the literature. The linearized stability of the black holes [15], +no hair theorems [16], black hole thermodynamics and Hawking radiation have +also been investigated. Mandelbrot [17] investigted the problem on the variability +of dimensions and describe how a ball of thin thread is seen as an observer changes +scale. An object which looks like a point object from a very large distance becomes +a three-dimensional ball visible at a closer distance. Therefore at various scales the +ball appears to change shape as an observer moves down. While the embedding +dimensions for the ball has not changed, the effective dimensions of the contents +however also remains same. It is possible that there are compact [18] or non- +compact [19] dimensions present at a certain point. In this case the (3 + 1) metric +is simply not true, although one obtains a valid description with general relativity. +We also probe the black hole solution in Rastall theory [20] which is prescribed +by a modification of GR to accommodate the present accelerating universe [21,22, +23]. +Regular (i.e. non-singular) black holes have been initiated by Bardeen [24] and +thereafter a number of black hole solutions in four dimensions have been obtained +[25,26,27,28,29,30,31,32,33]. In this case one can find metrics which are spher- +ically symmetric, static, asymptotically flat, with regular centres, and for which +the resulting Einstein tensor is physically reasonable, satisfying the weak energy +condition and having components which are bounded and fall off appropriately at +large distance. Dymnikova [25] obtained nonsingular Schwarzsclid black hole solu- +tion in vacuum and thereafter extended to obtain nonsingular cosmological black +hole [26] solutions to include the de Sitter solution in the usual four dimensions. +Formation and evaporation of non-singular black hole is also discussed [34] from +an initial vacuum region accommodating Bardeen-like static region supported by +finite density and pressures, subsequently its pressure vanishes rapidly at large ra- +dius which however behaves as a cosmological constant at a small radius. In 2019, + +Nonsingular Black Holes in Higher dimensions +3 +Event Horizon Telescope group captured the first ever image of a supermassive +black hole at the centre of the M87 galaxy which triggers the various possibili- +ties for the state of the compact object and opened up new horizon in theoretical +physics. The motivation of the paper is to obtain a nonsingular black hole solution +in higher dimensions and investigate the different features of such black holes in +GR and beyond GR. For this we consider Higher dimensional Einstein gravity +(GR) and Rastall gravity for a comparative study. +The paper is organised as follows: In sec. 2, the Einstein field equation in a +static higher dimensional metric is obtained. In sec. 3, non-singular black holes are +obtained in Rastall theory with extension of spacetime dimensions. In sec. 4, we +present analytical set up of the non-singular black hole solution to investigate the +shadow of the black hole. The effective potential and the shadow behaviour of the +black holes are analyzed in sec. 5. Finally we summarize in sec 6. +2 Einstein Field Equation in Higher Dimensions +We consider a higher dimensional gravitational action which is given by +I = − +1 +16πGD +� √−g dDx R + Im +(1) +where R is the Ricci scalar, GD is the D-dimensional gravitational constant and +Im represents the matter action. The Einstein field equation is given by +RAB − 1 +2gABR = κ2TAB +(2) +where A, B = 0, 1, ...D −1 and T A +B = (−ρ, Pr, P⊥, ...) the energy-momentum tensor, +ρ the energy density, Pr the radial pressure, P⊥ transverse pressure, κ2 = 8πGD +c2 +. +We consider D dimensional spacetime metric given by +ds2 = −eνdt2 + eλ dr2 + r2dΩ2 +D−2 +(3) +where ν and λ are functions of radial coordinate r and ΩD−2 is for unit sphere in +SD−2 dimensions. The components of the Einstein equations and the metric given +by eq. (26) are given by +T t +t = (D − 2) +2 +� +e−λ +� +D − 3 +r2 +− λ′ +r +� +− D − 3 +r2 +� +, +(4) +T r +r = (D − 2) +2 +� +e−λ +� +D − 3 +r2 ++ ν′ +r +� +− D − 3 +r2 +� +, +(5) +T θ1 +θ1 = −(D − 3)(D − 4) +2r2 ++ e−λ +2 × +� +ν′′ + ν′2 +2 − λ′ν′ +2 ++ (D − 3)(ν′ − λ′) +r ++ (D − 3)(D − 4) +r2 +� +, +(6) +T θ2 +θ2 = ...... = T θD−2 +θD−2 = T θ1 +θ1 +(7) +for simplicity we have taken κ2 = 8πGD +c2 += 1. The radial null vector lA can be +selected to have the components lt = eλ/2, lr = ±eν/2 and li = 0. The two + +4 +Bikash Chandra Paul +radial null-null components of the Ricci tensor are equal, and given by RABlAlB = +eλRtt + eνRrr = (D − 2) (eλ+ν)′ +2eλ+ν , which vanishes if and only if (λ + ν) is a constant. +A rescaling of the time coordinate can be set to make the sum of the terms equal +to zero for black hole solution and we write +λ + ν = 0 +(8) +Now substituting +λ = − ln f(r) +(9) +we obtain the following components of energy momentum tensors in terms of f(r), +which are given by +T t +t = T r +r = D − 2 +2 +� +f(r) +� +D − 3 +r2 ++ +f′ +rf(r) +� +− D − 3 +r2 +� +(10) +T θ1 +θ1 = f(r) +2 +� +f′′ +f + 2(D − 3)f′ +rf(r) ++ (D − 3)(D − 4) +r2 +� +− (D − 3)(D − 4) +2r2 +(11) +T θ2 +θ2 = ...... = T θD−2 +θD−2 = T θ1 +θ1 +(12) +where the prime denotes the derivative with respect to r. The source term satis- +fying +T t +t = T r +r , +and +T θ2 +θ2 = ...... = T θD−2 +θD−2 = T θ1 +θ1 +(13) +and the equation of state, T A +B ;A = 0,. +Assume the density profile in higher dimensions T t +t = −ρ as +ρ = −T t +t = ρ0 e +− rD−1 +rD−1 +∗ +(14) +where r∗ is a dimensional constant connected with a constant density ρ0. The +density ρ0 also permits a D dimensional de Sitter solution with its size given by +r2 +0 = (D − 1)(D − 2) +2Λ +, +(15) +where Λ = ρ0. Using the density profile given eq. (14) in eq. (10) we integrate and +obtain the metric potential which yields +f(r) = 1 − rD−3 +g +rD−3 + +2ρ0rD−1 +∗ +(D − 1)(D − 2) +1 +rD−3 e +− rD−1 +rD−1 +∗ +(16) +where rD−3 +g += +� +2ρ0 +(D−1)(D−2) +� +rD−1 +∗ +. The higher dimensional metric is now can be +written as +ds2 = − +� +1 − Rs(r) +rD−3 +� +dt2 + +dr2 +� +1 − Rs(r) +rD−3 +� + r2dΩ2 +D−2 +(17) +where we denote +Rs(r) = rD−3 +g +� +1 − exp +� +−rD−1 +rD−1 +∗ +�� +(18) + +Nonsingular Black Holes in Higher dimensions +5 +and +rD−1 +∗ += r2 +0 rD−3 +g +, +(19) +where r2 +0 = (D−1)(D−2) +2ρ0 +. This is an exact spherically symmetric solution of the +Einstein field equations in D-dimensions. For D = 4 the solution given by eq. +(17) reduces to the solution obtained by Dymnikova [25]. The other components +of energy momentum tensor can be obtained using the Einstein’s field equations +which are given by +T θ2 +θ2 = ... = T θD−2 +θD−2 = +� +D − 1 +D − 2 +� r +r∗ +�D−1 +− 1 +� +ρ0e +− rD−1 +rD−1 +∗ +. +(20) +It is evident that in the usual 4 dimensions Dymnikova [25] black hole solutions +recovered with anisotropic fluid distribution when r = r∗, which is true also in +higher dimensions. The nonsingular black hole (NSBH) solutions are permitted +with anisotropic fluid distributions in higher dimensions. The energy density and +radial pressure follow the vacuum configuration but the tangential pressures do +not. The tangential pressure is non-zero which remains positive definite for r > r∗. +At the center the tangential pressure is negative indicating existence of exotic mat- +ter (P⊥ < 0) at the center of the black hole. The nonsingular black hole solution +obtained by Dymnikova can not be described in lower dimension D = 2 + 1, how- +ever, we can extend the concept of NSBH in more than the usual four dimensions. +The generalization of the black hole solution in higher dimensions accommodates +a new class of NSBH solutions where the tangential pressure increases to a large +extent inside the non-singular black hole with a different feature but away from +the centre of the black hole it decreases exponentially. +The mass of a massive object in higher dimension is given by +m(r) = AD−2 +� r +0 +r′D−2ρ(r′)dr′ +(21) +where AD−2 = +2π +D−1 +2 +Γ ( D−1 +2 +) which at r → ∞ is connected to the whole mass M con- +nected with rD−3 +g +by the Schwarzschild relation. The modulus difference between +Rs(rg) and rD−3 +g +is rD−3 +g +e +− rD−1 +rD−1 +∗ +. The measure of the difference between the higher +dimensional Schwarzschild mass m(r) ∼ rD−3 +g +in a singular black hole and Rs of a +non-singular black hole is given by +M − m(r) +M += exp +� +−rD−1 +rD−1 +∗ +� +. +(22) +Here m(r) becomes M at infinite distance. It is found that the mass difference +decreases as the dimension in which black hole embedded increases. The metric +has two event horizons located at +r+ = rg +� +1 − O +� +exp +� +−r2g +r2 +0 +��� +, +r− = r0 +� +1 − O +� +exp +� +−r0 +rg +��� +. +(23) + +6 +Bikash Chandra Paul +Here r+ is the external event horizon. The metric evaluated at gtt(r+) = 0 de- +scribes an object with the similar properties properties of a black hole by a dis- +tant observer, it does not send light signals outside and could not interact with +its surroundings by the gravitational field. In four dimensions it is found that +both r+ and r− are removable singularities of the metric. The singularities can be +eliminated by an appropriate transformation. +3 Higher Dimensional Rastall gravity +In this section we explore NSBH solution in the Rastall theory of gravity for D ≥ 4 +dimensions. The Rastall theory [20] is based on the modification of the Einstein +field equation for a spacetime with Ricci scalar filled by an energy momentum +source as follows: +T AB; A = λRB +(24) +where λ is the Rastall parameter which is a measure for deviation from the stan- +dard GR conservation law. Consequently the Rastall field equation can be written +as +GAB + κ2λgABR = κ2TAB +(25) +where κ2 is the Rastall gravitational constant. The above field equation reduces +to that of GR in the limit λ → 0 and κ2 = 8πG. However, for a vanishing trace +of the energy-momentum tensor, for example the electrovacuum solution can be +obtained when λ = 1 +4 or R = 0. It is important to note that the former possibility +is not physically acceptable as the trace of the energy momentum tensor vanishes +T = 0 for any scalar field. Consequently the matter configuration where the energy- +momentum tensor has null trace, the relativistic solution obtained in Rastall theory +is same as that one obtains in the general theory of relativity (GR). This feature of +Rastall theory which is a modified GR led us to look for black holes solutions in a +background of matter/energy with non-vanishing trace. It may be pointed out here +that the Rastall gravity is widely used to accommodate acceptable explanation for +the current acceleration of the universe which has no solution in GR and for this +it is interesting to explore NSBH in Rastall theory. +We consider the metric for black hole solution in higher dimensions D ≥ 4: +ds2 = −f(r)dt2 + dr2 +f(r) + r2dΩ2 +D−2. +(26) +Using the metric, we obtain the non-vanishing components of the Rastall tensor +HAB = GAB + λgABR and κ2 = 1, +Ht +t = D − 2 +2r2 +� +rf′ − (D − 3) + (D − 3)f� + λR, +(27) +Hr +r = D − 2 +2r2 +� +rf′ − (D − 3) + (D − 3)f� + λR, +(28) +Hθi +θi = r2f′′ + (D − 3)(2rf′ + (D − 4)(f − 1) +2r2 ++ λR +(29) +where i = 1, 2, ..., (D − 2), and the Ricci scalar in D dimensions is given by +R = − 1 +r2 +� +r2f′′ + 2(D − 2)rf′ + (D − 2)(D − 3)(f − 1) +� +(30) + +Nonsingular Black Holes in Higher dimensions +7 +in the above we denote ()′ to represent derivative with respect to the radial co- +ordinate r. We solve the field equation to obtain higher dimensional non-singular +black holes in Rastall theory and for this Ht +t = T t +t and Hrr = T rr yield +Pr = D − 2 +2r2 +� +rf′ − (D − 3) + (D − 3)f� +− λ +r2 +� +r2f′′ + 2(D − 2)rf′ + (D − 2)(D − 3)(f − 1) +� +, +(31) +and also we consider Hθ1 +θ1 = T θ1 +θ1 , ... and HθD−2 +θD−2 = T θD−2 +θD−2 which yield +P⊥ = +1 +2r2 +� +r2f′′ + 2(D − 3)rf′ + (D − 3)(D − 4)(f − 1) +� +− λ +r2 +� +r2f′′ + 2(D − 2)rf′ + (D − 2)(D − 3)(f − 1) +� +. +(32) +In this case we explore the non-singular Black hole obtained in higher dimensional +Rastall gravity, the general solution of the metric is +f(r) = 1 − rD−3 +g +rD−3 + +2ρ0rD−1 +∗ +(D − 1)(D − 2) +1 +rD−3 e +− rD−1 +rD−1 +∗ +(33) +The energy density and radial pressure are +ρ = + + +D − 2 − 2λD + 2(D − 1)λ rD−1 +rD−1 +∗ +D − 2 + + ρ0e +− rD−1 +rD−1 +∗ +, +(34) +Pr = − + + +D − 2 − 2λD + 2(D − 1)λ rD−1 +rD−1 +∗ +D − 2 + + ρ0e +− rD−1 +rD−1 +∗ +, +(35) +the tangential pressure is given by +P⊥ = +� +(1 − 2λ)D − 1 +D − 2 +rD−1 +rD−1 +∗ +− D − 2 − 2λD +D − 2 +� +ρ0e +− rD−1 +rD−1 +∗ +. +(36) +The energy density and the transverse pressure in Rastall gravity framework ob- +tained in eqs. (34) and (36) reduces to the eqs. (14) and (20) in GR for λ → 0. +The modification introduced in GR by Rastall admits nonsingular Dymnikova +[25] black hole (NSBH) with normal matter while the radial pressure corresponds +to vacuum equation of state. At the center of the NSBH the energy density is +ρ = (D − 2 − 2λD)ρ0, which increases as the. number of spacetime dimension in- +creases for a given range of Rastall parameter λ < D−2 +2D . It is evident that for a +given dimension, NSBH admits greater mass for lower values of λ and the lower +limiting value for λ < D−2 +2D +and |λ| > D−2 +2D +(for negative λ). The corresponding +tangential pressure at the center P⊥ = (2Dλ + 2 − D)ρ0 is finite but negative. In +D = 4 dimensions, at the center of the black hole, ρ(r = 0) = 2(1 − 4λ)ρ0 and +tangential pressure P⊥ = −(1−4λ)ρ0 which indicates existence of exotic matter at +the centre in GR (as λ = 0) as well as in Rastall theory for λ > − 1 +4. Thus NSBH +can be realized with both central radial pressure and tangential pressure negative +and equal but an anisotropy in pressure develops away from the center in Rastall + +8 +Bikash Chandra Paul +gravity, normal matter exists when r > +� +D−2−2λD +(D−1)(1−2λ +�1/(D−1) +r∗. The tangential +pressure indicates black hole surrounded by exotic matter in Rastall gravity [35] +for the range 1 +4 < λ < 1 +2.. For r → ∞, the energy density and pressure vanishes +asymptotically. +When λ = D−2 +2D , we get the following : +ρ = ρ0 +� +D − 1 +D +rD−1 +rD−1 +∗ +� +e +− rD−1 +rD−1 +∗ +, +(37) +Pr = −ρ = ρ0 +� +D − 1 +D +rD−1 +rD−1 +∗ +� +e +− rD−1 +rD−1 +∗ +, +(38) +the tangential pressure is given by +P⊥ = 2ρ0 +� +D − 1 +D(D − 2) +rD−1 +rD−1 +∗ +� +e +− rD−1 +rD−1 +∗ +. +(39) +one obtains NSBH with ρ > 0, ρ + Pr = 0 and P⊥ > 0. For D = 4 dimensions, +λ = 1 +4 and the NSBH can be realized in Rastall gravity with normal matter which +however is not permitted in GR. Also we note that at the centre of the NSBH the +tangential pressure vanishes, admitting a perfect vacuum NSBH in the usual four +dimensions. The result obtained in this case is also applicable in higher dimensions. +This is a new result. +4 Analytical set up +The modified Schwarzschild metric for a non-singular black hole is given by +ds2 = −f(r) dt2 + f(r)−1 dr2 + r2dΩ2 +D−2 +(40) +where f(r) = 1 − � rg +r +�D−3 + � rg +r +�D−3 exp +� +− � r +r∗ +�D−1� +, making use of the as- +sumption κ2 = 8π made earlier, we write rg = +� +16πM +(D−2)AD−2 +� +1 +D−3 and the area +of D dimensional sphere AD−2 = +2π +D−1 +2 +Γ( D−1 +2 ), where M represents the mass of the +non-singular Black hole. The metric function gtt = f(r), whose sign determines +gravitational trapping [34], we plot to draw a sketch to study the existence of +black hole solutions. The metric potential f(r) is plotted with r in Fig. (1) for +D = 4 and Fig. (2) for D = 10. It is evident that both the extreme black hole +and non-extreme black holes can be obtained for a given set of values of rg and +r0. We note that extreme black hole exists for rg = 1.0 and r0 = 0.57 in D = 4 +and rg = 2.0 and r0 = 1.57 in D = 10. In the first case no black hole exist for +rg < 1.0 and the later case for r0 > 1.57. The photon radii are tabulated in Table-I +for D = 4 and Table-II for D = 10. It is found that for D = 4, it increases with +decrease of ρ0 ∼ 1/r2 +0 for a given mass but for a given ρ0, photon radius is found +to increase with mass. In D = 10 dimensions as ρ0 is decreases the photon radius +decreases then increases and decreases once again. In Fig (3) dimensional varia- +tion of the photon radius for M = 1 is plotted for non-singular black holes with + +Nonsingular Black Holes in Higher dimensions +9 +1 +2 +3 +4 +5 +6 +r +�1.0 +�0.5 +0.5 +1.0 +f�r� +Fig. 1 Radial variation of f(r) for rg = 0.8 (Red), 1.0 (Black), 1.5 (Blue) for r0 = 0.57 in +D = 4. +1 +2 +3 +4 +5 +6 +r +�1.0 +�0.5 +0.5 +1.0 +f�r� +Fig. 2 Radial variation of the metric function f(r) in D = 10 for r0 = 1.1 (Blue), 1.57 (Black) +and 2.0 (Red) with rg = 2. +dimensions. The photon radius is maximum at D = 4 and then decreases sharply +as the dimensions is increases and remains constant. +The Lagrangian is given by +L = 1 +2gAB ˙xA ˙xB. +(41) +2 +4 +6 +8 +10 +�1 +0 +1 +2 +3 +r +V +Fig. 3 Radial variation of the potential for J= 5 (Blue), 6 (Green), 8 (Dashed), 10 (Red) in +D = 4 for non-singular BH. + +10 +Bikash Chandra Paul +where ˙() = +d +dτ and τ is the affine parameter. Expanding eq. (41) we get +2L = −f(r)˙t2 + +1 +f(r) ˙r2 + (r2 ˙θ1 +2 + sin2θ1 ˙θ2 +2 + ......) +(42) +To obtain trajectory of light path, we set θi = π +2 where i = 1, ..., D − 3 and θD−2 +is a free parameter. The momenta are given by +Pt = ∂L +∂ ˙t = −f(r)˙t, +Pr = ∂L +∂ ˙r = +1 +f(r) ˙r, +Pθ1 = ∂L +∂ ˙θ1 += r2 ˙θ1, Pθ2 = ∂L +∂ ˙θ2 += r2sin2θ1 ˙θ2, .... +(43) +Now as defined above, θi = π +2 , and at the equatorial plane θ1 = π +2 , +∂L +∂ ˙t = constant +(44) +and we determine the energy (E) and angular momentum (J) at r → ∞ as +f(r)˙t = E, PθD−2 = r2 +˙ +θD−2 = J. +(45) +The Hamilton Jacobi equation is the most general method to find the geodesic +equation of motion around black hole or a compact object, we adopt the technique +to obtain the photon orbits. In higher dimensions we get +∂S +∂τ = H = −1 +2gAB ∂S +∂xA +∂S +∂xB +(46) +where gAB is the inverse of the metric and S is the Jacobian. The Jacobian is +given by +S = 1 +2m2τ − E + JθD−2 + Sr(r) + +D−3 +� +i=1 +Sθi(θi) +(47) +where Sr(r) and Sθi(θi) are functions of r and θi respectively and m is the mass +of the test particle, it is zero for photon. The Hamilton-Jacobi eq. (46) can be +written as +r4 +� +1 − +Rs +rD−3 +�2 � +∂S +∂τ +� += E2r4 − r2 +� +1 − +Rs +rD−3 +� +(K + J2) +(48) +D−3 +� +i=1 +1 +Πi−1 +n=1sin2θn +�∂Sθi +∂θi +�2 += K − ΠD−3 +i=1 J2cot2θi +(49) +where K is the Carter constant [36]. Using the above eq. (43) in eq. (46) we get +the following +˙t = +E +f(r), +˙θD−2 = +J +r2ΠD−3 +i=1 sin2θi +; +r2 ˙r = ± +√ +R, +r2 +D−3 +� +i=1 +Πi−1 +n=1sin2θn ˙θi = ± +� +Θi +(50) + +Nonsingular Black Holes in Higher dimensions +11 +in the above ”+” and ”-” sign corresponds to motion of photon in outgoing and +incoming radial direction and over dot represents derivative w.r.t to the affine +parameterτ. For the null curves the eqs.(49) can be expressed as +R(r) = E2r4 − r2f(r)(K2 + J2), +(51) +Θi(θi) = K − ΠD−3 +i=1 J2cot2θi. +(52) +The characteristics of photon near the black hole can be defined by two impact +parameters, which are functions of the constants E, J and K. For general orbit we +define the impact parameters ξ = J +E and η = +K +E2 . The boundary of the shadow of +a black hole can be estimated from the effective potential. The radial null geodesic +from eqs. (48) and (50) is given by +� +dr +dτ +�2 ++ Veff = 0, +(53) +where Veff is the effective potential, for radial motion we obtain +Veff = f(r) +r2 (K + J2) − E2 += 1 +r2 +� +1 − +�rg +r +�D−3 � +1 − e−( r +r∗ ) +D−1�� +(K + J2) − E2. +(54) +The effective potential is identical to the classical equation describing the motion +of a massless particle in a 1-dimensional potential V (r) provided its energy is +1 +2E2 (of course the true energy should be E), but we use this form to obtain an +expression for potential in our study. We plot radial variation of V (r) in Fig. (4) in +a four dimensional universe for singular as well as non-singular black hole. As the +angular velocity increases the photons heading towards the black hole are unstable. +In Fig. 2, it is found that there is no difference of the behaviour of the potential. +In Fig. (4) we plot radial variation of V (r) for different angular momentum, for a +non-singular BH, it is evident that as the angular momentum increases the photon +can approach near to the BH unbounded +The photon orbits are circular and unstable for a maximum value of the effec- +tive potential. The unstable circular orbit determines the boundary of the apparent +shape and can be maximized. The maximal value of the effective potential corre- +sponds to the circular orbits and the unstable photons satisfies +Veff +��� +r=rp += dVeff +dr +��� +r=rp += 0, +R(r) = dR(r) +dr +��� +r=rc += 0 +(55) +The impact parameters are now related as Using eqs. (54) and (55), we get +f(rp) +r2p +(K + J2) − E2 = 0 +rpf′(rp) − 2f(rp) +r3p +(K + J2) = 0. +(56) +In four dimensions the potential V (r) is plotted in Fig. (4) with different angular +momentum (J) for rg = 2 and E = 1. The particles are bounded for a radius +r < rmin and unbounded for the range rmin < r < rmax. The range of values + +12 +Bikash Chandra Paul +rp in +rp in +rp in +r0 +M = 2 +M = 5 +M = 10 +0.5 +0.8757 +1.0192 +1.1532 +0.6 +1.0221 +1.1851 +1.3389 +0.8 +1.3079 +1.5053 +1.6958 +1.0 +1.5880 +1.8142 +2.0383 +1.2 +6.0000 +2.1150 +2.3702 +1.5 +6.0000 +- +- +1.8 +5.990 +- +- +2 +2.9921 +- +- +Table 1 The variation of the photon radius (rp) in D = 4 with r0 = +� +(D−1)(D−2) +4ρ0 +and the +mass of the BH. +rp in +rp in +r0 +M = 5 +M = 10 +0.5 +0.9369 +1.0746 +0.6 +1.0035 +1.0002 +0.7 +1.0641 +0.9863 +0.8 +1.1203 +1.0613 +0.84 +1.1417 +24.6008 +0.85 +1.1470 +0.9472 +0.89 +1.178 +1.5069 +0.9 +1.1730 +25.2259 +0.91 +1.1781 +0.9598 +0.95 +1.1983 +1.1687 +1.0 +1.2229 +0.9772 +1.2 +1.8426 +1.0105 +2 +3.5942 +6.9106 +Table 2 The variation of the photon radius (rp) in D = 10 with r0 = +� +(D−1)(D−2) +4ρ0 +and the +mass of the BH. +3 +4 +5 +6 +7 +8 +9 +10 +D +2 +4 +6 +8 +10 +Rp +[t] +Fig. 4 Dimensional variation of the photon radius for M = 1 for a non-singular BH +can be determined from the sketch. It is found that rmin decreases as angular +momentum (J) increases. We note that the potentials for Schwarzschild black hole +(singular) and that for non-singular black holes overlaps for a set of similar values +of D, J and E. +We draw the shadow contour of non-singular black hole in Fig. (8) for a given +value of ρ0 (say, 0.16 unit) in all dimensions. It is shown that as the spacetime +dimensions increases the radius of the shadow decreases. + +Nonsingular Black Holes in Higher dimensions +13 +�10 +�5 +0 +5 +10 +�10 +�5 +0 +5 +10 +Fig. 5 Contour plot for an object having M = 2M⊙ with ρ0 = 0.16 unit for D = 4 (Red), +D = 5 (Dashed) and D = 6 (Green), D = 8 (Thick) +�40 +�20 +0 +20 +40 +�40 +�20 +0 +20 +40 +Fig. 6 Contour plot for an object for ρ0 = 0.04 unit in D = 4 with M = 2M⊙ (Black), +M = 4M⊙ (Green), M = 6M⊙ (Red), M = 10M⊙ (Blue) +5 Effective potential and shadow behaviour +The effective potential of the Schwarzschild-Tangherlini black holes exhibits a max- +imum for the photon sphere radius rp corresponding to the real and the positive +solution of the constraint obtained from eq. (56), +rpf′(rp) − 2f(rp) = 0. +(57) +Defining impact parameters η and ξ that are functions of the energy E, angular +momentum J and the Carter constant K as +ξ = J +E , +η = K +E2 +(58) +we get from eq. (55) corresponding to Veff +E2 += 0 and +R +E2 = 0, the following +η + ξ2 = +r2p +f(rp), η + ξ2 = +4r2p +rf′(rp) + 2f(rp). +(59) + +14 +Bikash Chandra Paul +Now we obtain +η + ξ2 = +5r2p +rpf′(rp) + 3f(rp), +(60) +where the right hand side corresponds to +r2 +p +f(rp), the observer’s frame the shadow +can be described properly making use of the celestial coordinates α and β as +introduced earlier [37]. Following the definition introduced by Subrahmanyan as +follows +α = +lim +rp→∞ +� +rpP θD−2 +P t +� +, +βi = +lim +rp→∞ +� +rpP θi +P t +� +, +(61) +where +i = 1, ...(D − 3). +For an observer on the equatorial plane, these equations reduced to +η + ξ2 = α2 + β2 = +r2p +f(rp) +(62) +the radius of the shadow is Rbhs = +rp +√ +f(rp). The form of f(r) is complex and +therefore we study numerically. The photon radius depends on the dimensions. +The photon radius is plotted in Fig. 4, it is evident that as the mass of the black +hole increases the radius decreases. It is maximum in D = 4 but decreases sharply +as the dimension increases but almost constant with the increase in dimension. The +fig (9) shows that as the mass increases the radius of the shadow also increases. +6 Discussion +We obtain non-singular black hole (NSBH) solutions in the higher dimensional +Einstein’s general theory of gravity (GR) and found that the methods in GR can +be adopted also in Rastall gravity. Considering a specific exponential form of the +energy density we obtain NSBH which reduces to the Dymnikova NSBH solution +[25] obtained in the usual four dimensions (D = 4). In 2+1 dimensions no black hole +solution exists. However, a non-rotating NSBH solutions obtained by Dymnikova +in four dimensional GR can be accommodated in higher dimensions. We obtained +NSBH in a vacuum described by T t +t + T rr = 0 with p⊥ < 0 (where i = 1, ..., D − 2) +near the center indicating requirement of exotic matter which however extends up +to certain height thereafter p⊥ > 0 for r > � D−2 +D−1 +� +1 +D−1 r∗ in GR. But in the Rastall +gravity p⊥ > 0 for r > +� +2−D+2λD +(D−1)(2λ−1) +� +1 +D−1 r∗ when λ ̸= D−2 +2D +and thereafter at a +large distance it vanishes because the pressure decreases rapidly i.e., exponentially. +Both in GR and modified gravity it indicates existence of exotic matter near the +center of the NSBH but in the later case the Rastall parameter plays an important +role in determining the distance from the centre where the normal matter exists in +the tangential direction. In the usual four dimensions at the center of the NSBH +in the modified theory we get the following estimations ρ(r = 0) = 2(1 − 4λ)ρ0 +and tangential pressure P⊥ = −2(1 − 4λ) which are determined by the Rastall +parameter λ. It is evident that exotic matter at the center of the black hole requires +both in GR (as λ = 0) as well as in Rastall theory with the lower limiting value +of the Rastall parameter λ < 1 +4. The tangential pressure is negative it indicates + +Nonsingular Black Holes in Higher dimensions +15 +NSBH surrounded by exotic matter in Rastall gravity, existence of BH with exotic +matter also reported in [35]. Thus NSBH is realized with both the central radial +pressure and tangential pressure negative and equal initially but an anisotropy +in pressure develops away from the center in Rastall gravity with normal matter +thereafter when r > +� +D−2+2λD +(1−2λ)(D−2) +� +1 +D−1 r∗. +However, we note a new and interesting result in Rastall theory that permits +a NSBH with normal matter in the usual four and in higher dimensions when +λ = D−2 +2D , which however is not permitted in GR. In It is also noted that away from +the center at a large distance, the tangential pressure remains positive definite at a +maximum radial distance which is P⊥ = (1−2λ) D−1 +D−2 +rD−1 +rD−1 +∗ +ρ0e +− rD−1 +rD−1 +∗ +. Thus one gets +a physically realistic NSBH for 1 +4 < λ < 1 +2 in D = 4 dimensions. For r → ∞, both +the energy density and pressure vanishes asymptotically. Thus the Rastall gravity +has rich structure which unearth the structure of non-singular black hole even with +normal matter for a restricted domain of the Rastall parameter depending on the +embedding spacetime dimensions. Thus, we see that for λ ̸= 0 the Rastall theory +plays an important role leading to distinct solutions relative to GR. +The sketch of the potentials permissible in the theory are plotted in Figs. (1) +and (2), which show that both extreme and non-extreme black holes exist. +The contour plots in Fig. (5) for NSBH shows that the circular shadow radius +decreases as the spacetime dimension is increased for a given mass. The circular +shadow radius in Fig. (6) show that the radii increases with the mass of the +compact objects for a given dimensions. The rotating NSBH will be taken up +elsewhere. +Acknowledgment The author would like to thank IUCAA , Pune and IUCAA +Centre for Astronomy Research and Development (ICARD), NBU for extending +research facilities and North Bengal University for a research grant. BCP acknowl- +edge the suggestions and constructive criticism of the anonymous Referee. +References +1. T. Kaluza, Sitz. Preuss. Acad. Wiss. F 1, 966 (1921) +2. O. Klein, Ann. Phys. 37, 895 (1926) +3. C. Cadeau and E. Woolgar, Class. Quant. Grav. 18 , 527, (2001) +4. A. Strominger and C. Vafa, Phys. Lett. B 379, 99 (1996) +5. O. Aharony, S. S. Gubser, J. M. Maldacena, H. 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Subrahmanyan, The mathematical theory of black holes, (Oxford University Press, +1992) + diff --git a/MNAyT4oBgHgl3EQfgfgN/content/tmp_files/load_file.txt b/MNAyT4oBgHgl3EQfgfgN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ade1260193f5569d66fdb1f6b16e9a28159effe1 --- /dev/null +++ b/MNAyT4oBgHgl3EQfgfgN/content/tmp_files/load_file.txt @@ -0,0 +1,609 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf,len=608 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='00358v1 [gr-qc] 1 Jan 2023 Noname manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (will be inserted by the editor) Nonsingular Black Holes in Higher dimensions Bikash Chandra Paul Received: date / Accepted: date Abstract We present a class of new nonsingular black holes in higher dimensional theories of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Assuming a specific form of the stress energy tensor exact an- alytic solutions of the field equation are generated in general theory of relativity (GR) and Rastall theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The non-singular black hole solutions are obtained with a finite pressure at the centre in D = 4 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For D > 4 the transverse pres- sure is found finite at the centre for a set of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In the later case the transverse pressure is more than that in the usual four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The exact ana- lytic solution of the field equations in higher dimensions for large r coincides with the Schwarzschild black hole solution in the usual four and in higher dimensions which is singularity free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The different features of the generalized non-singular black hole in GR and modified GR are explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' A new vacuum nonsingular black hole is found in Rastall gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We also study the motion of massive and massless particles around the black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 1 Introduction The idea that spacetime dimensions should be extended from four to higher dimen- sions came from the seminal work of Kaluza and Klein [1,2] who first tried to unify gravity with electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The Kaluza-Klein approach has been revived and considerably generalized after realizing that many interesting theories of particle interactions need spacetime dimensions more than four for their formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Dur- ing the last few decades considerable research activities in progress to understand the quantum properties of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The investigation seems to lead some people to believe that a consistent theory of quantum gravity cannot be obtained within the framework of point-field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For example, superstring theory is considered B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Paul E-mail: bcpaul@nbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='in Department of Physics, University of North Bengal, Siliguri, Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' : Darjeeling 734 013, West Bengal, India and IUCAA Centre for Astronomy Research and Development, North Bengal 2 Bikash Chandra Paul to be the promising candidate which may unify gravity with the other fundamen- tal forces in nature which requires ten dimensions for consistent formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The advent of string theory has opened up new and interesting possibilities in this con- text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The discovery that a supergravity theory coupled to Yang-Mills fields with a gauge group SO(32) or E8 ×E8 is anomaly-free in ten dimensions had inspired con- siderable activities in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Although the expected breakthrough has not yet come, worldwide hectic activities have served to focus on a number of issues which need further investigations in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The present ideas in dimensional reduction suggest that our cosmos may be a 3-brane evolving in a D-dimensional spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Cadeau and Woolgar [3] addressed this issue in the context of black holes which led to homogeneous but non- FRW-braneworld cosmologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Classical General Relativity (GR) in more than the usual four dimensions is thus a sub- ject of increasing attention in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' A successful development of counting of the five dimensional black hole entropy [4] and the AdS/CFT correspondence relates the properties of a D-dimensional black hole with those of a quantum field theory in (D − 1) dimensions [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' There has been a growing interest to investigate the physics of higher-dimensional black holes [6] which is markedly different, and much richer in structure compared to four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In connection with localized sources, higher dimensional generalization of the spherically symmetric Schwarzschild, Reisner-Nordstr¨om black holes, Kerr black holes can be found in the literature [7,8,9,10,11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The generalization of the rotating Kerr black hole [8,9,13,14] and black holes in compactified spacetime [7, 12] are also found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The linearized stability of the black holes [15], no hair theorems [16], black hole thermodynamics and Hawking radiation have also been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Mandelbrot [17] investigted the problem on the variability of dimensions and describe how a ball of thin thread is seen as an observer changes scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' An object which looks like a point object from a very large distance becomes a three-dimensional ball visible at a closer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Therefore at various scales the ball appears to change shape as an observer moves down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' While the embedding dimensions for the ball has not changed, the effective dimensions of the contents however also remains same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is possible that there are compact [18] or non- compact [19] dimensions present at a certain point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In this case the (3 + 1) metric is simply not true, although one obtains a valid description with general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We also probe the black hole solution in Rastall theory [20] which is prescribed by a modification of GR to accommodate the present accelerating universe [21,22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Regular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' non-singular) black holes have been initiated by Bardeen [24] and thereafter a number of black hole solutions in four dimensions have been obtained [25,26,27,28,29,30,31,32,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In this case one can find metrics which are spher- ically symmetric, static, asymptotically flat, with regular centres, and for which the resulting Einstein tensor is physically reasonable, satisfying the weak energy condition and having components which are bounded and fall off appropriately at large distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Dymnikova [25] obtained nonsingular Schwarzsclid black hole solu- tion in vacuum and thereafter extended to obtain nonsingular cosmological black hole [26] solutions to include the de Sitter solution in the usual four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Formation and evaporation of non-singular black hole is also discussed [34] from an initial vacuum region accommodating Bardeen-like static region supported by finite density and pressures, subsequently its pressure vanishes rapidly at large ra- dius which however behaves as a cosmological constant at a small radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In 2019, Nonsingular Black Holes in Higher dimensions 3 Event Horizon Telescope group captured the first ever image of a supermassive black hole at the centre of the M87 galaxy which triggers the various possibili- ties for the state of the compact object and opened up new horizon in theoretical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The motivation of the paper is to obtain a nonsingular black hole solution in higher dimensions and investigate the different features of such black holes in GR and beyond GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For this we consider Higher dimensional Einstein gravity (GR) and Rastall gravity for a comparative study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The paper is organised as follows: In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 2, the Einstein field equation in a static higher dimensional metric is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 3, non-singular black holes are obtained in Rastall theory with extension of spacetime dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 4, we present analytical set up of the non-singular black hole solution to investigate the shadow of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The effective potential and the shadow behaviour of the black holes are analyzed in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Finally we summarize in sec 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 2 Einstein Field Equation in Higher Dimensions We consider a higher dimensional gravitational action which is given by I = − 1 16πGD � √−g dDx R + Im (1) where R is the Ricci scalar, GD is the D-dimensional gravitational constant and Im represents the matter action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The Einstein field equation is given by RAB − 1 2gABR = κ2TAB (2) where A, B = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='D −1 and T A B = (−ρ, Pr, P⊥, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=') the energy-momentum tensor, ρ the energy density, Pr the radial pressure, P⊥ transverse pressure, κ2 = 8πGD c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We consider D dimensional spacetime metric given by ds2 = −eνdt2 + eλ dr2 + r2dΩ2 D−2 (3) where ν and λ are functions of radial coordinate r and ΩD−2 is for unit sphere in SD−2 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The components of the Einstein equations and the metric given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (26) are given by T t t = (D − 2) 2 � e−λ � D − 3 r2 − λ′ r � − D − 3 r2 � , (4) T r r = (D − 2) 2 � e−λ � D − 3 r2 + ν′ r � − D − 3 r2 � , (5) T θ1 θ1 = −(D − 3)(D − 4) 2r2 + e−λ 2 × � ν′′ + ν′2 2 − λ′ν′ 2 + (D − 3)(ν′ − λ′) r + (D − 3)(D − 4) r2 � , (6) T θ2 θ2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='. = T θD−2 θD−2 = T θ1 θ1 (7) for simplicity we have taken κ2 = 8πGD c2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The radial null vector lA can be selected to have the components lt = eλ/2, lr = ±eν/2 and li = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The two 4 Bikash Chandra Paul radial null-null components of the Ricci tensor are equal, and given by RABlAlB = eλRtt + eνRrr = (D − 2) (eλ+ν)′ 2eλ+ν , which vanishes if and only if (λ + ν) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' A rescaling of the time coordinate can be set to make the sum of the terms equal to zero for black hole solution and we write λ + ν = 0 (8) Now substituting λ = − ln f(r) (9) we obtain the following components of energy momentum tensors in terms of f(r), which are given by T t t = T r r = D − 2 2 � f(r) � D − 3 r2 + f′ rf(r) � − D − 3 r2 � (10) T θ1 θ1 = f(r) 2 � f′′ f + 2(D − 3)f′ rf(r) + (D − 3)(D − 4) r2 � − (D − 3)(D − 4) 2r2 (11) T θ2 θ2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='. = T θD−2 θD−2 = T θ1 θ1 (12) where the prime denotes the derivative with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The source term satis- fying T t t = T r r , and T θ2 θ2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='. = T θD−2 θD−2 = T θ1 θ1 (13) and the equation of state, T A B ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='A = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Assume the density profile in higher dimensions T t t = −ρ as ρ = −T t t = ρ0 e − rD−1 rD−1 ∗ (14) where r∗ is a dimensional constant connected with a constant density ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The density ρ0 also permits a D dimensional de Sitter solution with its size given by r2 0 = (D − 1)(D − 2) 2Λ , (15) where Λ = ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Using the density profile given eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (14) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (10) we integrate and obtain the metric potential which yields f(r) = 1 − rD−3 g rD−3 + 2ρ0rD−1 ∗ (D − 1)(D − 2) 1 rD−3 e − rD−1 rD−1 ∗ (16) where rD−3 g = � 2ρ0 (D−1)(D−2) � rD−1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The higher dimensional metric is now can be written as ds2 = − � 1 − Rs(r) rD−3 � dt2 + dr2 � 1 − Rs(r) rD−3 � + r2dΩ2 D−2 (17) where we denote Rs(r) = rD−3 g � 1 − exp � −rD−1 rD−1 ∗ �� (18) Nonsingular Black Holes in Higher dimensions 5 and rD−1 ∗ = r2 0 rD−3 g , (19) where r2 0 = (D−1)(D−2) 2ρ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' This is an exact spherically symmetric solution of the Einstein field equations in D-dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For D = 4 the solution given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (17) reduces to the solution obtained by Dymnikova [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The other components of energy momentum tensor can be obtained using the Einstein’s field equations which are given by T θ2 θ2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' = T θD−2 θD−2 = � D − 1 D − 2 � r r∗ �D−1 − 1 � ρ0e − rD−1 rD−1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (20) It is evident that in the usual 4 dimensions Dymnikova [25] black hole solutions recovered with anisotropic fluid distribution when r = r∗, which is true also in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The nonsingular black hole (NSBH) solutions are permitted with anisotropic fluid distributions in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The energy density and radial pressure follow the vacuum configuration but the tangential pressures do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The tangential pressure is non-zero which remains positive definite for r > r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' At the center the tangential pressure is negative indicating existence of exotic mat- ter (P⊥ < 0) at the center of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The nonsingular black hole solution obtained by Dymnikova can not be described in lower dimension D = 2 + 1, how- ever, we can extend the concept of NSBH in more than the usual four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The generalization of the black hole solution in higher dimensions accommodates a new class of NSBH solutions where the tangential pressure increases to a large extent inside the non-singular black hole with a different feature but away from the centre of the black hole it decreases exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The mass of a massive object in higher dimension is given by m(r) = AD−2 � r 0 r′D−2ρ(r′)dr′ (21) where AD−2 = 2π D−1 2 Γ ( D−1 2 ) which at r → ∞ is connected to the whole mass M con- nected with rD−3 g by the Schwarzschild relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The modulus difference between Rs(rg) and rD−3 g is rD−3 g e − rD−1 rD−1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The measure of the difference between the higher dimensional Schwarzschild mass m(r) ∼ rD−3 g in a singular black hole and Rs of a non-singular black hole is given by M − m(r) M = exp � −rD−1 rD−1 ∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (22) Here m(r) becomes M at infinite distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is found that the mass difference decreases as the dimension in which black hole embedded increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The metric has two event horizons located at r+ = rg � 1 − O � exp � −r2g r2 0 ��� , r− = r0 � 1 − O � exp � −r0 rg ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (23) 6 Bikash Chandra Paul Here r+ is the external event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The metric evaluated at gtt(r+) = 0 de- scribes an object with the similar properties properties of a black hole by a dis- tant observer, it does not send light signals outside and could not interact with its surroundings by the gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In four dimensions it is found that both r+ and r− are removable singularities of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The singularities can be eliminated by an appropriate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 3 Higher Dimensional Rastall gravity In this section we explore NSBH solution in the Rastall theory of gravity for D ≥ 4 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The Rastall theory [20] is based on the modification of the Einstein field equation for a spacetime with Ricci scalar filled by an energy momentum source as follows: T AB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' A = λRB (24) where λ is the Rastall parameter which is a measure for deviation from the stan- dard GR conservation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Consequently the Rastall field equation can be written as GAB + κ2λgABR = κ2TAB (25) where κ2 is the Rastall gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The above field equation reduces to that of GR in the limit λ → 0 and κ2 = 8πG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' However, for a vanishing trace of the energy-momentum tensor, for example the electrovacuum solution can be obtained when λ = 1 4 or R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is important to note that the former possibility is not physically acceptable as the trace of the energy momentum tensor vanishes T = 0 for any scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Consequently the matter configuration where the energy- momentum tensor has null trace, the relativistic solution obtained in Rastall theory is same as that one obtains in the general theory of relativity (GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' This feature of Rastall theory which is a modified GR led us to look for black holes solutions in a background of matter/energy with non-vanishing trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It may be pointed out here that the Rastall gravity is widely used to accommodate acceptable explanation for the current acceleration of the universe which has no solution in GR and for this it is interesting to explore NSBH in Rastall theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We consider the metric for black hole solution in higher dimensions D ≥ 4: ds2 = −f(r)dt2 + dr2 f(r) + r2dΩ2 D−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (26) Using the metric, we obtain the non-vanishing components of the Rastall tensor HAB = GAB + λgABR and κ2 = 1, Ht t = D − 2 2r2 � rf′ − (D − 3) + (D − 3)f� + λR, (27) Hr r = D − 2 2r2 � rf′ − (D − 3) + (D − 3)f� + λR, (28) Hθi θi = r2f′′ + (D − 3)(2rf′ + (D − 4)(f − 1) 2r2 + λR (29) where i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=', (D − 2), and the Ricci scalar in D dimensions is given by R = − 1 r2 � r2f′′ + 2(D − 2)rf′ + (D − 2)(D − 3)(f − 1) � (30) Nonsingular Black Holes in Higher dimensions 7 in the above we denote ()′ to represent derivative with respect to the radial co- ordinate r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We solve the field equation to obtain higher dimensional non-singular black holes in Rastall theory and for this Ht t = T t t and Hrr = T rr yield Pr = D − 2 2r2 � rf′ − (D − 3) + (D − 3)f� − λ r2 � r2f′′ + 2(D − 2)rf′ + (D − 2)(D − 3)(f − 1) � , (31) and also we consider Hθ1 θ1 = T θ1 θ1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' and HθD−2 θD−2 = T θD−2 θD−2 which yield P⊥ = 1 2r2 � r2f′′ + 2(D − 3)rf′ + (D − 3)(D − 4)(f − 1) � − λ r2 � r2f′′ + 2(D − 2)rf′ + (D − 2)(D − 3)(f − 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (32) In this case we explore the non-singular Black hole obtained in higher dimensional Rastall gravity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' the general solution of the metric is f(r) = 1 − rD−3 g rD−3 + 2ρ0rD−1 ∗ (D − 1)(D − 2) 1 rD−3 e − rD−1 rD−1 ∗ (33) The energy density and radial pressure are ρ = \uf8eb \uf8ed D − 2 − 2λD + 2(D − 1)λ rD−1 rD−1 ∗ D − 2 \uf8f6 \uf8f8 ρ0e − rD−1 rD−1 ∗ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (34) Pr = − \uf8eb \uf8ed D − 2 − 2λD + 2(D − 1)λ rD−1 rD−1 ∗ D − 2 \uf8f6 \uf8f8 ρ0e − rD−1 rD−1 ∗ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (35) the tangential pressure is given by P⊥ = � (1 − 2λ)D − 1 D − 2 rD−1 rD−1 ∗ − D − 2 − 2λD D − 2 � ρ0e − rD−1 rD−1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (36) The energy density and the transverse pressure in Rastall gravity framework ob- tained in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (34) and (36) reduces to the eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (14) and (20) in GR for λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The modification introduced in GR by Rastall admits nonsingular Dymnikova [25] black hole (NSBH) with normal matter while the radial pressure corresponds to vacuum equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' At the center of the NSBH the energy density is ρ = (D − 2 − 2λD)ρ0, which increases as the.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' number of spacetime dimension in- creases for a given range of Rastall parameter λ < D−2 2D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is evident that for a given dimension, NSBH admits greater mass for lower values of λ and the lower limiting value for λ < D−2 2D and |λ| > D−2 2D (for negative λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The corresponding tangential pressure at the center P⊥ = (2Dλ + 2 − D)ρ0 is finite but negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In D = 4 dimensions, at the center of the black hole, ρ(r = 0) = 2(1 − 4λ)ρ0 and tangential pressure P⊥ = −(1−4λ)ρ0 which indicates existence of exotic matter at the centre in GR (as λ = 0) as well as in Rastall theory for λ > − 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Thus NSBH can be realized with both central radial pressure and tangential pressure negative and equal but an anisotropy in pressure develops away from the center in Rastall 8 Bikash Chandra Paul gravity, normal matter exists when r > � D−2−2λD (D−1)(1−2λ �1/(D−1) r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The tangential pressure indicates black hole surrounded by exotic matter in Rastall gravity [35] for the range 1 4 < λ < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='. For r → ∞, the energy density and pressure vanishes asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' When λ = D−2 2D , we get the following : ρ = ρ0 � D − 1 D rD−1 rD−1 ∗ � e − rD−1 rD−1 ∗ , (37) Pr = −ρ = ρ0 � D − 1 D rD−1 rD−1 ∗ � e − rD−1 rD−1 ∗ , (38) the tangential pressure is given by P⊥ = 2ρ0 � D − 1 D(D − 2) rD−1 rD−1 ∗ � e − rD−1 rD−1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (39) one obtains NSBH with ρ > 0, ρ + Pr = 0 and P⊥ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For D = 4 dimensions, λ = 1 4 and the NSBH can be realized in Rastall gravity with normal matter which however is not permitted in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Also we note that at the centre of the NSBH the tangential pressure vanishes, admitting a perfect vacuum NSBH in the usual four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The result obtained in this case is also applicable in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' This is a new result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 4 Analytical set up The modified Schwarzschild metric for a non-singular black hole is given by ds2 = −f(r) dt2 + f(r)−1 dr2 + r2dΩ2 D−2 (40) where f(r) = 1 − � rg r �D−3 + � rg r �D−3 exp � − � r r∗ �D−1� , making use of the as- sumption κ2 = 8π made earlier, we write rg = � 16πM (D−2)AD−2 � 1 D−3 and the area of D dimensional sphere AD−2 = 2π D−1 2 Γ( D−1 2 ), where M represents the mass of the non-singular Black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The metric function gtt = f(r), whose sign determines gravitational trapping [34], we plot to draw a sketch to study the existence of black hole solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The metric potential f(r) is plotted with r in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (1) for D = 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (2) for D = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is evident that both the extreme black hole and non-extreme black holes can be obtained for a given set of values of rg and r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We note that extreme black hole exists for rg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 and r0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='57 in D = 4 and rg = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 and r0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='57 in D = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In the first case no black hole exist for rg < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 and the later case for r0 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The photon radii are tabulated in Table-I for D = 4 and Table-II for D = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is found that for D = 4, it increases with decrease of ρ0 ∼ 1/r2 0 for a given mass but for a given ρ0, photon radius is found to increase with mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In D = 10 dimensions as ρ0 is decreases the photon radius decreases then increases and decreases once again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In Fig (3) dimensional varia- tion of the photon radius for M = 1 is plotted for non-singular black holes with Nonsingular Black Holes in Higher dimensions 9 1 2 3 4 5 6 r �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 f�r� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 1 Radial variation of f(r) for rg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='8 (Red), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 (Black), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 (Blue) for r0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='57 in D = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 1 2 3 4 5 6 r �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 f�r� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 2 Radial variation of the metric function f(r) in D = 10 for r0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='1 (Blue), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='57 (Black) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 (Red) with rg = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The photon radius is maximum at D = 4 and then decreases sharply as the dimensions is increases and remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The Lagrangian is given by L = 1 2gAB ˙xA ˙xB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (41) 2 4 6 8 10 �1 0 1 2 3 r V Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 3 Radial variation of the potential for J= 5 (Blue), 6 (Green), 8 (Dashed), 10 (Red) in D = 4 for non-singular BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 10 Bikash Chandra Paul where ˙() = d dτ and τ is the affine parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Expanding eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (41) we get 2L = −f(r)˙t2 + 1 f(r) ˙r2 + (r2 ˙θ1 2 + sin2θ1 ˙θ2 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='.) (42) To obtain trajectory of light path, we set θi = π 2 where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=', D − 3 and θD−2 is a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The momenta are given by Pt = ∂L ∂ ˙t = −f(r)˙t, Pr = ∂L ∂ ˙r = 1 f(r) ˙r, Pθ1 = ∂L ∂ ˙θ1 = r2 ˙θ1, Pθ2 = ∂L ∂ ˙θ2 = r2sin2θ1 ˙θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='. (43) Now as defined above, θi = π 2 , and at the equatorial plane θ1 = π 2 , ∂L ∂ ˙t = constant (44) and we determine the energy (E) and angular momentum (J) at r → ∞ as f(r)˙t = E, PθD−2 = r2 ˙ θD−2 = J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (45) The Hamilton Jacobi equation is the most general method to find the geodesic equation of motion around black hole or a compact object, we adopt the technique to obtain the photon orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In higher dimensions we get ∂S ∂τ = H = −1 2gAB ∂S ∂xA ∂S ∂xB (46) where gAB is the inverse of the metric and S is the Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The Jacobian is given by S = 1 2m2τ − E + JθD−2 + Sr(r) + D−3 � i=1 Sθi(θi) (47) where Sr(r) and Sθi(θi) are functions of r and θi respectively and m is the mass of the test particle, it is zero for photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The Hamilton-Jacobi eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (46) can be written as r4 � 1 − Rs rD−3 �2 � ∂S ∂τ � = E2r4 − r2 � 1 − Rs rD−3 � (K + J2) (48) D−3 � i=1 1 Πi−1 n=1sin2θn �∂Sθi ∂θi �2 = K − ΠD−3 i=1 J2cot2θi (49) where K is the Carter constant [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Using the above eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (43) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (46) we get the following ˙t = E f(r), ˙θD−2 = J r2ΠD−3 i=1 sin2θi ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' r2 ˙r = ± √ R, r2 D−3 � i=1 Πi−1 n=1sin2θn ˙θi = ± � Θi (50) Nonsingular Black Holes in Higher dimensions 11 in the above ”+” and ”-” sign corresponds to motion of photon in outgoing and incoming radial direction and over dot represents derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='t to the affine parameterτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For the null curves the eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (49) can be expressed as R(r) = E2r4 − r2f(r)(K2 + J2), (51) Θi(θi) = K − ΠD−3 i=1 J2cot2θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (52) The characteristics of photon near the black hole can be defined by two impact parameters, which are functions of the constants E, J and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For general orbit we define the impact parameters ξ = J E and η = K E2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The boundary of the shadow of a black hole can be estimated from the effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The radial null geodesic from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (48) and (50) is given by � dr dτ �2 + Veff = 0, (53) where Veff is the effective potential, for radial motion we obtain Veff = f(r) r2 (K + J2) − E2 = 1 r2 � 1 − �rg r �D−3 � 1 − e−( r r∗ ) D−1�� (K + J2) − E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (54) The effective potential is identical to the classical equation describing the motion of a massless particle in a 1-dimensional potential V (r) provided its energy is 1 2E2 (of course the true energy should be E), but we use this form to obtain an expression for potential in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We plot radial variation of V (r) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (4) in a four dimensional universe for singular as well as non-singular black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' As the angular velocity increases the photons heading towards the black hole are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 2, it is found that there is no difference of the behaviour of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (4) we plot radial variation of V (r) for different angular momentum, for a non-singular BH, it is evident that as the angular momentum increases the photon can approach near to the BH unbounded The photon orbits are circular and unstable for a maximum value of the effec- tive potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The unstable circular orbit determines the boundary of the apparent shape and can be maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The maximal value of the effective potential corre- sponds to the circular orbits and the unstable photons satisfies Veff ��� r=rp = dVeff dr ��� r=rp = 0, R(r) = dR(r) dr ��� r=rc = 0 (55) The impact parameters are now related as Using eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (54) and (55), we get f(rp) r2p (K + J2) − E2 = 0 rpf′(rp) − 2f(rp) r3p (K + J2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (56) In four dimensions the potential V (r) is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (4) with different angular momentum (J) for rg = 2 and E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The particles are bounded for a radius r < rmin and unbounded for the range rmin < r < rmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The range of values 12 Bikash Chandra Paul rp in rp in rp in r0 M = 2 M = 5 M = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='8757 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0192 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='1532 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0221 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='1851 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='3389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='3079 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5053 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='6958 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5880 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='8142 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0383 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='1150 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='3702 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='990 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='9921 Table 1 The variation of the photon radius (rp) in D = 4 with r0 = � (D−1)(D−2) 4ρ0 and the mass of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' rp in rp in r0 M = 5 M = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='9369 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0035 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0002 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='8426 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0105 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5942 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='9106 Table 2 The variation of the photon radius (rp) in D = 10 with r0 = � (D−1)(D−2) 4ρ0 and the mass of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 3 4 5 6 7 8 9 10 D 2 4 6 8 10 Rp [t] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 4 Dimensional variation of the photon radius for M = 1 for a non-singular BH can be determined from the sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is found that rmin decreases as angular momentum (J) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We note that the potentials for Schwarzschild black hole (singular) and that for non-singular black holes overlaps for a set of similar values of D, J and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We draw the shadow contour of non-singular black hole in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (8) for a given value of ρ0 (say, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='16 unit) in all dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is shown that as the spacetime dimensions increases the radius of the shadow decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Nonsingular Black Holes in Higher dimensions 13 �10 �5 0 5 10 �10 �5 0 5 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 5 Contour plot for an object having M = 2M⊙ with ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='16 unit for D = 4 (Red), D = 5 (Dashed) and D = 6 (Green), D = 8 (Thick) �40 �20 0 20 40 �40 �20 0 20 40 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 6 Contour plot for an object for ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='04 unit in D = 4 with M = 2M⊙ (Black), M = 4M⊙ (Green), M = 6M⊙ (Red), M = 10M⊙ (Blue) 5 Effective potential and shadow behaviour The effective potential of the Schwarzschild-Tangherlini black holes exhibits a max- imum for the photon sphere radius rp corresponding to the real and the positive solution of the constraint obtained from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (56), rpf′(rp) − 2f(rp) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (57) Defining impact parameters η and ξ that are functions of the energy E, angular momentum J and the Carter constant K as ξ = J E , η = K E2 (58) we get from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (55) corresponding to Veff E2 = 0 and R E2 = 0, the following η + ξ2 = r2p f(rp), η + ξ2 = 4r2p rf′(rp) + 2f(rp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (59) 14 Bikash Chandra Paul Now we obtain η + ξ2 = 5r2p rpf′(rp) + 3f(rp), (60) where the right hand side corresponds to r2 p f(rp), the observer’s frame the shadow can be described properly making use of the celestial coordinates α and β as introduced earlier [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Following the definition introduced by Subrahmanyan as follows α = lim rp→∞ � rpP θD−2 P t � , βi = lim rp→∞ � rpP θi P t � , (61) where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='(D − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For an observer on the equatorial plane, these equations reduced to η + ξ2 = α2 + β2 = r2p f(rp) (62) the radius of the shadow is Rbhs = rp √ f(rp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The form of f(r) is complex and therefore we study numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The photon radius depends on the dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The photon radius is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 4, it is evident that as the mass of the black hole increases the radius decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is maximum in D = 4 but decreases sharply as the dimension increases but almost constant with the increase in dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The fig (9) shows that as the mass increases the radius of the shadow also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 6 Discussion We obtain non-singular black hole (NSBH) solutions in the higher dimensional Einstein’s general theory of gravity (GR) and found that the methods in GR can be adopted also in Rastall gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Considering a specific exponential form of the energy density we obtain NSBH which reduces to the Dymnikova NSBH solution [25] obtained in the usual four dimensions (D = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In 2+1 dimensions no black hole solution exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' However, a non-rotating NSBH solutions obtained by Dymnikova in four dimensional GR can be accommodated in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We obtained NSBH in a vacuum described by T t t + T rr = 0 with p⊥ < 0 (where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=', D − 2) near the center indicating requirement of exotic matter which however extends up to certain height thereafter p⊥ > 0 for r > � D−2 D−1 � 1 D−1 r∗ in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' But in the Rastall gravity p⊥ > 0 for r > � 2−D+2λD (D−1)(2λ−1) � 1 D−1 r∗ when λ ̸= D−2 2D and thereafter at a large distance it vanishes because the pressure decreases rapidly i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=', exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Both in GR and modified gravity it indicates existence of exotic matter near the center of the NSBH but in the later case the Rastall parameter plays an important role in determining the distance from the centre where the normal matter exists in the tangential direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In the usual four dimensions at the center of the NSBH in the modified theory we get the following estimations ρ(r = 0) = 2(1 − 4λ)ρ0 and tangential pressure P⊥ = −2(1 − 4λ) which are determined by the Rastall parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is evident that exotic matter at the center of the black hole requires both in GR (as λ = 0) as well as in Rastall theory with the lower limiting value of the Rastall parameter λ < 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The tangential pressure is negative it indicates Nonsingular Black Holes in Higher dimensions 15 NSBH surrounded by exotic matter in Rastall gravity, existence of BH with exotic matter also reported in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Thus NSBH is realized with both the central radial pressure and tangential pressure negative and equal initially but an anisotropy in pressure develops away from the center in Rastall gravity with normal matter thereafter when r > � D−2+2λD (1−2λ)(D−2) � 1 D−1 r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' However, we note a new and interesting result in Rastall theory that permits a NSBH with normal matter in the usual four and in higher dimensions when λ = D−2 2D , which however is not permitted in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In It is also noted that away from the center at a large distance, the tangential pressure remains positive definite at a maximum radial distance which is P⊥ = (1−2λ) D−1 D−2 rD−1 rD−1 ∗ ρ0e − rD−1 rD−1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Thus one gets a physically realistic NSBH for 1 4 < λ < 1 2 in D = 4 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For r → ∞, both the energy density and pressure vanishes asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Thus the Rastall gravity has rich structure which unearth the structure of non-singular black hole even with normal matter for a restricted domain of the Rastall parameter depending on the embedding spacetime dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Thus, we see that for λ ̸= 0 the Rastall theory plays an important role leading to distinct solutions relative to GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The sketch of the potentials permissible in the theory are plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (1) and (2), which show that both extreme and non-extreme black holes exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The contour plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (5) for NSBH shows that the circular shadow radius decreases as the spacetime dimension is increased for a given mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The circular shadow radius in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (6) show that the radii increases with the mass of the compact objects for a given dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The rotating NSBH will be taken up elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Acknowledgment The author would like to thank IUCAA , Pune and IUCAA Centre for Astronomy Research and Development (ICARD), NBU for extending research facilities and North Bengal University for a research grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' BCP acknowl- edge the suggestions and constructive criticism of the anonymous Referee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' References 1.' metadata={'source': 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a/MNE4T4oBgHgl3EQfKAyT/content/tmp_files/2301.04926v1.pdf.txt b/MNE4T4oBgHgl3EQfKAyT/content/tmp_files/2301.04926v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c5def8732611ef20a1beae1e4cb1a8b659887dbf --- /dev/null +++ b/MNE4T4oBgHgl3EQfKAyT/content/tmp_files/2301.04926v1.pdf.txt @@ -0,0 +1,1133 @@ +CLIP2Scene: Towards Label-efficient 3D Scene Understanding by CLIP +Runnan Chen1, Youquan Liu2, Lingdong Kong3, Xinge Zhu6, Yuexin Ma5, +Yikang Li4, Yuenan Hou4, Yu Qiao4, Wenping Wang7 +1The University of Hong Kong +2Hochschule Bremerhaven +3National University of Singapore +4Shanghai AI Lab +5ShanghaiTech University +6The Chinese University of Hong Kong +7Texas A&M University +Abstract +Contrastive +language-image +pre-training +(CLIP) +achieves promising results in 2D zero-shot and few-shot +learning. +Despite the impressive performance in 2D +tasks, applying CLIP to help the learning in 3D scene +understanding has yet to be explored. In this paper, we +make the first attempt to investigate how CLIP knowledge +benefits 3D scene understanding. To this end, we propose +CLIP2Scene, a simple yet effective framework that transfers +CLIP knowledge from 2D image-text pre-trained models to +a 3D point cloud network. We show that the pre-trained +3D network yields impressive performance on various +downstream tasks, i.e., annotation-free and fine-tuning +with labelled data for semantic segmentation. Specifically, +built upon CLIP, we design a Semantic-driven Cross-modal +Contrastive Learning framework that pre-trains a 3D +network via semantic and spatial-temporal consistency +regularization. For semantic consistency regularization, we +first leverage CLIP’s text semantics to select the positive +and negative point samples and then employ the contrastive +loss to train the 3D network. In terms of spatial-temporal +consistency regularization, we force the consistency be- +tween the temporally coherent point cloud features and +their corresponding image features. +We conduct experi- +ments on the nuScenes and SemanticKITTI datasets. For +the first time, our pre-trained network achieves annotation- +free 3D semantic segmentation with 20.8% mIoU. When +fine-tuned with 1% or 100% labelled data, our method +significantly outperforms other self-supervised methods, +with improvements of 8% and 1% mIoU, respectively. +Furthermore, we demonstrate its generalization capability +Semantic and +Spatial-Temporal +Consistency +Regularization +Image +Encoder +Annotation-free +1% annotation +CLIP2Scene +Text +Encoder +CLIP +How CLIP benefits +3D scene +understanding? +100% annotation +Semantic-driven Cross-modal Contrastive Learning +car, bus +pedestrian +car +Figure 1. We explore whether and how CLIP knowledge benefits +3D scene understanding. To this end, we propose CLIP2Scene, a +semantic-driven cross-modal contrastive learning framework that +leverages CLIP knowledge to pre-train a 3D point cloud seg- +mentation network via semantic and spatial-temporal consistency +regularization. +CLIP2Scene yields impressive performance on +annotation-free 3D semantic segmentation and significantly out- +performs other self-supervised methods when fine-tuning on an- +notated data. +for handling cross-domain datasets. +1. Introduction +3D scene understanding is fundamental in autonomous +driving, robot navigation, etc [24, 26]. +Current deep +arXiv:2301.04926v1 [cs.CV] 12 Jan 2023 + +learning-based methods have shown inspirational perfor- +mance on 3D point cloud data [37, 50, 29, 44, 15, 45]. +However, some drawbacks hinder their real-world applica- +tions. The first one comes from their heavy reliance on the +large collection of the annotated point clouds, especially +when high-quality 3D annotations are expensive to acquire +[34, 40]. Besides, they typically fail to recognize novel ob- +jects that are never seen in the training data [11, 35]. As +a result, it may need extra annotation efforts to train the +model on recognizing these novel objects, which is both te- +dious and time-consuming. +Contrastive Vision-Language Pre-training (CLIP) [38] +provides a new perspective that mitigates the above issues +in 2D vision. It was trained on large-scale free-available +image-text pairs from websites and built vision-language +correlation to achieve promising open-vocabulary recogni- +tion. MaskCLIP [49] further explores semantic segmen- +tation based on CLIP. With minimal modifications to the +CLIP pre-trained network, MaskCLIP can be directly used +for the semantic segmentation of novel objects without ad- +ditional training efforts. PointCLIP [48] reveals that the +zero-shot classification ability of CLIP can be generalized +from the 2D image to the 3D point cloud. It perspectively +projects a point cloud frame into different views of 2D depth +maps that bridge the modal gap between the image and +the point cloud. The above studies indicate the potential +of CLIP on enhancing the 2D segmentation and 3D clas- +sification performance. However, whether and how CLIP +knowledge benefits 3D scene understanding is still under- +explored. +In this paper, we explore how to leverage CLIP’s 2D +image-text pre-learned knowledge for 3D scene understand- +ing. Previous cross-modal knowledge distillation methods +[40, 34] suffer from the optimization-conflict issue, i.e., +some of the positive pairs are regarded as negative sam- +ples for contrastive learning, leading to unsatisfactory rep- +resentation learning and hammering the performance of +downstream tasks. Besides, they also ignore the tempo- +ral coherence of the multi-sweep point cloud, failing to +utilize the rich inter-sweep correspondence. +To handle +the mentioned problems, we propose a novel Semantic- +driven Cross-modal Contrastive Learning framework that +fully leverages CLIP’s semantic and visual information to +regularize a 3D network. Specifically, we propose Seman- +tic Consistency Regularization and Spatial-Temporal Con- +sistency Regularization. In semantic consistency regular- +ization, we utilize CLIP’s text semantics to select the posi- +tive and negative point samples for less-conflict contrastive +learning. For spatial-temporal consistency regularization, +we take CLIP’s image pixel feature to impose a soft con- +straint on points within local space and time. Such oper- +ation also prevents the network from degenerating due to +image-to-point calibration errors. +We conduct several downstream tasks on nuScenes to +verify how the pre-trained network benefits the 3D scene +understanding. +The first one is annotation-free semantic +segmentation. Following MaskCLIP, we place class names +into multiple hand-crafted templates as prompts and av- +erage the text embeddings generated by CLIP to conduct +the annotation-free segmentation. For the first time, our +method achieves 20.8% mIoU annotation-free 3D semantic +segmentation without any labelled data for training. Sec- +ondly, we compare with other self-supervised methods to +verify the superiority of our method in label-efficient learn- +ing. When fine-tuning the 3D network with 1% or 100% la- +belled data, our method significantly outperforms state-of- +the-art self-supervised methods, with improvements of 8% +and 1% mIoU, respectively. Besides, to verify the general- +ization capability, we pre-train the network on the nuScenes +dataset and evaluate it on the SemanticKITTI dataset. Our +method still significantly outperforms state-of-the-art meth- +ods. +The contributions of our work are summarized as fol- +lows. +• The first work that distils CLIP knowledge to a 3D net- +work for 3D scene understanding. +• We propose a novel Semantic-driven Cross-modal +Contrastive Learning framework that pre-trains a 3D +network via spatial-temporal and semantic consistency +regularization. +• We +propose +a +novel +Semantic-guided +Spatial- +Temporal Consistency Regularization that forces the +consistency between the temporally coherent point +cloud features and their corresponding image features. +• For the first time, our method achieves promising +performance on annotation-free 3D scene segmenta- +tion and significantly outperforms state-of-the-art self- +supervised methods when fine-tuning with labelled +data. +2. Related Work +Zero-shot Learning in 3D. The objective of zero-shot +learning (ZSL) is to recognize objects that are unseen in +the training set. Many efforts have been devoted to the 2D +recognition tasks [8, 30, 47, 36, 31, 1, 43, 32, 4, 2, 19, 33, +23], and few works concentrate on performing ZSL in the +3D domain [18, 11, 35, 16, 17]. [18] makes the first at- +tempt to apply ZSL to 3D tasks, where they train PointNet +[37] on ”seen” samples and test on ”unseen” samples. Sub- +sequent work [16] addresses the hubness problem caused +by the low-quality point cloud features. [17] proposes the +triplet loss to boost the performance under the transductive +setting, where the ”unseen” class is observed and unlabeled + +Spatial-Temporal Consistency Regularization +Image +Encoder +Text +Encoder +car, bus +Pedestrian +… +A +photo +of a { }; +This is +the { } +in the +scene; +… +Semantic Consistency Regularization +Point +Encoder +CLIP +pixel-to-text mapping +pixel-point-text pairs +pixel-to-point mapping +3D Network +point-text pairs +… … … +… +… +… +… +Multi-sweeps +calibration +… +… +grid 1 +grid 2 +grid 3 +pulling force +Semantic-guided fusion features +text embedding +point feature +text embedding +point feature +pixel feature +point feature +prompts +𝑃1 +𝑃2 +𝑃3 +image feature +… … +… +… +grid 1 +grid 2 +grid 3 +Figure 2. Illustration of the Semantic-driven Cross-modal Contrastive Learning. Firstly, we obtain the text embeddings ti, image pixel +feature xi, and point feature pi by text encoder, image encoder, and point encoder, respectively. Secondly, we leverage CLIP knowledge to +construct positive and negative samples for contrastive learning. Thus we obtain point-text pairs {xi, ti}M +i=1 and all pixel-point-text pairs +in a short temporal {ˆxk +i , ˆpk +i , tk +i } +ˆ +M,K +i=1,k=1. Here, {xi, ti}M +i=1 and {ˆxk +i , ˆpk +i , tk +i } +ˆ +M,K +i=1,k=1 are used for Semantic Consistency Regularization and +Spatial-Temporal Consistency Regularization, respectively. Lastly, we perform Semantic Consistency Regularization by pulling the point +features to their corresponding text embedding and Spatial-Temporal Consistency Regularization by mimicking the temporally coherent +point features to their corresponding pixel features. +in the training phase. Recently, some studies introduced +CLIP into zero-shot learning. MaskCLIP [49] investigates +the problem of utilizing CLIP to help the 2D dense pre- +diction tasks and exhibits encouraging zero-shot semantic +segmentation performance. PointCLIP [48] is the pioneer- +ing work that applies CLIP to 3D recognition. As opposed +to previous approaches that require training on the labelled +point cloud, PointCLIP is free from any 3D training and +shows impressive performance on zero-shot and few-shot +classification tasks. Our work takes a step further to inves- +tigate whether the rich semantic and visual knowledge in +CLIP can benefit the 3D semantic segmentation tasks. +Self-supervised Representation Learning. The purpose +of self-supervised learning is to learn a good representa- +tion that benefits the downstream tasks. The dominant ap- +proaches resort to contrastive learning to pre-train the net- +work [27, 25, 21, 20, 14, 13, 7, 10, 12, 9]. Recently, inspired +by the success of CLIP, leveraging the pre-trained model +of CLIP to the downstream tasks has raised the commu- +nity’s attention. DenseCLIP [39] utilizes the CLIP’s pre- +trained knowledge for dense image pixel prediction. Det- +CLIP [46] proposes a pre-training method equipped with +CLIP for open-world detection. In this paper, we make the +first attempt to pre-train a 3D network with CLIP’s knowl- +edge for 3D scene understanding. +Cross-modal Knowledge Distillation. +Recently, an in- +creasing number of researchers have focused on transferring +the knowledge in 2D images to 3D point cloud [34, 40]. +PPKT [34] proposes the contrastive pixel-to-point knowl- +edge transfer to utilize the rich information in image back- +bones. SLidR [40] resorts to the InfoNCE loss to help the +3D network distil rich knowledge from the 2D image back- +bone. Our work explores leveraging the image-text pre- +trained CLIP knowledge to help 3D scene understanding. +3. Methodology +Considering the impressive open-vocabulary perfor- +mance achieved by CLIP in image classification and seg- +mentation, natural curiosities have been raised. Can CLIP +endow the ability to a 3D network for annotation-free +scene understanding? +And further, will it promote the +network performance when fine-tuned on labelled data? +To answer the above questions, we study the cross-modal +knowledge transfer of CLIP for 3D scene understanding, +termed CLIP2Scene. Our work is a pioneer in exploiting +CLIP knowledge for 3D scene understanding. In what fol- +lows, we revisit the CLIP applied in 2D open-vocabulary +classification and semantic segmentation, then present our +CLIP2Scene in detail. +Our approach consists of three +major components: Semantic Consistency Regularization, +Semantic-Guided Spatial-Temporal Consistency Regular- +ization, and Switchable Self-Training Strategy. + +car +road +bicycle +... +building +Text embeddings +Figure 3. Illustration of the image pixel-to-text mapping. +The +dense pixel-text correspondence {xi, ti}M +i=1 is extracted by the +off-the-shelf method MaskCLIP [49]. +3.1. Revisiting CLIP +Contrastive Vision-Language Pre-training (CLIP) miti- +gates the following drawbacks that dominate the computer +vision field: 1. Deep models need a large amount of for- +matted and labelled training data, which is expensive to ac- +quire; 2. The model’s generalization ability is weak, mak- +ing it difficult to migrate to a new scenario with unseen +objects. CLIP consists of an image encoder (ResNet [28] +or ViT [6]) and a text encoder (Transformer [42]), both +respectively project the image and text representation to a +joint embedding space. During training, CLIP constructs +positive and negative samples from 400 million image-text +pairs to train both encoders with a contrastive loss, where +the large-scale image-text pairs are free-available from the +Internet and assumed to contain every class of images and +most concepts of text. Therefore, CLIP can achieve promis- +ing open-vocabulary recognition. +For 2D zero-shot classification, CLIP first places the +class name into a pre-defined template to generate the text +embeddings and then encodes images to obtain image em- +beddings. +Next, it calculates the similarity matrices be- +tween images and text embeddings to determine the class. +MaskCLIP further extends CLIP into 2D semantic segmen- +tation. Specifically, MaskCLIP modifies the attention pool- +ing layer of the CLIP’s image encoder, thus performing +pixel-level mask prediction instead of the global image- +level prediction. +3.2. CLIP2Scene +As shown in Fig. 2, we first leverage CLIP and 3D net- +work to respectively extract the text embeddings, image +pixel feature and point feature. +Secondly, we construct +positive and negative samples based on CLIP’s knowledge. +Lastly, we impose Semantic Consistency Regularization by +pulling the point features to their corresponding text embed- +ding. At the same time, we apply Spatial-Temporal Con- +sistency Regularization by forcing the consistency between +temporally coherent point features and their corresponding +pixel features. In what follows, we present the details and +insights. +3.2.1 +Semantic Consistency Regularization +As CLIP is pre-trained on 2D images and text, our first con- +cern is the domain gap between 2D images and the 3D point +cloud. To this end, we build dense pixel-point correspon- +dence and transfer image knowledge to the 3D point cloud +via the pixel-point pairs. Specifically, we calibrate the Li- +DAR point cloud with corresponding images captured by +six cameras. Therefore, the dense pixel-point correspon- +dence {xi, pi}M +i=1 can be obtained accordingly, where xi +and pi indicates i-th paired image feature and point feature, +which are respectively extracted by the CLIP’s image en- +coder and the 3D network. M is the number of pairs. Note +that it is an online operation and is irreverent to the image +and point data augmentation. +Previous methods [40, 34] provide a promising solution +to cross-modal knowledge transfer. +They first construct +positive pixel-point pairs {xi, pi}M +i=1 and negative pairs +{xi, pj}(i ̸= j), and then pull in the positive pairs while +pushing away the negative pairs in the embedding space via +the InfoNCE loss. Despite the encourageable performance +of previous methods in transferring cross-modal knowl- +edge, they are both confronted with the same optimization- +conflict issue. For example, suppose i-th pixel xi and j-th +point pj are in the different positions of the same instance +with the same semantics. However, the InfoNCE loss will +try to push them away, which is unreasonable and ham- +mer the performance of the downstream tasks [40]. In light +of this, we propose a Semantic Consistency Regularization +that leverages the CLIP’s semantic information to allevi- +ate this issue. Specifically, we generate the dense pixel- +text pairs {xi, ti}M +i=1 by following the off-the-shelf method +MaskCLIP [49] (Fig. 3), where ti is the text embedding gen- +erated from the CLIP’s text encoder. Note that the pixel-text +mappings are free-available from CLIP without any addi- +tional training. We then transfer pixel-text pairs to point- +text pairs {pi, ti}M +i=1 and utilize the text semantics to se- +lect the positive and negative point samples for contrastive + +Image 𝐼 +Pixel-to-point mapping +𝑃1 +𝑃2 +𝑃3 +Multi-sweeps calibration +… … … +… +… +grid 1 +grid 2 +grid 3 +… +… +grid 1 +grid 2 +grid 3 +{𝑓𝑛}𝑛=1 +𝑁 +ො𝑥𝑖 +𝑘, ො𝑝𝑖 +𝑘 +𝑛=1,𝑘=1 +𝑁,𝐾 +Text embedding +Figure 4. Illustration of the image pixel-to-point mapping (left) +and semantic-guided fusion feature generation (right). We build +the grid-wise correspondence between an image I and the tem- +porally coherent LiDAR point cloud {Pk}K +k=1 within S seconds +and generate semantic-guided fusion features for individual grids. +Both {ˆxk +i , ˆpk +i } +ˆ +M,K +i=1,k=1 and {fn}N +n=1 are used to perform Spatial- +Temporal Consistency Regularization. +learning. The objective function is as follows: +LS info = − +C +� +c=1 +log +� +ti∈c,pi exp(D(ti, pi)/τ) +� +ti∈c,tj /∈c,pj exp(D(ti, pj)/τ), +(1) +where ti ∈ c indicates that ti is generated by c-th classes +name, and C is the number of classes. D denotes the scalar +product operation and τ is a temperature term (τ > 0). +Since the text is composed of class names placed into +pre-defined templates, the text embedding represents the se- +mantic information of the corresponding class. Therefore, +those points with the same semantics will be restricted near +the same text embedding, and those with different semantics +will be pushed away. To this end, our Semantic Consistency +Regularization causes less conflict in contrastive learning. +3.2.2 +Semantic-guided Spatial-temporal Consistency +Regularization +Besides semantic consistency regularization, we consider +how image pixel features help to regularize a 3D network. +The natural alternative directly pulls in the point feature +with its corresponding pixel in the embedding space. How- +ever, after trial and error, we observe that the network easily +degenerates and achieves poor performance in the down- +stream tasks when following the aforementioned strategy. +The main reason lies in the noise-assigned semantics of the +image pixel and the imperfect pixel-point mapping caused +by the calibration errors. To this end, we propose a novel +semantic-guided Spatial-Temporal Consistency Regulariza- +tion to alleviate the problem by imposing a soft constraint +on points within local space and time. +Specifically, given an image I and temporally coherent +LiDAR point cloud {Pk}K +k=1, where K is the number of +sweeps within S seconds. Note that the image is matched +to the first frame of the point cloud P1 with pixel-point pairs +{ˆx1 +i , ˆp1 +i } ˆ +M +i=1. We register the rest of the point cloud to the +first frame via the calibration matrices and map them to the +image (Fig. 4). Thus we obtain all pixel-point-text pairs +in a short temporal {ˆxk +i , ˆpk +i , tk +i } +ˆ +M,K +i=1,k=1. Next, we divide +the entire stitched point cloud into regular grids {gn}N +n=1, +where the temporally coherent points are located in the +same grid. We impose the spatial-temporal consistency con- +straint within individual grids by the following objective +function: +LSSR = +� +gn +� +(ˆi,ˆk)∈gn +(1 − sigmoid(D(ˆp +ˆk +ˆi , fn)))/N, (2) +where (ˆi, ˆk) ∈ gn indicates the pixel-point pair {ˆxk +i , ˆpk +i } +is located in the n-th grid. {fn}N +n=1 is a semantic-guided +cross-modal fusion feature formulated by: +fn = +� +(ˆi,ˆk)∈gn +a +ˆk +ˆi ∗ ˆx +ˆk +ˆi + b +ˆk +ˆi ∗ ˆp +ˆk +ˆi , +(3) +where aˆk +ˆi and bˆk +ˆi are attention weight calculated by: +a +ˆk +ˆi = +exp(D(ˆxˆk +ˆi , t1 +ˆi )/λ) +� +(ˆi,ˆk)∈gn exp(D(ˆxˆk +ˆi , t1 +ˆi )/λ) + exp(D(ˆpˆk +ˆi , t1 +ˆi )/λ) +, +b +ˆk +ˆi = +exp(D(ˆpˆk +ˆi , t1 +ˆi )/λ) +� +(ˆi,ˆk)∈gn exp(D(ˆxˆk +ˆi , t1 +ˆi )/λ) + exp(D(ˆpˆk +ˆi , t1 +ˆi )/λ) +, +(4) +where λ is the temperature term. +Actually, those pixel and point features within the local +grid gn are restricted near a dynamic centre fn. Thus, such a +soft constraint alleviates the noisy prediction and calibration +error issues. At the same time, it imposes Spatio-Temporal +Regularization on the temporally coherent point features. +3.2.3 +Switchable Self-training Strategy +We combine the loss function LS info and LSSR to end- +to-end train the whole network, where the CLIP’s image +and text encoder backbone are frozen during training. We +find that method worked only when the pixel-point feature +{xi, pi}M +i=1 and {ˆxk +i , ˆpk +i } +ˆ +M,K +i=1,k=1, which are used in LS info +and LSSR, are generated from different learnable linear +layer. On top of that, we further put forward an effective +strategy to promote performance. Specifically, after con- +trastive learning of the 3D network for a few epochs, we +randomly switch the point labels between the paired im- +age pixel’s labels and their own predictions for self-training. +Merely training the 3D network with their own predictions +yields satisfactory performance. Essentially, such a Switch- +able Self-Training Strategy (S3) increases the number of + +Table 1. Ablation study experiments on the nuScenes validation +dataset for annotation-free semantic segmentation. +Ablation target +Settings +mIoU(%) +- +baseline +15.1 +Prompts +nuScenes +15.1 +semanticKITTI +13.9 +Cityscapes +11.3 +Regularization +w/o SCR +19.8 +KL +0 +Training Strategies +w/o S3 +18.8 +ST +10.1 +Sweeps +1 sweep +18.7 +3 sweeps +20.8 +5 sweeps +20.6 +merged +18.6 +- +CLIP2Scene +20.8 +positive and negative samples by switching the point pseudo +labels, which benefits cross-modal knowledge distillation. +4. Experiments +Datasets. +We conduct experiments on two large-scale +outdoor +LiDAR +segmentation +benchmarks, +i.e., +Se- +manticKITTI [3] and nuScenes [5, 22]. +The nuScenes +dataset contains 700 scenes for training, 150 scenes for +validation and 150 scenes for testing, where 16 classes +are utilized for LiDAR semantic segmentation. As to Se- +manticKITTI, it contains 19 classes for training and evalu- +ation. It has 22 sequences, where sequences 00 to 10, 08 +and 11 to 21 are used for training, validation and testing, +respectively. +Implementation Details. +We use the nuScenes [5, 22] +dataset to pre-train the network. +Following SLidR, we +pre-train the network on all key frames from 600 scenes. +Besides, we fine-tune the pre-trained network on Se- +manticKITTI [3] to verify the generalization ability. We +leverage CLIP’s image encoder and text encoder to gener- +ate image features and text embedding, respectively. Fol- +lowing MaskCLIP, we modify the attention pooling layer of +the CLIP’s image encoder, thus extracting the dense pixel- +text correspondences. We take SPVCNN [41] as the 3D +network to produce the point-wise feature. The whole net- +work is trained on the PyTorch platform. The training time +is about 40 hours for 20 epochs on two NVIDIA Tesla A100 +GPUs. For the switchable self-training strategy, we ran- +domly switch the point supervision signal after 10 epochs. +The optimizer is SGD with a cosine scheduler. We set the +temperature λ and τ to be 1 and 0.5, respectively. +The +sweep number is set to be 3 empirically. We apply sev- +eral data augmentations in contrastive learning, including +random rotation around the z-axis and random flip on the +Table 2. Comparison of different self-supervised methods for se- +mantic segmentation on the nuScenes and SemanticKITTI valida- +tion datasets. +Initialization +nuScenes +semanticKITTI +1% +100% +1% +Random +42.2 +69.1 +32.5 +PPKT [34] +48.0 +70.1 +39.1 +SLidR [40] +48.2 +70.4 +39.6 +CLIP2Scene +56.3 +71.5 +42.6 +point cloud, random horizontal flip and random crop-resize +on the image. +4.1. Annotation-free Semantic Segmentation +After pre-training the network, we show the performance +of the 3D network when it is not fine-tuned on any annota- +tions. As no previous method reports the 3D annotation-free +segmentation performance, we compare our method with +different setups (Table 1). In what follows, we describe the +experimental settings and give insights into our method and +the different settings. +Settings. We conduct experiments on the nuScenes dataset +to evaluate the annotation-free semantic segmentation per- +formance. Following MaskCLIP [49], we place the class +name into 85 hand-craft prompts and feed it into the CLIP’s +text encoder to produce multiple text features. We then av- +erage the text features and feed the averaged features to the +classifier for point-wise prediction. Besides, to explore how +to effectively transfer CLIP’s knowledge to the 3D network +for annotation-free segmentation, We conduct the following +experiments to highlight the effectiveness of different mod- +ules in our framework. +Baseline. The input of the 3D network is only one sweep, +and we pre-train the framework via semantic consistency +regularization. +Prompts (nuScenes, semanticKITTI, Cityscapes). Based +on the baseline, we respectively replace the nuScenes, se- +manticKITTI, and Cityscapes class names into the prompts +to produce the text embedding. +Regularization (w/o STR, KL). Based on the full method, +we remove the Spatial-temporal Consistency Regulariza- +tion (w/o SCR). Besides, we abuse both SR and SCR +and distill the image feature to the point cloud by Kull- +back–Leibler (KL) divergence loss. +Training Strategies (w/o S3, ST). We abuse the Switchable +Self-Training Strategy (w/o S3) in the full method. Besides, +we show the performance of only training the 3D network +by their own predictions after ten epochs (ST). +Sweeps Number (1 sweep, 3 sweeps, 5 sweeps, and +merged). We set the sweep number K to be 1, 3, and 5, re- +spectively. Besides, we also take three sweeps of the point +cloud as the input to pre-train the network. +Effect of Different Prompts. +To verify how text em- + +Ground truth +Ours* +Ours +Bus +Motorcycle +Car +Truck +Figure 5. Qualitative results of annotation-free semantic segmentation on nuScenes dataset. Note that we show the results by individual +class. From the left to the right column are the bus, motorcycle, car and truck, respectively. The first row is the ground truth; The second +row (ours*) is our prediction of the highlighted target; the third row is our prediction of full classes (ours). +bedding affects the performance, we generate various text +embeddings by the class name from different datasets +(nuScenes, SemanticKITT, and Cityscapes) for pre-training +the framework. +As shown in Table 1, we find that +even learning with other datasets’ text embedding (se- +manticKITT and Cityscapes), the 3D network could still +recognize the nuScenes’s objects with decent performance +(13.9 and 11.3 mIoU, respectively). The result shows that +the 3D network is capable of open-vocabulary recognition. +Effect of Semantic and Spatial-temporal Consistency +Regularization. We remove Spatial-temporal Consistency +Regularization (w/o SCR) from our method. Experiments +show that the performance is dramatically decreased, indi- +cating the effectiveness of our design. Besides, we also dis- +till the image feature to the point cloud by KL divergence +loss, where the text embeddings calculate the logits. How- +ever, such a method fails to transfer the semantic informa- +tion from the image. The main reason is the noise-assigned +semantics of the image pixel and the imperfect pixel-point +correspondence due to the calibration error. +Effect of Switchable Self-training Strategy. To examine +the effect of the Switchable Self-Training Strategy, we ei- +ther train the network with image supervision (w/o S3) or +train the 3D network by their own predictions. Both tri- +als witness the performance drop, indicating our Switch- +able Self-Training Strategy is efficient in cross-modal self- +supervised learning. The main reason is that the number of +positive and negative samples is enlarged by switching the +supervision signal. +Effect of Sweep Numbers. Intuitively, the performance of +our method benefits from more sweeps information. There- +fore, we also show the performance when restricting sweep +size to 1, 3, and 5, respectively. However, we observe that +the performance of 5 sweeps is similar to 3 sweeps but is +more computationally expensive. Thus, we empirically set +the sweep number to be 3. +Qualitative Evaluation. We show the qualitative evalua- +tion in Fig. 5. Note that we show the results by individ- +ual class (construction vehicle, truck, and car). The results +show that our method is able to perceive the objects without +any annotation training data. However, we also observe the +false positive predictions around the ground truth objects. +We will resolve this issue in future work. +4.2. Annotation-efficient Semantic Segmentation +Besides annotation-free semantic segmentation, the pre- +trained 3D network also boosts the performance when it +is fine-tuned on labelled data. To the best of our knowl- +edge, only one published method SLidR studies image-to- +Lidar self-supervised representation distillation. We also +compared our method with another self-supervised method +PPKT [34] for 3D network pre-training. +In the follow- +ings, we first introduce SLidR [40] and PPKT, then compare +them in detail. +PPKT. PPKT is a cross-modal self-supervised method for +the RGB-D dataset. It performs 2D-to-3D knowledge dis- +tillation via pixel-to-point contrastive loss. Since there is +no public code, we re-implement it for a fair comparison. + +Input +Ground Truth +SLidR +Ours +Figure 6. Qualitative results of fine-tuning on 1% nuScenes dataset. From the first row to the last row are the input Lidar scan, ground truth, +prediction of SLidR, and our prediction, respectively. Note that we show the results by error map, where the red point indicates the wrong +prediction. Apparently, our method achieves decent performance. +Specifically, we use the same 3D network and training pro- +tocol but replace our semantic and Spatio-Temporal Reg- +ularization with InfoNCE loss. The framework is trained +on 4, 096 randomly selected image-to-point pairs for 50 +epochs. +SLidR. SLidR is an image-to-Lidar self-supervised method +for autonomous driving data. +Compared with PPKT, +it introduces image super-pixel into cross-modal self- +supervised learning. For a fair comparison, we replace our +loss function with their superpixel-driven contrastive loss. +Performance. As shown in Table 2, our method signifi- +cantly outperforms the state-of-the-art methods when fine- +tuned on 1% and 100% data, with the improvement of +8.1% and 1.1%, respectively. +Compared with the ran- +dom initialization, the improvement is 14.1% and 2.4%, re- +spectively, indicating the efficiency of our semantic-driven +cross-modal contrastive learning framework. The qualita- +tive results are shown in Fig. 6. Besides, we also verify the +cross-domain generalization ability of our method. When +pre-training the 3D network on the nuScenes dataset and +fine-tuning on 1% SemanticKITTI dataset, our method sig- +nificantly outperforms other state-of-the-art self-supervised +methods. +Discussions. PPKT and SLidR reveal that contrastive loss +is promising for transferring knowledge from image to point +cloud. Like self-supervised learning, constructing the pos- +itive and negative samples is vital to unsupervised cross- +modal knowledge distillation. +However, previous meth- +ods suffer from the optimization-conflict issue, i.e., some +of the negative paired samples are actually positive pairs. +For example, the road occupies a large proportion of the +point cloud in a scene and is supposed to have the same +semantics in the semantic segmentation task. When ran- +domly selecting training samples, most negatively defined +road-road points are actually positive. When feedforward- +ing such training samples into contrastive learning, the con- +trastive loss will push them away in the embedding space, +leading to unsatisfactory representation learning and ham- +mering the downstream tasks’ performance. +SLidR in- +troduces superpixel-driven contrastive learning to alleviate +such issues. The motivation is that the visual representation +of the image pixel and the projected points are consistent +intra-superpixel. Although avoiding selecting the negative +image-point pairs from the same superpixel, the conflict is- +sue still exists inter-superpixel. In our CLIP2Scene, we in- +troduce the free-available dense pixel-text correspondence +to alleviate the optimization conflicts. The text embedding +represents the semantic information and can be used to se- +lect more reasonable training samples for contrastive learn- +ing. +Besides training sample selection, the previous method +also ignores the temporal coherence of the multi-sweep +point cloud. Similar to multi-view consistency, multi-sweep +consistency emphasizes inter-sweep consistency along time +series. That is, for those LiDAR points mapping to the same +image pixel, their feature should be the same. Besides, con- +sidering the sparsity of the LiDAR scan and the calibration +error between the LiDAR scan and the camera image. We + +专 +. - relax the pixel-to-point mapping to image grid-to-point grid +mapping and calculate the dynamic centre within the indi- +vidual grid for consistency regularization. To this end, our +Spatial-temporal consistency regularization leads to a more +comprehensive point representation. +Last but not least, the previous method typically enlarges +the number of training samples by data augmentation. In +our CLIP2Scene, we find that randomly switching the su- +pervision signal benefits self-supervised learning. Essen- +tially, our Switchable Self-Training Strategy enlarges the +training samples and prevents the network from deteriorat- +ing. +5. Conclusion +We explored how CLIP knowledge benefits 3D scene +understanding in this paper, termed CLIP2Scene. To ef- +ficiently transfer CLIP’s image feature and text feature +to a 3D network, we propose a novel Semantic-driven +Cross-modal Contrastive Learning framework including Se- +mantic Regularization and Spatial-Temporal Regulariza- +tion. For the first time, our pre-trained 3D network achieves +annotation-free 3D semantic segmentation with decent per- +formance. 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In IEEE Conference + +on Computer Vision and Pattern Recognition, pages 9939– +9948, 2021. 2 + diff --git a/MNE4T4oBgHgl3EQfKAyT/content/tmp_files/load_file.txt b/MNE4T4oBgHgl3EQfKAyT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1fff1baa6b4ef8299b73fa9ddb445568ae961959 --- /dev/null +++ b/MNE4T4oBgHgl3EQfKAyT/content/tmp_files/load_file.txt @@ -0,0 +1,840 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf,len=839 +page_content='CLIP2Scene: Towards Label-efficient 3D Scene Understanding by CLIP Runnan Chen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Youquan Liu2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Lingdong Kong3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Xinge Zhu6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Yuexin Ma5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Yikang Li4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Yuenan Hou4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Yu Qiao4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Wenping Wang7 1The University of Hong Kong 2Hochschule Bremerhaven 3National University of Singapore 4Shanghai AI Lab 5ShanghaiTech University 6The Chinese University of Hong Kong 7Texas A&M University Abstract Contrastive language-image pre-training (CLIP) achieves promising results in 2D zero-shot and few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Despite the impressive performance in 2D tasks, applying CLIP to help the learning in 3D scene understanding has yet to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' In this paper, we make the first attempt to investigate how CLIP knowledge benefits 3D scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' To this end, we propose CLIP2Scene, a simple yet effective framework that transfers CLIP knowledge from 2D image-text pre-trained models to a 3D point cloud network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We show that the pre-trained 3D network yields impressive performance on various downstream tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=', annotation-free and fine-tuning with labelled data for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Specifically, built upon CLIP, we design a Semantic-driven Cross-modal Contrastive Learning framework that pre-trains a 3D network via semantic and spatial-temporal consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' For semantic consistency regularization, we first leverage CLIP’s text semantics to select the positive and negative point samples and then employ the contrastive loss to train the 3D network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' In terms of spatial-temporal consistency regularization, we force the consistency be- tween the temporally coherent point cloud features and their corresponding image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We conduct experi- ments on the nuScenes and SemanticKITTI datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' For the first time, our pre-trained network achieves annotation- free 3D semantic segmentation with 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='8% mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' When fine-tuned with 1% or 100% labelled data, our method significantly outperforms other self-supervised methods, with improvements of 8% and 1% mIoU, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Furthermore, we demonstrate its generalization capability Semantic and Spatial-Temporal Consistency Regularization Image Encoder Annotation-free 1% annotation CLIP2Scene Text Encoder CLIP How CLIP benefits 3D scene understanding?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 100% annotation Semantic-driven Cross-modal Contrastive Learning car, bus pedestrian car Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We explore whether and how CLIP knowledge benefits 3D scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' To this end, we propose CLIP2Scene, a semantic-driven cross-modal contrastive learning framework that leverages CLIP knowledge to pre-train a 3D point cloud seg- mentation network via semantic and spatial-temporal consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' CLIP2Scene yields impressive performance on annotation-free 3D semantic segmentation and significantly out- performs other self-supervised methods when fine-tuning on an- notated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' for handling cross-domain datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Introduction 3D scene understanding is fundamental in autonomous driving, robot navigation, etc [24, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Current deep arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='04926v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='CV] 12 Jan 2023 learning-based methods have shown inspirational perfor- mance on 3D point cloud data [37, 50, 29, 44, 15, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' However, some drawbacks hinder their real-world applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The first one comes from their heavy reliance on the large collection of the annotated point clouds, especially when high-quality 3D annotations are expensive to acquire [34, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Besides, they typically fail to recognize novel ob- jects that are never seen in the training data [11, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' As a result, it may need extra annotation efforts to train the model on recognizing these novel objects, which is both te- dious and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Contrastive Vision-Language Pre-training (CLIP) [38] provides a new perspective that mitigates the above issues in 2D vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' It was trained on large-scale free-available image-text pairs from websites and built vision-language correlation to achieve promising open-vocabulary recogni- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' MaskCLIP [49] further explores semantic segmen- tation based on CLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' With minimal modifications to the CLIP pre-trained network, MaskCLIP can be directly used for the semantic segmentation of novel objects without ad- ditional training efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' PointCLIP [48] reveals that the zero-shot classification ability of CLIP can be generalized from the 2D image to the 3D point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' It perspectively projects a point cloud frame into different views of 2D depth maps that bridge the modal gap between the image and the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The above studies indicate the potential of CLIP on enhancing the 2D segmentation and 3D clas- sification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' However, whether and how CLIP knowledge benefits 3D scene understanding is still under- explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' In this paper, we explore how to leverage CLIP’s 2D image-text pre-learned knowledge for 3D scene understand- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Previous cross-modal knowledge distillation methods [40, 34] suffer from the optimization-conflict issue, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=', some of the positive pairs are regarded as negative sam- ples for contrastive learning, leading to unsatisfactory rep- resentation learning and hammering the performance of downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Besides, they also ignore the tempo- ral coherence of the multi-sweep point cloud, failing to utilize the rich inter-sweep correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' To handle the mentioned problems, we propose a novel Semantic- driven Cross-modal Contrastive Learning framework that fully leverages CLIP’s semantic and visual information to regularize a 3D network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Specifically, we propose Seman- tic Consistency Regularization and Spatial-Temporal Con- sistency Regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' In semantic consistency regular- ization, we utilize CLIP’s text semantics to select the posi- tive and negative point samples for less-conflict contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' For spatial-temporal consistency regularization, we take CLIP’s image pixel feature to impose a soft con- straint on points within local space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Such oper- ation also prevents the network from degenerating due to image-to-point calibration errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We conduct several downstream tasks on nuScenes to verify how the pre-trained network benefits the 3D scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The first one is annotation-free semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Following MaskCLIP, we place class names into multiple hand-crafted templates as prompts and av- erage the text embeddings generated by CLIP to conduct the annotation-free segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' For the first time, our method achieves 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='8% mIoU annotation-free 3D semantic segmentation without any labelled data for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Sec- ondly, we compare with other self-supervised methods to verify the superiority of our method in label-efficient learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' When fine-tuning the 3D network with 1% or 100% la- belled data, our method significantly outperforms state-of- the-art self-supervised methods, with improvements of 8% and 1% mIoU, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Besides, to verify the general- ization capability, we pre-train the network on the nuScenes dataset and evaluate it on the SemanticKITTI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Our method still significantly outperforms state-of-the-art meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The contributions of our work are summarized as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The first work that distils CLIP knowledge to a 3D net- work for 3D scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We propose a novel Semantic-driven Cross-modal Contrastive Learning framework that pre-trains a 3D network via spatial-temporal and semantic consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We propose a novel Semantic-guided Spatial- Temporal Consistency Regularization that forces the consistency between the temporally coherent point cloud features and their corresponding image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' For the first time, our method achieves promising performance on annotation-free 3D scene segmenta- tion and significantly outperforms state-of-the-art self- supervised methods when fine-tuning with labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Related Work Zero-shot Learning in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The objective of zero-shot learning (ZSL) is to recognize objects that are unseen in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Many efforts have been devoted to the 2D recognition tasks [8, 30, 47, 36, 31, 1, 43, 32, 4, 2, 19, 33, 23], and few works concentrate on performing ZSL in the 3D domain [18, 11, 35, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' [18] makes the first at- tempt to apply ZSL to 3D tasks, where they train PointNet [37] on ”seen” samples and test on ”unseen” samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Sub- sequent work [16] addresses the hubness problem caused by the low-quality point cloud features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' [17] proposes the triplet loss to boost the performance under the transductive setting, where the ”unseen” class is observed and unlabeled Spatial-Temporal Consistency Regularization Image Encoder Text Encoder car, bus Pedestrian … A photo of a { };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' This is the { } in the scene;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' … Semantic Consistency Regularization Point Encoder CLIP pixel-to-text mapping pixel-point-text pairs pixel-to-point mapping 3D Network point-text pairs … … … … … … … Multi-sweeps calibration … … grid 1 grid 2 grid 3 pulling force Semantic-guided fusion features text embedding point feature text embedding point feature pixel feature point feature prompts 𝑃1 𝑃2 𝑃3 image feature … … … … grid 1 grid 2 grid 3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Illustration of the Semantic-driven Cross-modal Contrastive Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Firstly, we obtain the text embeddings ti, image pixel feature xi, and point feature pi by text encoder, image encoder, and point encoder, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Secondly, we leverage CLIP knowledge to construct positive and negative samples for contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Thus we obtain point-text pairs {xi, ti}M i=1 and all pixel-point-text pairs in a short temporal {ˆxk i , ˆpk i , tk i } ˆ M,K i=1,k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Here, {xi, ti}M i=1 and {ˆxk i , ˆpk i , tk i } ˆ M,K i=1,k=1 are used for Semantic Consistency Regularization and Spatial-Temporal Consistency Regularization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Lastly, we perform Semantic Consistency Regularization by pulling the point features to their corresponding text embedding and Spatial-Temporal Consistency Regularization by mimicking the temporally coherent point features to their corresponding pixel features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' in the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Recently, some studies introduced CLIP into zero-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' MaskCLIP [49] investigates the problem of utilizing CLIP to help the 2D dense pre- diction tasks and exhibits encouraging zero-shot semantic segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' PointCLIP [48] is the pioneer- ing work that applies CLIP to 3D recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' As opposed to previous approaches that require training on the labelled point cloud, PointCLIP is free from any 3D training and shows impressive performance on zero-shot and few-shot classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Our work takes a step further to inves- tigate whether the rich semantic and visual knowledge in CLIP can benefit the 3D semantic segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Self-supervised Representation Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The purpose of self-supervised learning is to learn a good representa- tion that benefits the downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The dominant ap- proaches resort to contrastive learning to pre-train the net- work [27, 25, 21, 20, 14, 13, 7, 10, 12, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Recently, inspired by the success of CLIP, leveraging the pre-trained model of CLIP to the downstream tasks has raised the commu- nity’s attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' DenseCLIP [39] utilizes the CLIP’s pre- trained knowledge for dense image pixel prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Det- CLIP [46] proposes a pre-training method equipped with CLIP for open-world detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' In this paper, we make the first attempt to pre-train a 3D network with CLIP’s knowl- edge for 3D scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Cross-modal Knowledge Distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Recently, an in- creasing number of researchers have focused on transferring the knowledge in 2D images to 3D point cloud [34, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' PPKT [34] proposes the contrastive pixel-to-point knowl- edge transfer to utilize the rich information in image back- bones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' SLidR [40] resorts to the InfoNCE loss to help the 3D network distil rich knowledge from the 2D image back- bone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Our work explores leveraging the image-text pre- trained CLIP knowledge to help 3D scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Methodology Considering the impressive open-vocabulary perfor- mance achieved by CLIP in image classification and seg- mentation, natural curiosities have been raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Can CLIP endow the ability to a 3D network for annotation-free scene understanding?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' And further, will it promote the network performance when fine-tuned on labelled data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' To answer the above questions, we study the cross-modal knowledge transfer of CLIP for 3D scene understanding, termed CLIP2Scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Our work is a pioneer in exploiting CLIP knowledge for 3D scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' In what fol- lows, we revisit the CLIP applied in 2D open-vocabulary classification and semantic segmentation, then present our CLIP2Scene in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Our approach consists of three major components: Semantic Consistency Regularization, Semantic-Guided Spatial-Temporal Consistency Regular- ization, and Switchable Self-Training Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' car road bicycle .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' building Text embeddings Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Illustration of the image pixel-to-text mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The dense pixel-text correspondence {xi, ti}M i=1 is extracted by the off-the-shelf method MaskCLIP [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Revisiting CLIP Contrastive Vision-Language Pre-training (CLIP) miti- gates the following drawbacks that dominate the computer vision field: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Deep models need a large amount of for- matted and labelled training data, which is expensive to ac- quire;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The model’s generalization ability is weak, mak- ing it difficult to migrate to a new scenario with unseen objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' CLIP consists of an image encoder (ResNet [28] or ViT [6]) and a text encoder (Transformer [42]), both respectively project the image and text representation to a joint embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' During training, CLIP constructs positive and negative samples from 400 million image-text pairs to train both encoders with a contrastive loss, where the large-scale image-text pairs are free-available from the Internet and assumed to contain every class of images and most concepts of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Therefore, CLIP can achieve promis- ing open-vocabulary recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' For 2D zero-shot classification, CLIP first places the class name into a pre-defined template to generate the text embeddings and then encodes images to obtain image em- beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Next, it calculates the similarity matrices be- tween images and text embeddings to determine the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' MaskCLIP further extends CLIP into 2D semantic segmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Specifically, MaskCLIP modifies the attention pool- ing layer of the CLIP’s image encoder, thus performing pixel-level mask prediction instead of the global image- level prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' CLIP2Scene As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 2, we first leverage CLIP and 3D net- work to respectively extract the text embeddings, image pixel feature and point feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Secondly, we construct positive and negative samples based on CLIP’s knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Lastly, we impose Semantic Consistency Regularization by pulling the point features to their corresponding text embed- ding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' At the same time, we apply Spatial-Temporal Con- sistency Regularization by forcing the consistency between temporally coherent point features and their corresponding pixel features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' In what follows, we present the details and insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='1 Semantic Consistency Regularization As CLIP is pre-trained on 2D images and text, our first con- cern is the domain gap between 2D images and the 3D point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' To this end, we build dense pixel-point correspon- dence and transfer image knowledge to the 3D point cloud via the pixel-point pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Specifically, we calibrate the Li- DAR point cloud with corresponding images captured by six cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Therefore, the dense pixel-point correspon- dence {xi, pi}M i=1 can be obtained accordingly, where xi and pi indicates i-th paired image feature and point feature, which are respectively extracted by the CLIP’s image en- coder and the 3D network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' M is the number of pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Note that it is an online operation and is irreverent to the image and point data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Previous methods [40, 34] provide a promising solution to cross-modal knowledge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' They first construct positive pixel-point pairs {xi, pi}M i=1 and negative pairs {xi, pj}(i ̸= j), and then pull in the positive pairs while pushing away the negative pairs in the embedding space via the InfoNCE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Despite the encourageable performance of previous methods in transferring cross-modal knowl- edge, they are both confronted with the same optimization- conflict issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' For example, suppose i-th pixel xi and j-th point pj are in the different positions of the same instance with the same semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' However, the InfoNCE loss will try to push them away, which is unreasonable and ham- mer the performance of the downstream tasks [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' In light of this, we propose a Semantic Consistency Regularization that leverages the CLIP’s semantic information to allevi- ate this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Specifically, we generate the dense pixel- text pairs {xi, ti}M i=1 by following the off-the-shelf method MaskCLIP [49] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 3), where ti is the text embedding gen- erated from the CLIP’s text encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Note that the pixel-text mappings are free-available from CLIP without any addi- tional training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We then transfer pixel-text pairs to point- text pairs {pi, ti}M i=1 and utilize the text semantics to se- lect the positive and negative point samples for contrastive Image 𝐼 Pixel-to-point mapping 𝑃1 𝑃2 𝑃3 Multi-sweeps calibration … … … … … grid 1 grid 2 grid 3 … … grid 1 grid 2 grid 3 {𝑓𝑛}𝑛=1 𝑁 ො𝑥𝑖 𝑘, ො𝑝𝑖 𝑘 𝑛=1,𝑘=1 𝑁,𝐾 Text embedding Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Illustration of the image pixel-to-point mapping (left) and semantic-guided fusion feature generation (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We build the grid-wise correspondence between an image I and the tem- porally coherent LiDAR point cloud {Pk}K k=1 within S seconds and generate semantic-guided fusion features for individual grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Both {ˆxk i , ˆpk i } ˆ M,K i=1,k=1 and {fn}N n=1 are used to perform Spatial- Temporal Consistency Regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The objective function is as follows: LS info = − C � c=1 log � ti∈c,pi exp(D(ti, pi)/τ) � ti∈c,tj /∈c,pj exp(D(ti, pj)/τ), (1) where ti ∈ c indicates that ti is generated by c-th classes name, and C is the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' D denotes the scalar product operation and τ is a temperature term (τ > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Since the text is composed of class names placed into pre-defined templates, the text embedding represents the se- mantic information of the corresponding class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Therefore, those points with the same semantics will be restricted near the same text embedding, and those with different semantics will be pushed away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' To this end, our Semantic Consistency Regularization causes less conflict in contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='2 Semantic-guided Spatial-temporal Consistency Regularization Besides semantic consistency regularization, we consider how image pixel features help to regularize a 3D network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The natural alternative directly pulls in the point feature with its corresponding pixel in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' How- ever, after trial and error, we observe that the network easily degenerates and achieves poor performance in the down- stream tasks when following the aforementioned strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The main reason lies in the noise-assigned semantics of the image pixel and the imperfect pixel-point mapping caused by the calibration errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' To this end, we propose a novel semantic-guided Spatial-Temporal Consistency Regulariza- tion to alleviate the problem by imposing a soft constraint on points within local space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Specifically, given an image I and temporally coherent LiDAR point cloud {Pk}K k=1, where K is the number of sweeps within S seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Note that the image is matched to the first frame of the point cloud P1 with pixel-point pairs {ˆx1 i , ˆp1 i } ˆ M i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We register the rest of the point cloud to the first frame via the calibration matrices and map them to the image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Thus we obtain all pixel-point-text pairs in a short temporal {ˆxk i , ˆpk i , tk i } ˆ M,K i=1,k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Next, we divide the entire stitched point cloud into regular grids {gn}N n=1, where the temporally coherent points are located in the same grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We impose the spatial-temporal consistency con- straint within individual grids by the following objective function: LSSR = � gn � (ˆi,ˆk)∈gn (1 − sigmoid(D(ˆp ˆk ˆi , fn)))/N, (2) where (ˆi, ˆk) ∈ gn indicates the pixel-point pair {ˆxk i , ˆpk i } is located in the n-th grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' {fn}N n=1 is a semantic-guided cross-modal fusion feature formulated by: fn = � (ˆi,ˆk)∈gn a ˆk ˆi ∗ ˆx ˆk ˆi + b ˆk ˆi ∗ ˆp ˆk ˆi , (3) where aˆk ˆi and bˆk ˆi are attention weight calculated by: a ˆk ˆi = exp(D(ˆxˆk ˆi , t1 ˆi )/λ) � (ˆi,ˆk)∈gn exp(D(ˆxˆk ˆi , t1 ˆi )/λ) + exp(D(ˆpˆk ˆi , t1 ˆi )/λ) , b ˆk ˆi = exp(D(ˆpˆk ˆi , t1 ˆi )/λ) � (ˆi,ˆk)∈gn exp(D(ˆxˆk ˆi , t1 ˆi )/λ) + exp(D(ˆpˆk ˆi , t1 ˆi )/λ) , (4) where λ is the temperature term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Actually, those pixel and point features within the local grid gn are restricted near a dynamic centre fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Thus, such a soft constraint alleviates the noisy prediction and calibration error issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' At the same time, it imposes Spatio-Temporal Regularization on the temporally coherent point features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='3 Switchable Self-training Strategy We combine the loss function LS info and LSSR to end- to-end train the whole network, where the CLIP’s image and text encoder backbone are frozen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We find that method worked only when the pixel-point feature {xi, pi}M i=1 and {ˆxk i , ˆpk i } ˆ M,K i=1,k=1, which are used in LS info and LSSR, are generated from different learnable linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' On top of that, we further put forward an effective strategy to promote performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Specifically, after con- trastive learning of the 3D network for a few epochs, we randomly switch the point labels between the paired im- age pixel’s labels and their own predictions for self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Merely training the 3D network with their own predictions yields satisfactory performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Essentially, such a Switch- able Self-Training Strategy (S3) increases the number of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Ablation study experiments on the nuScenes validation dataset for annotation-free semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Ablation target Settings mIoU(%) baseline 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='1 Prompts nuScenes 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='1 semanticKITTI 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='9 Cityscapes 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='3 Regularization w/o SCR 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='8 KL 0 Training Strategies w/o S3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='8 ST 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='1 Sweeps 1 sweep 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='7 3 sweeps 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='8 5 sweeps 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='6 merged 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='6 CLIP2Scene 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='8 positive and negative samples by switching the point pseudo labels, which benefits cross-modal knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Experiments Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We conduct experiments on two large-scale outdoor LiDAR segmentation benchmarks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=', Se- manticKITTI [3] and nuScenes [5, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The nuScenes dataset contains 700 scenes for training, 150 scenes for validation and 150 scenes for testing, where 16 classes are utilized for LiDAR semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' As to Se- manticKITTI, it contains 19 classes for training and evalu- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' It has 22 sequences, where sequences 00 to 10, 08 and 11 to 21 are used for training, validation and testing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We use the nuScenes [5, 22] dataset to pre-train the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Following SLidR, we pre-train the network on all key frames from 600 scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Besides, we fine-tune the pre-trained network on Se- manticKITTI [3] to verify the generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We leverage CLIP’s image encoder and text encoder to gener- ate image features and text embedding, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Fol- lowing MaskCLIP, we modify the attention pooling layer of the CLIP’s image encoder, thus extracting the dense pixel- text correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We take SPVCNN [41] as the 3D network to produce the point-wise feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The whole net- work is trained on the PyTorch platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The training time is about 40 hours for 20 epochs on two NVIDIA Tesla A100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' For the switchable self-training strategy, we ran- domly switch the point supervision signal after 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The optimizer is SGD with a cosine scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We set the temperature λ and τ to be 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The sweep number is set to be 3 empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We apply sev- eral data augmentations in contrastive learning, including random rotation around the z-axis and random flip on the Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Comparison of different self-supervised methods for se- mantic segmentation on the nuScenes and SemanticKITTI valida- tion datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Initialization nuScenes semanticKITTI 1% 100% 1% Random 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='2 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='1 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='5 PPKT [34] 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='1 SLidR [40] 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='6 CLIP2Scene 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='6 point cloud, random horizontal flip and random crop-resize on the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Annotation-free Semantic Segmentation After pre-training the network, we show the performance of the 3D network when it is not fine-tuned on any annota- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' As no previous method reports the 3D annotation-free segmentation performance, we compare our method with different setups (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' In what follows, we describe the experimental settings and give insights into our method and the different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We conduct experiments on the nuScenes dataset to evaluate the annotation-free semantic segmentation per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Following MaskCLIP [49], we place the class name into 85 hand-craft prompts and feed it into the CLIP’s text encoder to produce multiple text features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We then av- erage the text features and feed the averaged features to the classifier for point-wise prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Besides, to explore how to effectively transfer CLIP’s knowledge to the 3D network for annotation-free segmentation, We conduct the following experiments to highlight the effectiveness of different mod- ules in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The input of the 3D network is only one sweep, and we pre-train the framework via semantic consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Prompts (nuScenes, semanticKITTI, Cityscapes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Based on the baseline, we respectively replace the nuScenes, se- manticKITTI, and Cityscapes class names into the prompts to produce the text embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Regularization (w/o STR, KL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Based on the full method, we remove the Spatial-temporal Consistency Regulariza- tion (w/o SCR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Besides, we abuse both SR and SCR and distill the image feature to the point cloud by Kull- back–Leibler (KL) divergence loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Training Strategies (w/o S3, ST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We abuse the Switchable Self-Training Strategy (w/o S3) in the full method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Besides, we show the performance of only training the 3D network by their own predictions after ten epochs (ST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Sweeps Number (1 sweep, 3 sweeps, 5 sweeps, and merged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We set the sweep number K to be 1, 3, and 5, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Besides, we also take three sweeps of the point cloud as the input to pre-train the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Effect of Different Prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' To verify how text em- Ground truth Ours* Ours Bus Motorcycle Car Truck Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Qualitative results of annotation-free semantic segmentation on nuScenes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Note that we show the results by individual class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' From the left to the right column are the bus, motorcycle, car and truck, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The first row is the ground truth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The second row (ours*) is our prediction of the highlighted target;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' the third row is our prediction of full classes (ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' bedding affects the performance, we generate various text embeddings by the class name from different datasets (nuScenes, SemanticKITT, and Cityscapes) for pre-training the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' As shown in Table 1, we find that even learning with other datasets’ text embedding (se- manticKITT and Cityscapes), the 3D network could still recognize the nuScenes’s objects with decent performance (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='9 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='3 mIoU, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The result shows that the 3D network is capable of open-vocabulary recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Effect of Semantic and Spatial-temporal Consistency Regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We remove Spatial-temporal Consistency Regularization (w/o SCR) from our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Experiments show that the performance is dramatically decreased, indi- cating the effectiveness of our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Besides, we also dis- till the image feature to the point cloud by KL divergence loss, where the text embeddings calculate the logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' How- ever, such a method fails to transfer the semantic informa- tion from the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The main reason is the noise-assigned semantics of the image pixel and the imperfect pixel-point correspondence due to the calibration error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Effect of Switchable Self-training Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' To examine the effect of the Switchable Self-Training Strategy, we ei- ther train the network with image supervision (w/o S3) or train the 3D network by their own predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Both tri- als witness the performance drop, indicating our Switch- able Self-Training Strategy is efficient in cross-modal self- supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The main reason is that the number of positive and negative samples is enlarged by switching the supervision signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Effect of Sweep Numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Intuitively, the performance of our method benefits from more sweeps information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' There- fore, we also show the performance when restricting sweep size to 1, 3, and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' However, we observe that the performance of 5 sweeps is similar to 3 sweeps but is more computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Thus, we empirically set the sweep number to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Qualitative Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We show the qualitative evalua- tion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Note that we show the results by individ- ual class (construction vehicle, truck, and car).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The results show that our method is able to perceive the objects without any annotation training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' However, we also observe the false positive predictions around the ground truth objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We will resolve this issue in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Annotation-efficient Semantic Segmentation Besides annotation-free semantic segmentation, the pre- trained 3D network also boosts the performance when it is fine-tuned on labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' To the best of our knowl- edge, only one published method SLidR studies image-to- Lidar self-supervised representation distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We also compared our method with another self-supervised method PPKT [34] for 3D network pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' In the follow- ings, we first introduce SLidR [40] and PPKT, then compare them in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' PPKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' PPKT is a cross-modal self-supervised method for the RGB-D dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' It performs 2D-to-3D knowledge dis- tillation via pixel-to-point contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Since there is no public code, we re-implement it for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Input Ground Truth SLidR Ours Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Qualitative results of fine-tuning on 1% nuScenes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' From the first row to the last row are the input Lidar scan, ground truth, prediction of SLidR, and our prediction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Note that we show the results by error map, where the red point indicates the wrong prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Apparently, our method achieves decent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Specifically, we use the same 3D network and training pro- tocol but replace our semantic and Spatio-Temporal Reg- ularization with InfoNCE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The framework is trained on 4, 096 randomly selected image-to-point pairs for 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' SLidR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' SLidR is an image-to-Lidar self-supervised method for autonomous driving data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Compared with PPKT, it introduces image super-pixel into cross-modal self- supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' For a fair comparison, we replace our loss function with their superpixel-driven contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' As shown in Table 2, our method signifi- cantly outperforms the state-of-the-art methods when fine- tuned on 1% and 100% data, with the improvement of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='1% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='1%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Compared with the ran- dom initialization, the improvement is 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='1% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='4%, re- spectively, indicating the efficiency of our semantic-driven cross-modal contrastive learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The qualita- tive results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Besides, we also verify the cross-domain generalization ability of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' When pre-training the 3D network on the nuScenes dataset and fine-tuning on 1% SemanticKITTI dataset, our method sig- nificantly outperforms other state-of-the-art self-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' PPKT and SLidR reveal that contrastive loss is promising for transferring knowledge from image to point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Like self-supervised learning, constructing the pos- itive and negative samples is vital to unsupervised cross- modal knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' However, previous meth- ods suffer from the optimization-conflict issue, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=', some of the negative paired samples are actually positive pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' For example, the road occupies a large proportion of the point cloud in a scene and is supposed to have the same semantics in the semantic segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' When ran- domly selecting training samples, most negatively defined road-road points are actually positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' When feedforward- ing such training samples into contrastive learning, the con- trastive loss will push them away in the embedding space, leading to unsatisfactory representation learning and ham- mering the downstream tasks’ performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' SLidR in- troduces superpixel-driven contrastive learning to alleviate such issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The motivation is that the visual representation of the image pixel and the projected points are consistent intra-superpixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Although avoiding selecting the negative image-point pairs from the same superpixel, the conflict is- sue still exists inter-superpixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' In our CLIP2Scene, we in- troduce the free-available dense pixel-text correspondence to alleviate the optimization conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' The text embedding represents the semantic information and can be used to se- lect more reasonable training samples for contrastive learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Besides training sample selection, the previous method also ignores the temporal coherence of the multi-sweep point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Similar to multi-view consistency, multi-sweep consistency emphasizes inter-sweep consistency along time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' That is, for those LiDAR points mapping to the same image pixel, their feature should be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Besides, con- sidering the sparsity of the LiDAR scan and the calibration error between the LiDAR scan and the camera image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' We 专 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' - relax the pixel-to-point mapping to image grid-to-point grid mapping and calculate the dynamic centre within the indi- vidual grid for consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' To this end, our Spatial-temporal consistency regularization leads to a more comprehensive point representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Last but not least, the previous method typically enlarges the number of training samples by data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' In our CLIP2Scene, we find that randomly switching the su- pervision signal benefits self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Essen- tially, our Switchable Self-Training Strategy enlarges the training samples and prevents the network from deteriorat- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Conclusion We explored how CLIP knowledge benefits 3D scene understanding in this paper, termed CLIP2Scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' To ef- ficiently transfer CLIP’s image feature and text feature to a 3D network, we propose a novel Semantic-driven Cross-modal Contrastive Learning framework including Se- mantic Regularization and Spatial-Temporal Regulariza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' For the first time, our pre-trained 3D network achieves annotation-free 3D semantic segmentation with decent per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Besides, our method significantly outperforms state-of-the-art self-supervised methods when fine-tuning the 3D network with labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Potential Negative Impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Although our approach im- proves the 3D semantic segmentation performance in gen- eral, its effectiveness under adversarial attack is not con- sidered, which could be safety-critical in practical applica- tions, such as autonomous driving and robot navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' References [1] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} +page_content=' Akata, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfKAyT/content/2301.04926v1.pdf'} 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0000000000000000000000000000000000000000..e21b25c9e830910b7959b2e9833f6ad581a52182 --- /dev/null +++ b/NNE2T4oBgHgl3EQfqwgI/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6c6fd7b0af282e705bf2ea42f85f32e7422507d5cea297d75842e5e9ba38e741 +size 5898285 diff --git a/NtE4T4oBgHgl3EQf9g4i/content/tmp_files/2301.05355v1.pdf.txt b/NtE4T4oBgHgl3EQf9g4i/content/tmp_files/2301.05355v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d016a59b4544945aef48e19c576609be60cfd9b8 --- /dev/null +++ b/NtE4T4oBgHgl3EQf9g4i/content/tmp_files/2301.05355v1.pdf.txt @@ -0,0 +1,1646 @@ +arXiv:2301.05355v1 [nucl-th] 13 Jan 2023 +Possible interpretation of the complex expectation values associated with resonances +Takayuki Myo∗1, 2 and Kiyoshi Kat¯o†3 +1General Education, Faculty of Engineering, Osaka Institute of Technology, Osaka, Osaka 535-8585, Japan +2Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan +3Nuclear Reaction Data Centre, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan +(Dated: January 16, 2023) +We propose a possible scheme to interpret the complex expectation values associated with reso- +nances having the complex eigenenergies. Using the Green’s function for resonances, the expectation +value is basically described by the Breit-Wigner distribution as a function of the real excitation en- +ergy. In the expression of the complex expectation values for resonances, the real part brings the +integral value of the distribution, while the imaginary part produces the deviation from the Breit- +Wigner distribution, which explains a shift of the peak in the strength from the resonance energy. +We apply the present scheme to the several nuclear resonances of 12C including the Hoyle state, and +neutron/proton-rich nuclei of 6He, 6Be, 8He, and 8C. In these nuclei, many-body resonances are +obtained as the complex-energy eigenstates under the correct boundary condition using the com- +plex scaling method, and their nuclear radii are uniquely evaluated. We discuss the peculiar energy +dependence of the strength function of the square radius for the resonances in these nuclei. +PACS numbers: +21.60.Gx, 21.10.Pc, 21.10.Dr, 27.20.+n +I. +INTRODUCTION +Resonance is a general phenomenon occurring in var- +ious kinds of physical systems [1–5]. In nuclear physics, +many kinds of resonances are observed such as by the α- +decays and the nuclear reactions to excite single-particle, +collective, and compound states. In unstable nuclei con- +sisting of the stable core nucleus and a few excess nu- +cleons, the excess nucleons often form a weakly binding +state. The low-lying excited states in unstable nuclei can +be observed above the threshold energies of the particle +emission as resonances [6]. +The spectroscopy of reso- +nances provides useful information on the knowledge of +the properties of unstable nuclei. In nuclei, some of the +nucleons can be localized spatially and form a cluster +such as an α particle. A typical case is the Hoyle state of +12C and this 0+ +2 state is a resonance located just above +the threshold energy of α+α+α. The cluster states are +often observed as resonances near and above the thresh- +old energy of the α particle emission [7], some of which +play a decisive role in nucleosynthesis. +The resonance can be defined as a decaying state and +is described as imposing the boundary condition of the +outgoing wave, which is the so-called Siegert condition +[8, 9]. Under this condition, the resonance has a com- +plex eigenenergy Er −iΓ/2. Similar to the eigenenergies, +it is known that the expectation values of an Hermite +operator for resonances can be a complex number. The +physical interpretation of the complex expectation values +is a long-standing problem [10–17]. Berggren attempted +to evaluate the uncertainty of the expectation value of an +operator using its imaginary part [11], however this idea +∗takayuki.myo@oit.ac.jp +†kato@nucl.sci.hokudai.ac.jp +is limited to the operators commuting with the Hamil- +tonian and not settled yet. There is a discussion on this +problem in the aspect of the time dependence of the ex- +pectation value; for the resonance with a small decay +width Γ, the imaginary part can be related to the disper- +sion rate over time in the measurement [17]. +In this paper, we propose a possible scheme to inter- +pret the complex expectation values of an Hermite op- +erator for resonances; we utilize the Green’s function of +resonances in the strength function and the complex ex- +pectation value of the operator becomes the source of the +strength function on the real energy axis. We formulate +the general expression to utilize the complex expecta- +tion values in the strength function and discuss the roles +of the real and imaginary parts of the expectation val- +ues to determine the structure of the strength function. +This is a general framework for any physical operator +in many-body systems and is similar to the Morimatsu- +Yazaki method [18], which is used to calculate the energy +spectrum of the formation of the hadron resonances in +the two-body scattering process. +We apply this scheme to the radius of resonances, the +interpretation of which has been discussed in various +physics fields [12–16]. We show the numerical results of +the nuclear resonances with complex-energy eigenstates +obtained using the complex scaling method [1, 2, 19–23], +which enables us to describe many-body resonances un- +der the damping boundary condition [24]. In this study, +we choose five nuclei, 12C, 6He, 6Be, 8He, and 8C, which +are described assuming the α cluster. For 12C, we adopt +the α+α+α model and discuss the effect of the 0+ reso- +nances including the Hoyle state on the radius. For the +other four nuclei, we adopt the α+N+N+N+N model. +We describe many-body resonances in these nuclei using +the complex scaling and calculate the radii of the reso- +nances being the complex number. Using the complex- +scaled Green’s function, we evaluate the resonance com- + +2 +ponents of the strength functions of the square radius and +discuss the behavior of their distributions. The present +analysis becomes a basis to utilize the complex expecta- +tion values of various operators for resonances. +In section II, we explain the framework to utilize the +complex expectation values for resonances, the complex +scaling to obtain the many-body resonances, and nuclear +models with α cluster. +In section III, we discuss the +results of the strength functions of the nuclear radius. In +section IV, a summary is given. +II. +METHOD +A. +Framework +We explain a framework to utilize the complex expec- +tation values associated with resonances. +We start to +consider the two-body system with a single channel and +define the resonance wave function ΦR as the Gamow +decaying state satisfying the Siegert boundary condition +of the outgoing wave [8, 9]. This state has a complex +eigenenergy ER = Er − iΓ/2, where Er is a resonance +energy measured from the lowest threshold energy of the +particle emissions and Γ is a decay width. The corre- +sponding momentum is given as kR = κ − iγ. +The adjoint state of the resonance is the so-called the +anti-resonance ΦAR, which has a boundary condition of +incoming wave with the eigenenergy of EAR = Er + +iΓ/2 = E∗ +R and the momentum kAR = −κ − iγ = −k∗ +R. +This state is also called a capturing state or a growing +state [3, 25]. The resonance and anti-resonance form the +bi-orthogonal relation [26] and their radial components +have a relation of ΦAR,rad = Φ∗ +R,rad, and one often uses +the notation of ΦAR as �ΦR. For the continuum state Φk +with a complex momentum k, the adjoint state �Φk has +the momentum k∗ [21, 26]. +The completeness relation is extendable by separat- +ing the scattering states with real energy and momen- +tum into the resonances and the remaining non-resonant +continuum states orthogonal to the resonances. This is +so-called the extended completeness relation (ECR) [26] +and is expressed using the solutions of the bound (B), +resonant (R), and non-resonant continuum (k) states. +1 = +� +B +|ΦB⟩⟨�ΦB| + +� +R +|ΦR⟩⟨�ΦR| + +� +dk|Φk⟩⟨�Φk|(1) += +� +ν +� +|Φν⟩⟨�Φν|, +(2) +where ν is the unified index both for the discrete and +continuous states. +We start from the transition matrix elements such as +the electromagnetic type, from the bound state to the +resonance. The transition operator is ˆOTR and the corre- +sponding matrix element MTR becomes complex in gen- +eral, defined as +MTR = ⟨�Φ0| ˆO† +TR|ΦR⟩⟨�ΦR| ˆOTR|Φ0⟩, +(3) +where Φ0 is the initial bound state. +So far, we have +discussed the cases of monopole, dipole, and quadrupole +transitions for nuclei and investigate the contributions of +resonances [27–32]. +We explain the general procedure to calculate the +strength function S(E) as a function of the real scatter- +ing energy E using the ECR. We first define the Green’s +function of the system with the outgoing-wave condition: +G(E+) = +1 +E+ − H += +� +ν +� +|Ψν⟩⟨�Ψν| +E+ − Eν +, +(4) +where E+ = E +iǫ with a real positive number ǫ and one +imposes ǫ → 0 in the final stage of the calculation. The +strength function S(E) of the transition operator ˆOTR is +represented using the Green’s function and ECR as +S(E) = +� +ν +� +⟨�Ψ0| ˆO† +TR|Ψν⟩⟨�Ψν| ˆOTR|Ψ0⟩ δ(E − Eν)(5) += − 1 +π Im +� +⟨�Ψ0| ˆO† +TRG(E+) ˆOTR|Ψ0⟩ +� +(6) += +� +ν +� +Sν(E), +(7) +Sν(E) = − 1 +π Im +� +⟨�Ψ0| ˆO† +TR|Ψν⟩⟨�Ψν| ˆOTR|Ψ0⟩ +E+ − Eν +� +, (8) +where Sν(E) is the contribution of the specific state ν +such as resonances, to the strength function. The reso- +nance contribution SR(E) is explicitly written as +SR(E) = − 1 +π Im +� MTR +E − ER +� +, +(9) +where the resonance has a complex eigenenergy ER with +a negative imaginary part and then ǫ can be set to zero. +This form of the strength function is common for every +component including non-resonant continuum states [33]. +We have also applied this framework to many-body un- +bound states including the coupled channel case using +the complex scaling and have shown the validity of the +method [21, 22]. +Similarly to the transition case, we formulate the ex- +pectation value MEV of the arbitrary Hermite operator +ˆO for the resonance, which can be complex and is defined +as +MEV = ⟨�ΦR| ˆO|ΦR⟩ = MR + iMI. +(10) +We can express the strength function S(E) for the expec- +tation value of the operator ˆO using the Green’s function + +3 +as +S(E) = +� +ν +� +⟨�Ψν| ˆO|Ψν⟩ δ(E − Eν) +(11) += +� +ν +� +Sν(E), +(12) +Sν(E) = − 1 +π Im +� +⟨�Ψν| ˆO|Ψν⟩ +E+ − Eν +� +. +(13) +In the total strength S(E), the resonance contribution +SR(E) is written as +SR(E) = − 1 +π Im +� MEV +E − ER +� +(14) += 1 +π +MRΓ/2 − MI(E − Er) +(E − Er)2 + Γ2/4 +. +(15) +The integration of SR(E) over the energy gives the real +part of the expectation value for resonance. +� ∞ +−∞ +SR(E) dE = MR. +(16) +From the property of Eq. (15), one can understand the +roles of MR and MI in the complex expectation value of +MEV for resonance on the strength function SR(E), some +of which are useful and summarized as follows, where we +assume the finite value of MI: +1. The real part MR determines the amount of the ex- +pectation value for resonance, corresponding to the +integration of the strength function. For the term +including MR in Eq. (15), the strength distribution +obeys the well-known Breit-Wigner form with the +centroid energy Er. +2. The imaginary part MI produces the deviation +from the Breit-Wigner distribution with an odd +function measured from the energy of Er. The en- +ergy at the peak of the strength SR(E) shifts from +Er due to MI. +The width of the distribution is +affected by MI in the following relation. +SR(Er ± Γ/2) = 1 +π +MR ∓ MI +Γ +. +(17) +3. The energy Emax at the maximum strength and the +energy Emin at the minimum strength are given as +respectively, +Emax = Er + Γ +2 +MR − |MEV| +MI +, +(18) +Emin = Er + Γ +2 +MR + |MEV| +MI +. +(19) +From Eq. (18), the peak energy of the strength +function shifts from the resonance energy Er due +to the presence of MI. At the two energies of Emax +and Emin, the strength function shows the maxi- +mum and minimum values, respectively, as follows: +SR(Emax) = 1 +π +2 +Γ +|MEV|M 2 +I +(MR − |MEV|)2 + M 2 +I +, +(20) +SR(Emin) = 1 +π +2 +Γ +−|MEV|M 2 +I +(MR + |MEV|)2 + M 2 +I +. +(21) +The strength function becomes zero at the energy of +Er +Γ/2·MR/MI, which is a middle point between +Emax and Emin, namely, +SR +�Emax + Emin +2 +� += 0. +(22) +From these formulas, one can understand the role +of the imaginary part MI in the complex expectation +value of MEV to determine the energy distribution of +the strength function for resonances. This formulation is +general and one can apply this scheme to the resonances +in various physical systems. In this study, we show the +applications to many-body resonances of nuclei. +In the actual calculation, not only resonances but also +the non-resonant continuum states contribute to the to- +tal strength function S(E) in Eq. (12), and these com- +ponents are superposed to determine the distribution of +S(E). +It is noted that the total strength S(E) is the +observable and can be the positive definite depending on +the operators such as radius. +On the other hand, the +component Sν(E) is not necessary to keep the positive +definite, different from S(E), because the resonance and +non-resonance components are not observable and they +are allowed to show the negative value at some energies. +In addition, the resonance component SR(E) can show +the strength below the lowest threshold energy and the +remaining continuum component cancels this strength, +and in total, a zero value is obtained in S(E) [21, 33]. +B. +Complex scaling +We show several cases of the strength distributions of +resonances for many-body nuclear systems. For this pur- +pose, we describe many-body resonances using the com- +plex scaling method [1, 2, 4, 20–22]. +In the complex +scaling, the particle coordinates {ri} and the conjugate +momenta {pi} are transformed using a common scaling +angle θ as +ri → ri eiθ, +pi → pi e−iθ. +(23) +The Schr¨odinger equation is expressed using the complex- +scaled Hamiltonian Hθ as +HθΨθ = EθΨθ. +(24) +We solve the eigenvalue problem of Eq. (24) and obtain +the complex-scaled wave function Ψθ. The energy eigen- +values Eθ are obtained for bound, resonant, and contin- +uum states in the complex energy plane for a positive + +4 +θ. +For the resonance wave function, it is proved that +its asymptotic condition becomes the damping form if +2θ > | arg(ER)| (θ > | arg(kR)|) in the complex energy +plane [24]. +Using the complex-scaled solutions of Ψθ, one can in- +troduce the complex-scaled Green’s function Gθ(E) as a +function of the real energy E: +Gθ(E) = +1 +E − Hθ = +� +ν +� +|Ψθ +ν⟩⟨�Ψθ +ν| +E − Eθν +, +(25) +where considering the unbound states, Eθ +ν has a negative +imaginary part with a positive θ and ǫ is set to be zero +in Gθ(E). We apply the complex scaling to the strength +function and use Gθ(E) in Eq. (25). The strength func- +tion of the specific state ν is given as similarly to Eq. (13) +Sν(E) = − 1 +π Im +� +⟨�Ψθ +ν| ˆOθ|Ψθ +ν⟩ +E − Eθν +� +. +(26) +One can extract the contributions of the state ν, Sν(E), +in the total strength S(E) and classify S(E) in terms +of the ECR in Eq. (2). It is noted that Sν(E) is inde- +pendent of θ [21, 30, 33]. This is because the state ν is +uniquely classified in the ECR in Eq. (2) and then Sν(E) +is also uniquely obtained. In the numerical calculation, +we choose the value of θ to obtain stable solutions such +as the resonance eigenenergies in each nucleus. +C. +Nuclear models +We explain the nuclear models of 12C, 6He, 6Be, 8He, +and 8C where the α cluster is commonly assumed with +the s-wave configuration of two-proton and two-neutron +in a harmonic oscillator basis state. For 12C, this nu- +cleus is described by the three α (3α) clusters with the +orthogonality condition model [34, 35]. The total wave +function ΨJ with spin J for 12C is represented by the su- +perposition of the 3α configurations ΨJ +p with the weight +CJ +p as +ΨJ = +� +p +CJ +p ΨJ +p, +(27) +ΨJ +p = +3 +� +c=1 +ΨJ +c,LN,ℓn +3 +� +i=1 +φint(αi), +(28) +ΨJ +c,LN,ℓn = [ΦLN(Rc), φℓn(rc)]J , +(29) +where φint(α) is the internal wave function of the α clus- +ter. The index c indicates three kinds of the rearrange- +ment channels with different Jacobi coordinates as shown +in Fig. 1, which are superposed to make a symmetric +state with respect to the exchange of any two αs among +3α. +In each of the rearrangement channels, the basis +function is written as ΨJ +c,LN,ℓn and we expand it using the +available partial wave components ΦLN(R) and φℓn(r) +α +α +α +r1 +R1 +α +α +α +α +α +α +c = 1 +c = 2 +c = 3 +R2 +r2 +R3 +r3 +FIG. 1: Coordinate system of the α+α+α model for 12C with +three rearrangement channels. +α + N + N +r1 +r2 +r1 +r3 +r2 +r4 +α +α +N +N +N +N +N +N +α + N + N + N + N +FIG. 2: Coordinate systems of α+N+N for 6He, 6Be, and +α+N+N+N+N for 8He and 8C. +for each Jacobi coordinate, which are coupled with a total +spin J. In each partial wave component with the orbital +angular momentum L (ℓ) for the coordinate R (r), the +radial wave function is expanded by the Gaussian basis +functions having various range parameters with an index +of N (n) [37]. The index p is the set of {L, N, ℓ, n} to dis- +tinguish the basis states. The corresponding expansion +coefficients CJ +p are determined by solving the eigenvalue +problem of the Hamiltonian matrix of 12C. Using the ob- +tained CJ +p , one can evaluate the expectation value of an +operator for each eigenstate. +The 3α Hamiltonian for 12C is the same as used in the +previous studies [34, 35]: +H = +3 +� +i=1 +tαi − TG + +3 +� +i0). The +number of energy levels does not exceed the number of potential wells N. Other states +of the negative ion belong to the continuum. The energies of the bound states of the +molecular electron +2 +2 + + + +N +E +. +To illustrate the capability of the above formulas, let us consider a family of +carbon clusters consisting of two-, three- (equilateral triangle) and four (tetrahedron) +carbon atoms equidistant from each other. These model carbon structures exhaust all +possible configurations, in which all center-to-center distances are the same and equal +to R. In terms of the notation +R +e +A +R / + + + + and + +  + +B +, the determinant W of the +system of two equations reads +) +( +) +( +2 +2 +2 +A +B +B +A +A +B +W + + + + +. + + + + + +(5) +For the system of 3 equations the determinant is +) +2 +( +) +( +) +( +2 +3 +B +A +B +A +B +A +A +A +B +A +A +A +B +W + + + + + +. + + + + +(6) +The system of 4 equations leads to the determinant +) +3 +( +) +( +) +( +3 +4 +B +A +B +A +B +A +A +A +A +B +A +A +A +A +B +A +A +A +A +B +W + + + + + +. + + + + +(7) +Evaluating the determinants (5) - (7), we use the inter-atomic distance R = 2.751 +atomic units and +3055 +.0 +2. +27 +/ +27 +.1 + + + + a.u. (here 1.27 eV is the electron affinity of +С-). The graphs of the determinants versus the wave numbers  are presented in +Figure 1. The values of  at which the curves Wi(κ) cross the X axis are collected in +Table 1 along with the corresponding energy levels EN. + +Table 1. Zeros of the determinant of the system of +equations (3). + +Determinant +Zeros of +determinants, +a.u. +Energies +2 +2 + + + +N +E +, eV +W2(κ) +κ=0.42 +-2.40 eV +W3(κ) +κ=0.49 +-3.26 eV +W4(κ) +κ=0.55 +-4.11 eV + + + +W3(κ) chain +κ=0.26 +-0.919 eV + + +3 +W3(κ) chain +κ=0.46 +-2.878 eV + +Along with the equilateral triangular configuration, one can consider a linear chain of +three carbon atoms with the distance R/2 between the closest centers. The determinant +corresponding to tру configuration of atoms has the form [5] + +) +2 +)( +( +chain +) +( +2 +2 +3 +C +AB +B +B +A +B +C +A +C +B +C +A +C +B +W + + + + + + +, + + +(8) +where A and B are the same as for the previous determinants, C = 2e-R/2/R. Zeroes of +еру determinant (8) are shown in two lower lines in Table 1 and in Figure 1. New +energy levels at κ = 0.26 a.u. and at κ =0.46 a.u. appear for the chain structure. As we +can see, the number of energy levels and their positions are defined by the geometry +of the molecule constituents. + +Conclusion +In atomic physics, the simplest system that can be modeled by a particle in the field of +the zero-range wells is a negative ion of an atom with an external weakly bounded +electron in the s-state. Most of the time the outer electron spends outside the potential +well, where it can be considered free. The zero-range potential approach opens +perspectives for a study of the electron structure of more complex carbon structures +such as fullerenes and their derivatives. The wave functions and energies of the +attached electron were calculated earlier in [6] for single-cage (CN)- and multi-cage +(onion-like) (CM@CN@...)- singly charged fullerene anions using the model of +spherical potential well. This approach is only applicable for spherically symmetric +structures. The model described here can be used for carbon structures of arbitrary +geometries. The input information for such calculations includes the coordinates of +carbon atoms (for example in C20 [7]) and the binding energy of a single carbon +anion. The determinant of the system of equations (3) can be calculated numerically +using standard software packages such as MATLAB [8]. + +References +1. Orden A. V. and Saykally R.J. (1998) Small Carbon Clusters: Spectroscopy, +Structure, and Energetics, Chem. Rev., 98, 2313-2357. +2. Stasyuk A. J, Stasyuk O. A., Sola M. and Alexander A. Voityuk A. A. (2020) +Chem. Commun., 56, 352-355. +3. Drukarev G. F. (1978) The zero-range potential model and its application in +atomic and molecular physics, Adv. Quantum Chem., 11, 251-274. +4. Demkov Yu. N. and Ostrovskii V. N., “Zero-Range Potentials and Their +Applications in Atomic Physics”, 1st ed.; Plenum Press: New York, NY, USA, +1988. +5. Amusia M. Ya., Baltenkov A. S., and Woiciechowski I. (2022) Wigner time +delay of particles elastically scattered by a cluster of zero-range potentials, +Phys. Rev. A 105, 012807-18. +6. Dolmatov V. K. and Manson S. T. (2022), A glimpse into photodetachment +spectra of giant and nested fullerene anions, Atoms, 10, 99-110. +7. Baltenkov A. S. (2018), Spherical coordinates of carbon atoms in C20 fullerene +cage. arXiv:1812.07878v1. +8. MATLAB 2022a, The MathWorks, Inc., Natick, Massachusetts, United States. + + +4 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +-0.8 +-0.4 +0.0 +0.4 +0.8 +1.2 + + +Determinants +Wave vector , au + W2() + W3() + W4() + W3() line form + +Figure 1. Determinants of the system of equations (3) vs the wave vector  for carbon +molecules C2, C3, C4. + diff --git a/QtE3T4oBgHgl3EQfyQtr/content/tmp_files/load_file.txt b/QtE3T4oBgHgl3EQfyQtr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..13c17ac3ec9ba3254e7558fe0c7b98cac01fe2e6 --- /dev/null +++ b/QtE3T4oBgHgl3EQfyQtr/content/tmp_files/load_file.txt @@ -0,0 +1,150 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf,len=149 +page_content='1 About the connection of the electron binding energy of a single carbon anion with binding energies of an electron attached to carbon molecules A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Baltenkov1 and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Woiciechowski2 1Arifov Institute of Ion-Plasma and Laser Technologies, 100125, Tashkent, Uzbekistan 2Alderson Broaddus University, 101 College Hill Drive, Philippi, 26416, WV, USA Abstract We demonstrate that the model of zero-range potentials can be successfully employed for the description of attached electrons in atomic and molecular anions, for example, negatively charged carbon clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' To illustrate the capability of the model we calculate the energies of the attached electron for the family of carbon cluster anions consisting of two-, three- (equilateral triangle), and four (tetrahedron) carbon atoms equidistant from each other as well as for a C3 molecule having a chain structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' The considered approach can be easily extended to carbon clusters containing an arbitrary number of atoms arranged in an arbitrary configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' One of the tasks of quantum chemistry is determining the energy levels of an electron in a potential field, which is a superposition of centrally symmetric potentials with centers at the positions of atoms in a molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Solutions to quantum mechanical problems of this type are extremely difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Reliable results of ab initio calculations are only available for clusters with relatively small numbers of constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' For example, the structure- and energy calculations of carbon cluster anions CN- are reported for only up to N = 18 in refs [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' There exists, however, such a field, for which the problem of electron motion has a simple analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' This is the field formed by zero-range potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Within the approach described in [3, 4], the molecular constituents are replaced by a set of N zero-range potential wells located at the positions Rj (j =1, 2, 3…N) of the nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' The goal of the present communication is a demonstration that the model of zero-range potentials [3, 4] can be successfully employed for the description of attached electrons in atomic and molecular anions, for example, negatively charged carbon clusters with arbitrary number of constituents arranged in an arbitrary configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' We briefly outline the model in the next paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' The wave function of the electron ) (r \uf079 in a multicenter field, being a solution of the Schrödinger equation everywhere in space except the points Rj and decreasing at r → ∞, is a superposition of the wave functions of single carbon atoms \uf0e5 \uf03d \uf02d \uf03d N j j r j r e c j 1 ) ( \uf06b \uf079 r , j j R r r \uf02d \uf03d , (1) where \uf06b is the magnitude of the wave vector of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' At the positions r = Rj, function (1) has to satisfy the boundary conditions ] | | 1 [ 0 | | \uf061 \uf079 \uf02d \uf02d \uf03d \uf0ae \uf02d j j b j R r R r , (2) where 0 2E \uf03d \uf061 , 0 E is the binding energy of the electron in a single zero-range potential well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' We use the atomic units throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Application of the boundary conditions (2) to the wave function (1) results in a system of homogeneous linear equations for the coefficients cj in equation (1) 2 0 ) ( 1 \uf03d \uf02b \uf02d \uf0e5 \uf0b9 \uf03d \uf02d N i j j ij R j i R e c c ij \uf06b \uf06b \uf061 , i=1, 2 ,3…N, j i ij R R R \uf02d \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' (3) The system of equations (3) has nontrivial solutions when its determinant, being a function of \uf06b, W(\uf06b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' (4) Equation (4) is a transcendent equation connecting the wave vector κ with the wave vector α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' It has several solutions that correspond to discrete energy levels (κ>0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' The number of energy levels does not exceed the number of potential wells N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Other states of the negative ion belong to the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' The energies of the bound states of the molecular electron 2 2 \uf06b \uf02d \uf03d N E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' To illustrate the capability of the above formulas, let us consider a family of carbon clusters consisting of two-, three- (equilateral triangle) and four (tetrahedron) carbon atoms equidistant from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' These model carbon structures exhaust all possible configurations, in which all center-to-center distances are the same and equal to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' In terms of the notation R e A R / \uf06b \uf02d \uf03d and \uf06b \uf061 \uf02d \uf03d B , the determinant W of the system of two equations reads ) ( ) ( 2 2 2 A B B A A B W \uf02d \uf03d \uf03d \uf06b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' (5) For the system of 3 equations the determinant is ) 2 ( ) ( ) ( 2 3 B A B A B A A A B A A A B W \uf02b \uf02d \uf03d \uf03d \uf06b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' (6) The system of 4 equations leads to the determinant ) 3 ( ) ( ) ( 3 4 B A B A B A A A A B A A A A B A A A A B W \uf02b \uf02d \uf03d \uf03d \uf06b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' (7) Evaluating the determinants (5) - (7), we use the inter-atomic distance R = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='751 atomic units and 3055 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' 27 / 27 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='1 \uf03d \uf03d \uf061 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' (here 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='27 eV is the electron affinity of С-).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' The graphs of the determinants versus the wave numbers \uf06b are presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' The values of \uf06b at which the curves Wi(κ) cross the X axis are collected in Table 1 along with the corresponding energy levels EN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Zeros of the determinant of the system of equations (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Determinant Zeros of determinants, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Energies 2 2 \uf06b \uf02d \uf03d N E , eV W2(κ) κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='40 eV W3(κ) κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='49 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='26 eV W4(κ) κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='11 eV W3(κ) chain κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='919 eV 3 W3(κ) chain κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='878 eV Along with the equilateral triangular configuration, one can consider a linear chain of three carbon atoms with the distance R/2 between the closest centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' The determinant corresponding to tру configuration of atoms has the form [5] ) 2 )( ( chain ) ( 2 2 3 C AB B B A B C A C B C A C B W \uf02d \uf02b \uf02d \uf03d \uf03d \uf06b , (8) where A and B are the same as for the previous determinants, C = 2e-\uf06bR/2/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Zeroes of еру determinant (8) are shown in two lower lines in Table 1 and in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' New energy levels at κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='26 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' and at κ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='46 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' appear for the chain structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' As we can see, the number of energy levels and their positions are defined by the geometry of the molecule constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Conclusion In atomic physics, the simplest system that can be modeled by a particle in the field of the zero-range wells is a negative ion of an atom with an external weakly bounded electron in the s-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Most of the time the outer electron spends outside the potential well, where it can be considered free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' The zero-range potential approach opens perspectives for a study of the electron structure of more complex carbon structures such as fullerenes and their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' The wave functions and energies of the attached electron were calculated earlier in [6] for single-cage (CN)- and multi-cage (onion-like) (CM@CN@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=')- singly charged fullerene anions using the model of spherical potential well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' This approach is only applicable for spherically symmetric structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' The model described here can be used for carbon structures of arbitrary geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' The input information for such calculations includes the coordinates of carbon atoms (for example in C20 [7]) and the binding energy of a single carbon anion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' The determinant of the system of equations (3) can be calculated numerically using standard 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Woiciechowski I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' (2022) Wigner time delay of particles elastically scattered by a cluster of zero-range potentials, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' A 105, 012807-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Dolmatov V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' K.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' (2018), Spherical coordinates of carbon atoms in C20 fullerene cage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='07878v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' MATLAB 2022a, The MathWorks, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=', Natick, Massachusetts, United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='5 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content='2 Determinants Wave vector \uf06b, au W2(\uf06b) W3(\uf06b) W4(\uf06b) W3(\uf06b) line form Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} +page_content=' Determinants of the system of equations (3) vs the wave vector \uf06b for carbon molecules C2, C3, C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfyQtr/content/2301.04718v1.pdf'} diff --git a/TtFJT4oBgHgl3EQfMiyz/content/2301.11474v1.pdf b/TtFJT4oBgHgl3EQfMiyz/content/2301.11474v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2f5b4126e97b12971d8f672b425456300de61360 --- /dev/null +++ b/TtFJT4oBgHgl3EQfMiyz/content/2301.11474v1.pdf @@ -0,0 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Received January 30, 2023) +We investigate the intraband-transitions contribution to the optical properties of metallic carbon +nanotubes both experimentally and theoretically. The experimental dielectric constant for metallic +carbon nanotubes samples is derived from reflectance measurements using among others Kramers- +Kronig relations. +It is found in pretty good agreement with the predictions based on ab initio +calculations but departs from those based on an analytical model, the surface conductivity model, +widely used in the literature. We report new theoretical predictions for the plasma frequency of +metallic carbon nanotubes. They are in notable agreement with experimental observations. +Plasmonics became in the last years an active field of +research with the promise of improvements in sensing[1], +in particular for biological applications[2]. +The poten- +tial of highly integrated photonic devices[3], more effi- +cient solar cells,[4] and the enhancement of the radia- +tive yield of fluorophores[5] were also reported. In an ef- +fort of miniaturization, metallic Single-Wall Carbon Nan- +oTubes (m-SWCNT) could be envisioned as the smallest +metallic wire at the nanoscale with exceptional electri- +cal properties. +In contrast to nanowires based on no- +ble metals[6], individual m-SWCNT are defect-free and +present the highest-known DC-conductance[7]. +Conse- +quently, it would be relevant to consider them as build- +ing blocks for plasmonic devices down to the nanometric +scale. +The knowledge of m-SWCNT complex dielectric con- +stant is fundamental to engineer practical nanotube- +based optical devices. The experimental measurements +of the dielectric constant are mostly limited to unsorted +samples mixing semiconducting and metallic SWCNTs +in the visible and the near-infrared ranges[8, 9]. A wider +range of frequencies can be found in ref.[10]. +The in- +fluence of the synthesis technique (HiPCO, arc-grown, +CoMoCAT...) +on the optical properties of SWCNTs +was also studied[11] but only reports the real part of +the refractive index, missing the imaginary part. +The +most complete report to date[12] has measured the com- +plex electrical-conductivity for metallic and semiconduct- +ing SWCNT, but only for a narrow frequency range (0- +40 THz, 0-0.16 eV) which does not allow to design m- +SWCNT based optical antennas in the visible or near- +infrared range. +m-SWCNT’s dielectric constant could +also be known from theoretical calculations. +Most of +the ab initio studies focused on the optical properties of +semiconducting SWCNT, being interested in predicting +their band-gap[13–15]. Some work concerns m-SWCNTs +but the contribution of the intraband transitions, namely +the metallic behaviour of m-SWCNTs, is missing[16]. In +their work T. Movlarooy et al.[17] computes the dielec- +tric constants for chiralities (8,8) and (15,0) but does +not give the trend of variations of the optical proper- +ties depending on the diameter. +An analytical model, +known as the surface conductivity model, has also been +developed[18] for predicting m-SWCNT’s optical proper- +ties. To be treatable at the analytical level it relies on +approximations that have not been validated by a con- +frontation with experimental measurements. +This pa- +per aims at filling these lacks by determining the com- +plex dielectric constant over a wide spectral range (0.05 +to 5 eV) and comparing measurements with theoretical +predictions. +We focus on the metallic behavior of m- +SWCNT, highlighting the Drude model contribution to +the optical properties. We report new theoretical predic- +tions for the m-SWCNT plasma frequency, depending on +the m-SWCNT chirality, that are in agreement with the +experimental measurements. +HiPCO carbon nanotubes (Nano Integris) are sorted +by column chromatography following the procedure de- +scribed by Tanaka et al. +[19] to obtain enriched m- +SWCNT suspensions. They are filtered onto nitrocellu- +lose membranes to form thin films which are then trans- +ferred onto calcium fluoride CaF2 substrates. Samples +are further annealed under high vacuum (10−6 Torr) at +250oC for 4 hours to remove solvent and impurities[11, +12]. The film’s thickness, d1, is determined by Atomic +Force Microscopy (AFM) to be 30±5 nm (Fig.1(a)). Ra- +man spectra (Fig.1(c)) performed at 532 and 633 nm fea- +ture a strong peak at 1590 cm−1, the so-called G-band, +typical of SWCNTs. The broad asymmetric Fano line- +shape below the G-band is a key characteristic of metallic +carbon nanotubes[20], assessing the sorting quality. The +inset in Fig.1(c) displays a mapping of the intensity of +the G-band, underlying in conjunction with the AFM +data the nanotube film homogeneity in terms of thick- +ness, density, and nanotube orientation. The m-SWCNT +diameter distribution is relatively broad but could be +bracketed between 0.7 and 1.2 nm (Fig S1 in supplemen- +tary material). +arXiv:2301.11662v1 [physics.optics] 27 Jan 2023 + +2 +1200 +1300 +1400 +1500 +1600 +1700 +Raman shift [cm-1] +Intensity [a.u.] +633 nm +532 nm +substrate +m-SWCNT +20μm +Intensity at IG +0 +1 +nm +λ = 633 +IG(1590 + cm−1) +0 +5 +10 +15 +20 +25 +30 +10 +20 +30 +40 +Thickness [nm] +[ m] +5 m +μ +0.02 +0.05 0.1 +0.2 +0.5 +1 +2 3 +5 +Energy [eV] +0 +20 +40 +60 +80 +100 +Reflectance (%) +RCaF2 +Rm-SWCNT/CaF2 +0.7 +1 +2 +3 +4 +5 +6 +7 +8 +9 +(a) +(b) +(c) +m-SWCNT + film + +substrate +CaF2 +Air +α +˜n1 +˜n2 +d1 +d2 +n0 +Ei +r02Ei +(d) +0 +(0) +(1) +(2) +FIG. 1. (a) Upper panel: AFM image of a typical m-SWCNT +film. +Lower panel: mean thickness obtained by averaging +on a 5 µm window. +(b) Schematic of the m-SWCNT film +deposited on a thick CaF2 substrate. d1 and d2 denotes the +thicknesses of the m-SWCNT film and substrate, respectively. +˜n1 and ˜n2 are the complex refractive indices of the m-SWCNT +film and the CaF2 substrate. (c) Typical Raman spectra for +excitation at 633 nm (red curve) and 532 nm (green curve). +The inset shows a mapping of the G band at 1590 cm−1 over +the m-SWCNT film (excitation at 633nm). (d) Experimental +reflectance of a m-SWCNT film on CaF2 substrate (red curve) +and of the reference CaF2 substrate (dot curve). The inset is +a zoom on the excitonic transitions. +Broadband reflectance spectra are measured using an +FTIR spectrometer (Bruker IFS 66v/S) adapted with a +combined transmission and specular reflection apparatus +from 0.02 eV up to 0.6 eV, and a UV-Vis-NIR spectrom- +eter (Cary Varian 5000) adapted with a VW absolute +specular reflectance accessory from 0.6 eV up to 5 eV. +Measurements are done at near-normal incidence in both +cases. Great care is taken to acquire the reflectance spec- +tra which requires an extremely fine alignment procedure +to achieve a precision on the order of 1 %. Measurements +are calibrated on materials whose optical properties are +known. Fig.1(d) displays the m-SWCNT film reflectance +deposited on a CaF2 substrate (red curve), while the bare +substrate reflectance is the dotted-gray curve. The inset +is a zoom on the 0.7 to 3 eV range displaying the car- +bon nanotube excitonic transitions contribution[11, 21]. +While the M11 transitions (near 2 eV) contribute to the +reflectance largely above the level due to the substrate +contribution, the S11 transitions (0.7-1.2 eV) contribute +marginally to this reflectance, underlining again sorting +efficiency (see Fig. S2 in supplementary material for ad- +ditional comparison of the absorption spectra before and +after sorting). +The complex dielectric constant of the m-SWCNT +layer is extracted from the reflectance measurements us- +ing Kramers-Kronig relations[22–24] relating the phase +θ of a complex number z = |z|eiθ to its modulus |z|. +The procedure requires the determination of the com- +plex Fresnel coefficient r02 which describes light reflec- +tion by the m-SWCNT layer deposited on a semi-infinite +CaF2 substrate (Fig.1(b)). The reflectance experiment +does not measure |r02|2 directly. While thick compared +to any wavelength probes in this letter, the finite size +of the substrate (d2 = 1 mm) has to be taken into ac- +count, particularly in the spectral range where CaF2 is +transparent. Supplementary material S3 explains how to +relate the measured reflectance R to the quantity |r02|2. +Once the modulus |r02| and the phase θ02 of r02 are +known, the m-SWCNT refractive index could be ex- +tracted by solving: +r02[˜n1(ω)] = |r02|eiθ02(ω) +(1) +where r02[˜n1(ω)] is the analytical expression of the Fres- +nel coefficient. +The unknown to be found is the m- +SWCNT complex refractive-index, ˜n1(ω). The quantity +on the right side of Eq. (1) is derived from the experi- +ment. +This experimental refractive index, and the corre- +sponding dielectric constant, found from this procedure +are plotted in Fig.2, real and imaginary parts respec- +tively on the left and right side. The experimental con- +fidence interval is represented by a shaded area. +This +area includes the uncertainties on the film thickness, the +substrate refractive index, the reflectance spectral join +between the NIR / MIR range and the evaluation of θ, +the phase of the Fresnel coefficient. +The main source +of errors comes from the uncertainty on the film thick- +ness. To check the accuracy of the experimentally de- +termined refractive index, the sample transmittance was +measured in the 0.05 to 4 eV range and compared with +predictions based on this experimental refractive index +as shown in section S4 in supplementary materials. This +measurement, independent from the one used to extract +the complex refractive index, shows a good agreement +with the predictions. +It is a blind test to check the +consistency of the refractive index derived from the re- +flectance measurements. The interband transitions above +1 eV are of particular interest. As displayed in Fig.2(b), +strong sharp peaks are observed corresponding to the m- +SWCNT absorption peaks in solution (Fig. S2). From an +optical point of view, this means the m-SWNT thin film +behaves as a collection of individual nanotubes without +perceptible bundle effects. Therefore, it is of interest to +compare the experimental dielectric constant with calcu- +lations done on single m-SWCNTs. +Particularly relevant for plasmonic applications is the +cut-off energy E⋆ below which a film made of m-SWCNT +behaves like a metal. It is defined as the energy for which +the real part of the dielectric constant is null and below +which it is negative. This quantity is not clearly reported +in the literature but can be extracted from our experi- +ments: we found E⋆ = 0.55±0.03 eV for the m-SWCNT + +1160 - +1170- +1180 - +1190- +1200 +1210 - +1220 +1230 +1240 +1250 +1260 - +20μm +1250 +1300 +1350 +14003 +0.2 +0.5 +1 +2 +4 +Energy [eV] +-10 +-8 +-6 +-4 +-2 +0 +2 +4 +0.2 +0.5 +1 +2 +4 +Energy [eV] +0 +2 +4 +6 +8 +10 +12 +0.2 +0.5 +1 +2 +4 +Energy [eV] +-10 +-8 +-6 +-4 +-2 +0 +2 +4 +0.2 +0.5 +1 +2 +4 +Energy [eV] +0 +2 +4 +6 +8 +10 +12 +0.05 +0.1 +0.2 +0.5 +1 +2 +4 +Energy [eV] +0.5 +1 +2 +4 +5 +8 +0.05 +0.1 +0.2 +0.5 +1 +2 +4 +Energy [eV] +0.1 +0.2 +0.5 +1 +2 +4 +8 +12 +(a) +(b) +(c) +(d) +(e) +(f) +DFT +DFT +A. M. +A. M. +0.2 +0.5124 +Energy [eV] +0 +2 +4 +6 +8 +10 +12 +$\varepsilon_i$, imaginary part +(6,6) +(9,9) +(9,0) +(12,3) +Measurement +FIG. 2. Experimental measurement of the (a) real part and +(b) imaginary part of the complex refractive index of the m- +SWCNT layer (light-red shaded area, including confidence in- +terval). (c)(e) Real part and (d)(f) imaginary part of the di- +electric constant of the m-SWCNT layer, compared with pre- +dictions from (c)(d) the analytical model (A.M.) and (e)(f) +DFT calculations. Experimental measurements in light-red +shaded area, theoretical predictions for various (n,m) chirali- +ties in lines. +thin film. In the energy range below E⋆, the experimen- +tal dielectric constant shows an excellent agreement with +a Drude model[25] demonstrating that the m-SWCNT +film indeed behaves like a metal (Supplementary mate- +rial S5). +We now aim at comparing the experimental optical +properties of m-SWCNTs with theoretical predictions. +For this, SWCNT could be described as a rolled-up +graphene sheet with a diameter of ∼ 1 nm and a length +between 500 nm and 1 µm. +Due to this high aspect +ratio they are anisotropic materials. +Their dielectric +constant is a tensor with two different contributions: +the axial dielectric-constant ε∥ along the nanotube axis +and the transverse dielectric constant ε⊥ for the plane +perpendicular to the axis. +Both intraband and inter- +band transitions[26] contribute to the dielectric constant. +They could be expressed as a function of the light angu- +lar frequency ω by: ε∥(ω) = ε∥ +intra(ω) + ε∥ +inter(ω) and +ε⊥(ω) = ε⊥ +intra(ω) + ε⊥ +inter(ω). Assuming orientational +randomness (cf Fig.1(a)), the m-SWNT layer behaves as +a material with a uniform and isotropic dielectric con- +stant ε(ω) that reads: +ε(ω) = 2 +3ε⊥(ω) + 1 +3ε∥(ω) +(2) +This quantity is extracted from reflectance measure- +ments and plotted in Fig.2(c)-(f). Two competing mod- +els are used to predict the complex dielectric constant: an +analytical model and an ab initio model based on Density +Functional Theory (DFT). The analytical model[18, 27] +relies on the Boltzmann kinetic equation. Based on tight- +binding approximations[28], it shares many similarities +with the analytical model of graphene[29]. Importantly, +this model approximates m-SWCNT as infinitely thin +wire, a pure 1D object, meaning that the transverse di- +electric constant ε⊥(ω) is equal to one. The axial compo- +nent is computed with the help of the electrical conduc- +tance Σ∥ +B(ω) along the SWCNT axis. It is necessary to +use a characteristic length-scale t⋆ to convert the electri- +cal conductance into electrical conductivity, a quantity +related to the dielectric constant. Following reference[30] +the chosen characteristic length-scale is the SWCNT di- +ameter t⋆ = 2RCNT where RCNT is the nanotube radius. +Therefore, ε∥(ω) is related to the conductance derived +from the Boltzmann formula Σ∥ +K(ω) through the equa- +tion: +ε∥(ω) = 1 + iΣ∥ +B(ω) +ε0t⋆ω +(3) +where ε0 is the vacuum permittivity. +The relaxation +effects are incorporated phenomenologically[31, p.154] +through a damping rate γ = 2π/τ, where τ = 35 fs is +the relaxation time[27, 31]. The numerical value is ex- +tracted from the fit with the Drude model performed +on the experimental results (See supplementary material +S6). This value is in agreement with the literature[27]. +Fig.2(c) and (d) compare the measurements with pre- +dictions based on the analytical model shown as solid +lines. We focus the comparison on the range [0.2,4] eV +but our conclusions are valid for the whole energy range. +Indeed at low energy, the absolute value of the dielectric +constant is large shrinking the curves variations if the +whole energy range [0.05,4] eV is plotted. The experi- +mental sample is described by a distribution of metallic +nanotube species: armchair (n,n), zigzag (n,0) and chi- +ral (n,m), in the diameter range [0.7,1.2] nm determined +from the Radial Breathing Modes study (See supplemen- +tary material S1). To compare, we chose the (9,9) and +(9,0) nanotubes, with a respective diameter of 1.24 nm +and 0.71 nm, which are the extrema that bound the di- +ameter distribution. The chirality (6,6) whose diameter + +4 +of 0.81 nm is close to the center of the distribution, is +chosen as a representative species. This comparison does +not include any chiral species since the analytical solu- +tion for the interband transitions is not given explicitly +for this type of m-SWCNT[18, 27]. Results show that +the comparison is only qualitative. +It under-estimates +the real-part of the dielectric constant for energies larger +than 0.3 eV while it over-estimates the imaginary-part +of the dielectric constant in the energy range between +0.3 and 0.6 eV. This analytical model does not predict +accurately the dielectric constant of a film of m-SWCNT. +The optical properties of m-SWCNT can also be com- +puted from a first-principles approach based on the Den- +sity Functional Theory (DFT). (See supplementary ma- +terial S6) This method relaxes some assumptions inher- +ent to the analytical model. +Indeed, the electron dy- +namics takes into account both the σ-electrons and the +π-electrons and is not limited to the first three adjacent +atoms in the hexagonal structure. Then, SWCNT are +no more assumed to be infinitely thin perpendicular to +its axis (1D-object), so that ε⊥(ω) ̸= 1. Finally, DFT +takes into account curvature effects, which lacks in the +analytical model. +In this paper, we compute both intraband and inter- +band transitions to the complex dielectric constant for +a large range of chiral indexes. +Concerning interband +transitions, both ε⊥(ω) and ε∥(ω) are characterized by a +series of absorption lines in the high-energy range. It is +worth highlighting that ε⊥(ω) differs from unity. Arm- +chairs are the only species to be truly metallic. Zigzag +and chiral m-SWCNT are semi-metallic, with a low en- +ergy gap arising from curvature[11, 32]. This gap arises +at very low energy (e < 0.05 eV) below our measurement +range, but the tail of their contribution could contribute +to the dielectric constant in our measurement range. +The intraband transition contributes to the dielectric +constant as a Drude-like term[25] with a damping factor +γ = 2π/τ (τ = 35 fs for all nanotube species) to take into +account electronic relaxation. +The comparison between the experimental dielectric +constant and DFT calculations is shown in Fig.2(e) and +(f). The chiral nanotube (12,3) of diameter 1.09 nm was +also included. +The experimental measurements fall in +the range bounded by the two species (9,0) and (9,9). It +follows quite closely the predictions for the chirality (6,6) +that is close to the center of the diameter distribution. +Overall, a very good agreement is achieved between the +DFT calculations and the experimental values. +The cut-off energy E⋆ allows a quantitative assess- +ment of the theoretical predictions compared to the ex- +perimental reality. +Fig.3(a) displays E⋆ as a function +of m-SWCNT diameter. Experimental results are repre- +sented as a black shaded area, with a confidence inter- +val of ± 0.03 eV. DFT calculations for various chiralities +inside the experimental diameter distribution are repre- +sented by solid circles, while the value predicted by the +DFT +Analytics +Measurement +● +● +● +● +● +● +● +● +■ +■ +■ +■ +■ +■ +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +CNT diameter (nm) +E★ (eV) +DFT: (m,m) +A.M.: (m,m) +DFT: (m,0) +DFT: 2m+n=3q +A.M.: (m,0) +A.M.: 2m+n=3q +a) +b) +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +● +● +● +● +● +● +● +● +● +● +● +● +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +0 +1 +2 +3 +4 +5 +CNT diameter (nm) +ℏωp (eV) +FIG. 3. (a) Cut-off energy as a function of the m-SWCNT +diameter. Filled circles are predictions from the DFT calcu- +lations. They are delimited by the blue-shaded area. Filled +squares are predictions from the analytical model. They are +delimited by the red-shaded area. Armchair (m,m), zigzag +(m,0) and chiral (2m+n=3q) species are specified by the color +blue, red or black, respectively. (b) Plasma frequency as a +function of the m-SWCNT diameters predicted from the DFT +calculations (filled circles) and the analytical model (filled +squares). +Results from DFT are fitted with a linear curve +and results from the analytical model follow a hyperbola. +analytical model are represented by solid squares. Ex- +perimental values are clearly within the interval of DFT +calculations and clearly outside the values predicted by +the analytical model, its lowest value being 0.17 eV too +high. +To understand more deeply the discrepancies between +those two models, the Drude model can be considered. +Based on eq. (2), the cut-off energy E⋆ reads (see sup- +plementary material S7): +E⋆ = ℏ +� +ω2p +ε∥ +b + 2ε⊥ +b +− γ2 +(4) +with the plasma frequency ωp, the loss rate γ and the +background dielectric constant ε∥ +b and ε⊥ +b , respectively +parallel and perpendicular to the nanotube axis. Except +for the loss rate that is considered to be the same in +the two models, all other quantities differ. +The quan- +tity ε∥ +b differs marginally in the two theories by roughly +10-20 %. This difference can not explain the large dis- +crepancy between the experimental value and the theory. +As a consequence, we focus the analysis on the two other +quantities ωp and ε⊥ +b . +The plasma energy ℏωp is plotted in Fig.3(b) for a +wide distribution of nanotube diameters, wider than ex- +perimentally available. DFT calculations are represented +by solid circles while the analytical model values are dis- +played as solid squares. They strikingly differ for small +diameter nanotubes and seem to converge toward the +largest diameters. The nanotube diameter dependence +differs. For the analytical model, ℏωp varies as the in- +verse of the nanotube diameter[18, 27], while it seems + +5 +rather to decrease linearly for the DFT results . This +is independent of the nanotube type: armchair, zigzag +and chiral SWCNT align on the same line. This is a new +result leading to qualitative and quantitative differences +between the two theories. +The second parameter that +differs in the two model’s predictions is ε⊥ +b . The analyt- +ical model assumes ε⊥ +b = 1 whereas DFT predicts that +m-SWCNTs are also polarizable objects in the transverse +directions despite their small diameter compared to the +light wavelength. For energy smaller than 0.5 eV, the +DFT calculations predict a value ranging between 2 and +3, significantly different from 1. +Last but not least, DFT calculations take into account +the curvature-induced low-energy gap existing in non- +armchair SWCNT while the analytical model does not. +This effect is not captured by the Drude model. As ob- +served in Fig.3, it leads to higher E⋆ for zigzag and chi- +ral compared to armchair nanotubes, independently of +the diameter effect. Consequently, it could not explain +the discrepancy between the measurement and the pre- +dictions of the analytical model. +We reported the measurement of the complex dielectric +constant of metallic carbon nanotubes films over a wide +range of energies ( 0.05 to 4 eV) and the comparison with +predictions from two theoretical models. This allowed us +to investigate intraband transitions in m-SWCNT and in +particular to extract the fundamental quantities that are +the cut-off frequency E⋆ = 0.55 ± 0.03 eV, below which +a m-SWCNT film behaves like a metal and plasma fre- +quencies for a large range of chiralities. We highlight a re- +markable agreement with DFT calculations and exclude +prediction based on the analytical model. A similar con- +clusion was reached for graphene[29]. The assumptions +performed in the analytical model[18] are too crude to +describe correctly the optical properties of m-SWCNT. +Our methodology strengthens the need for ab initio cal- +culations to predict accurately the electronic and optical +properties of SWCNT. Based on this enhanced knowl- +edge of m-SWCNT optical properties, we envision the +use of metallic carbon nanotubes as fundamental build- +ing blocks for nanometric scale plasmonic devices in the +near infrared range, in particular telecom relevant optical +windows. +ER and NI contributed equally to this work. +They +initiated and lead this research. The samples were sorted +by DB under the supervision of NI with insights from +SC. The reflectance measurements were performed by +DB under the supervision of NI and JLB. The Kramers- +Kronig analysis were performed by DB and ER. The +DFT calculations were performed by PH. ER and NI +wrote a draft of the paper. +All the authors discussed +the results, agreed with the conclusions, and contributed +to the final version of the letter. +Authors thanks M. +Ramonda from CTM Montpellier for his help with +AFM measurements. +Reflectance measurements were +performed on the technological platform IRRAMAN of +the University of Montpellier. +∗ emmanuel.rousseau@umontpellier.fr; +nico- +las.izard@umontpellier.fr +[1] M. Butt, S. Khonina and N. Kazanskiy, Fiber and Inte- +grated Optics 40, 14 (2021) +[2] M. Li, S.K. Cushing and N. Wu, Analyst 140, 386 (2015) +[3] S.A. Maier, IEEE Journal of Selected Topics in Quantum +Electronics 12, 1671 (2006) +[4] M.M. Olaimat, L. Yousefi and O.M. Ramahi, J. Opt. Soc. +Am. B 38, 638 (2021) +[5] J.R. Lakowicz, K. Ray, M. Chowdhury, H. Szmacinski, +Y. Fu, J. Zhang and K. Nowaczyk, Analyst 133, 1308 +(2008) +[6] A.C. Pearson, J. Liu, E. Pound, B. Uprety, A.T. Wooley, +R.C. Davis and J.N. Harb, J. Phys. Chem. 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Gharbavi and H. Badehian, AIP Advances 5 077155 +(2015) +[17] T. Movlarooy Chinese Physics Letters 30 077301 (2013) +[18] G.Y. Slepyan, S.A. Maksimenko, A. Lakhtakia, O. Yevt- +shenko and A.V. Gusakovs Phys. Rev. B 60 17136 (1999) +[19] T. Tanaka, H. Liu, S. Fujii and H. Kataura, Phys. Status +Solidi RLL 9 301 (2011) +[20] E.H. Hasdeo, A.R.T. Nugraha, K. Sato, M.S. Dresselhaus +and R. Saito, Phys. Rev. B 88 115107 (2013) +[21] Y. Miyata, K. Yanagi, Y. Maniwa and H. Kataura phys- +ica status solidi (b) 245 2233 (2008) +[22] F. C. Jahoda, Physical Review 107 1261(1957) +[23] P. L. Nash and R.J. Bell and R. Alexander Journal of +Modern Optics 42 1837 (1995) +[24] E. Rousseau, N. Izard, J.L. Bantignies and D. Felbacq, +Spectroc. Acta A 259 119849 (2021) +[25] N. +W. +Aschcroft +and +N. +D +Mermin, +Solid +State +Physics,Saunders College Publishing, Harcourt Brace Jo- +vanovich College Publishers, 1976 +[26] M.G. Burdanova, A.P. Tsapenko, M.V. Kharlamova, E.I. +Kauppinen, B.P. Gorshunov, J. Kono and J. Lloyd- +Hughes, Advanced Optical Materials 9, 2101042 (2021) + +6 +[27] A. M. Nemilentsau Journal of Nanophotonics 5 050401 +(2011) +[28] S. Maksimenko and G. Slepyan, The Handbook of Nan- +otechnology. Nanometer Structures: Theory, Modeling, +and Simulation (2004) +[29] Cheon, Sosan and Kihm, Kenneth David, Complex Re- +fractive Index (RI) of Graphene, John Wiley & Sons, Ltd +[30] G.Y. Slepyan, M.V. Shuba, S.A. Maksimenko, C. Thom- +sen and A. Lakhtakia, Phys. Rev. B 81 205423 (2010) +[31] The Handbook of Nanotechnology. Nanometer Struc- +tures: Theory, Modeling, and Simulation, SPIE (2004) +[32] M. Ouyang, J.-L. Huang, C. L. Cheung and C. M. Lieber +Science 292 702 (2001) + diff --git a/VdFJT4oBgHgl3EQf3i1y/content/tmp_files/load_file.txt b/VdFJT4oBgHgl3EQf3i1y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1153201ddeae3050821a0f4db5adfecd7c288bc --- /dev/null +++ b/VdFJT4oBgHgl3EQf3i1y/content/tmp_files/load_file.txt @@ -0,0 +1,510 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf,len=509 +page_content='Probing optical properties of metallic carbon nanotubes: beyond the standard semi-empirical model for plasmonic predictions Domitille Baux1, Patrick Hermet2, St´ephane Campidelli3, Jean-Louis Bantignies1, Emmanuel Rousseau1,∗, Nicolas Izard1,∗ 1-Laboratoire Charles Coulomb, UMR5521, CNRS-Universit´e de Montpellier, 34095 Montpellier, France 2-ICGM, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Montpellier, CNRS, ENSCM, Montpellier, France and 3-Universit´e Paris-Saclay, CEA, CNRS, NIMBE, LICSEN, 91191, Gif-sur-Yvette, France∗ (Dated: Received January 30, 2023) We investigate the intraband-transitions contribution to the optical properties of metallic carbon nanotubes both experimentally and theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The experimental dielectric constant for metallic carbon nanotubes samples is derived from reflectance measurements using among others Kramers- Kronig relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' It is found in pretty good agreement with the predictions based on ab initio calculations but departs from those based on an analytical model, the surface conductivity model, widely used in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' We report new theoretical predictions for the plasma frequency of metallic carbon nanotubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' They are in notable agreement with experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Plasmonics became in the last years an active field of research with the promise of improvements in sensing[1], in particular for biological applications[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The poten- tial of highly integrated photonic devices[3], more effi- cient solar cells,[4] and the enhancement of the radia- tive yield of fluorophores[5] were also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' In an ef- fort of miniaturization, metallic Single-Wall Carbon Nan- oTubes (m-SWCNT) could be envisioned as the smallest metallic wire at the nanoscale with exceptional electri- cal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' In contrast to nanowires based on no- ble metals[6], individual m-SWCNT are defect-free and present the highest-known DC-conductance[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Conse- quently, it would be relevant to consider them as build- ing blocks for plasmonic devices down to the nanometric scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The knowledge of m-SWCNT complex dielectric con- stant is fundamental to engineer practical nanotube- based optical devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The experimental measurements of the dielectric constant are mostly limited to unsorted samples mixing semiconducting and metallic SWCNTs in the visible and the near-infrared ranges[8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' A wider range of frequencies can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The in- fluence of the synthesis technique (HiPCO, arc-grown, CoMoCAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=') on the optical properties of SWCNTs was also studied[11] but only reports the real part of the refractive index, missing the imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The most complete report to date[12] has measured the com- plex electrical-conductivity for metallic and semiconduct- ing SWCNT, but only for a narrow frequency range (0- 40 THz, 0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='16 eV) which does not allow to design m- SWCNT based optical antennas in the visible or near- infrared range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' m-SWCNT’s dielectric constant could also be known from theoretical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Most of the ab initio studies focused on the optical properties of semiconducting SWCNT, being interested in predicting their band-gap[13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Some work concerns m-SWCNTs but the contribution of the intraband transitions, namely the metallic behaviour of m-SWCNTs, is missing[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' In their work T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Movlarooy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' [17] computes the dielec- tric constants for chiralities (8,8) and (15,0) but does not give the trend of variations of the optical proper- ties depending on the diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' An analytical model, known as the surface conductivity model, has also been developed[18] for predicting m-SWCNT’s optical proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' To be treatable at the analytical level it relies on approximations that have not been validated by a con- frontation with experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' This pa- per aims at filling these lacks by determining the com- plex dielectric constant over a wide spectral range (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='05 to 5 eV) and comparing measurements with theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' We focus on the metallic behavior of m- SWCNT, highlighting the Drude model contribution to the optical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' We report new theoretical predic- tions for the m-SWCNT plasma frequency, depending on the m-SWCNT chirality, that are in agreement with the experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' HiPCO carbon nanotubes (Nano Integris) are sorted by column chromatography following the procedure de- scribed by Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' [19] to obtain enriched m- SWCNT suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' They are filtered onto nitrocellu- lose membranes to form thin films which are then trans- ferred onto calcium fluoride CaF2 substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Samples are further annealed under high vacuum (10−6 Torr) at 250oC for 4 hours to remove solvent and impurities[11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The film’s thickness, d1, is determined by Atomic Force Microscopy (AFM) to be 30±5 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Ra- man spectra (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='1(c)) performed at 532 and 633 nm fea- ture a strong peak at 1590 cm−1, the so-called G-band, typical of SWCNTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The broad asymmetric Fano line- shape below the G-band is a key characteristic of metallic carbon nanotubes[20], assessing the sorting quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='1(c) displays a mapping of the intensity of the G-band, underlying in conjunction with the AFM data the nanotube film homogeneity in terms of thick- ness, density, and nanotube orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The m-SWCNT diameter distribution is relatively broad but could be bracketed between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='7 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 nm (Fig S1 in supplemen- tary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='11662v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='optics] 27 Jan 2023 2 1200 1300 1400 1500 1600 1700 Raman shift [cm-1] Intensity [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='] 633 nm 532 nm substrate m-SWCNT 20μm Intensity at IG 0 1 nm λ = 633 IG(1590 cm−1) 0 5 10 15 20 25 30 10 20 30 40 Thickness [nm] [ m] 5 m μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='5 1 2 3 5 Energy [eV] 0 20 40 60 80 100 Reflectance (%) RCaF2 Rm-SWCNT/CaF2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='7 1 2 3 4 5 6 7 8 9 (a) (b) (c) m-SWCNT film substrate CaF2 Air α ˜n1 ˜n2 d1 d2 n0 Ei r02Ei (d) 0 (0) (1) (2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' (a) Upper panel: AFM image of a typical m-SWCNT film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Lower panel: mean thickness obtained by averaging on a 5 µm window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' (b) Schematic of the m-SWCNT film deposited on a thick CaF2 substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' d1 and d2 denotes the thicknesses of the m-SWCNT film and substrate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' ˜n1 and ˜n2 are the complex refractive indices of the m-SWCNT film and the CaF2 substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' (c) Typical Raman spectra for excitation at 633 nm (red curve) and 532 nm (green curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The inset shows a mapping of the G band at 1590 cm−1 over the m-SWCNT film (excitation at 633nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' (d) Experimental reflectance of a m-SWCNT film on CaF2 substrate (red curve) and of the reference CaF2 substrate (dot curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The inset is a zoom on the excitonic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Broadband reflectance spectra are measured using an FTIR spectrometer (Bruker IFS 66v/S) adapted with a combined transmission and specular reflection apparatus from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='02 eV up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='6 eV, and a UV-Vis-NIR spectrom- eter (Cary Varian 5000) adapted with a VW absolute specular reflectance accessory from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='6 eV up to 5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Measurements are done at near-normal incidence in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Great care is taken to acquire the reflectance spec- tra which requires an extremely fine alignment procedure to achieve a precision on the order of 1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Measurements are calibrated on materials whose optical properties are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='1(d) displays the m-SWCNT film reflectance deposited on a CaF2 substrate (red curve), while the bare substrate reflectance is the dotted-gray curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The inset is a zoom on the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='7 to 3 eV range displaying the car- bon nanotube excitonic transitions contribution[11, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' While the M11 transitions (near 2 eV) contribute to the reflectance largely above the level due to the substrate contribution, the S11 transitions (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 eV) contribute marginally to this reflectance, underlining again sorting efficiency (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' S2 in supplementary material for ad- ditional comparison of the absorption spectra before and after sorting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The complex dielectric constant of the m-SWCNT layer is extracted from the reflectance measurements us- ing Kramers-Kronig relations[22–24] relating the phase θ of a complex number z = |z|eiθ to its modulus |z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The procedure requires the determination of the com- plex Fresnel coefficient r02 which describes light reflec- tion by the m-SWCNT layer deposited on a semi-infinite CaF2 substrate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The reflectance experiment does not measure |r02|2 directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' While thick compared to any wavelength probes in this letter, the finite size of the substrate (d2 = 1 mm) has to be taken into ac- count, particularly in the spectral range where CaF2 is transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Supplementary material S3 explains how to relate the measured reflectance R to the quantity |r02|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Once the modulus |r02| and the phase θ02 of r02 are known, the m-SWCNT refractive index could be ex- tracted by solving: r02[˜n1(ω)] = |r02|eiθ02(ω) (1) where r02[˜n1(ω)] is the analytical expression of the Fres- nel coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The unknown to be found is the m- SWCNT complex refractive-index, ˜n1(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The quantity on the right side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' (1) is derived from the experi- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' This experimental refractive index, and the corre- sponding dielectric constant, found from this procedure are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2, real and imaginary parts respec- tively on the left and right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The experimental con- fidence interval is represented by a shaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' This area includes the uncertainties on the film thickness, the substrate refractive index, the reflectance spectral join between the NIR / MIR range and the evaluation of θ, the phase of the Fresnel coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The main source of errors comes from the uncertainty on the film thick- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' To check the accuracy of the experimentally de- termined refractive index, the sample transmittance was measured in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='05 to 4 eV range and compared with predictions based on this experimental refractive index as shown in section S4 in supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' This measurement, independent from the one used to extract the complex refractive index, shows a good agreement with the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' It is a blind test to check the consistency of the refractive index derived from the re- flectance measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The interband transitions above 1 eV are of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' As displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2(b), strong sharp peaks are observed corresponding to the m- SWCNT absorption peaks in solution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' From an optical point of view, this means the m-SWNT thin film behaves as a collection of individual nanotubes without perceptible bundle effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Therefore, it is of interest to compare the experimental dielectric constant with calcu- lations done on single m-SWCNTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Particularly relevant for plasmonic applications is the cut-off energy E⋆ below which a film made of m-SWCNT behaves like a metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' It is defined as the energy for which the real part of the dielectric constant is null and below which it is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' This quantity is not clearly reported in the literature but can be extracted from our experi- ments: we found E⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='55±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='03 eV for the m-SWCNT 1160 - 1170- 1180 - 1190- 1200 1210 - 1220 1230 1240 1250 1260 - 20μm 1250 1300 1350 14003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='5 1 2 4 Energy [eV] 10 8 6 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='5 1 2 4 Energy [eV] 0 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='5 1 2 4 Energy [eV] 10 8 6 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='5 1 2 4 Energy [eV] 0 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='5 1 2 4 Energy [eV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='5 1 2 4 5 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='5 1 2 4 Energy [eV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='5 1 2 4 8 12 (a) (b) (c) (d) (e) (f) DFT DFT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='5124 Energy [eV] 0 2 4 6 8 10 12 $\\varepsilon_i$, imaginary part (6,6) (9,9) (9,0) (12,3) Measurement FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Experimental measurement of the (a) real part and (b) imaginary part of the complex refractive index of the m- SWCNT layer (light-red shaded area, including confidence in- terval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' (c)(e) Real part and (d)(f) imaginary part of the di- electric constant of the m-SWCNT layer, compared with pre- dictions from (c)(d) the analytical model (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=') and (e)(f) DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Experimental measurements in light-red shaded area, theoretical predictions for various (n,m) chirali- ties in lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' In the energy range below E⋆, the experimen- tal dielectric constant shows an excellent agreement with a Drude model[25] demonstrating that the m-SWCNT film indeed behaves like a metal (Supplementary mate- rial S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' We now aim at comparing the experimental optical properties of m-SWCNTs with theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' For this, SWCNT could be described as a rolled-up graphene sheet with a diameter of ∼ 1 nm and a length between 500 nm and 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Due to this high aspect ratio they are anisotropic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Their dielectric constant is a tensor with two different contributions: the axial dielectric-constant ε∥ along the nanotube axis and the transverse dielectric constant ε⊥ for the plane perpendicular to the axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Both intraband and inter- band transitions[26] contribute to the dielectric constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' They could be expressed as a function of the light angu- lar frequency ω by: ε∥(ω) = ε∥ intra(ω) + ε∥ inter(ω) and ε⊥(ω) = ε⊥ intra(ω) + ε⊥ inter(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Assuming orientational randomness (cf Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='1(a)), the m-SWNT layer behaves as a material with a uniform and isotropic dielectric con- stant ε(ω) that reads: ε(ω) = 2 3ε⊥(ω) + 1 3ε∥(ω) (2) This quantity is extracted from reflectance measure- ments and plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2(c)-(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Two competing mod- els are used to predict the complex dielectric constant: an analytical model and an ab initio model based on Density Functional Theory (DFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The analytical model[18, 27] relies on the Boltzmann kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Based on tight- binding approximations[28], it shares many similarities with the analytical model of graphene[29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Importantly, this model approximates m-SWCNT as infinitely thin wire, a pure 1D object, meaning that the transverse di- electric constant ε⊥(ω) is equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The axial compo- nent is computed with the help of the electrical conduc- tance Σ∥ B(ω) along the SWCNT axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' It is necessary to use a characteristic length-scale t⋆ to convert the electri- cal conductance into electrical conductivity, a quantity related to the dielectric constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Following reference[30] the chosen characteristic length-scale is the SWCNT di- ameter t⋆ = 2RCNT where RCNT is the nanotube radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Therefore, ε∥(ω) is related to the conductance derived from the Boltzmann formula Σ∥ K(ω) through the equa- tion: ε∥(ω) = 1 + iΣ∥ B(ω) ε0t⋆ω (3) where ε0 is the vacuum permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The relaxation effects are incorporated phenomenologically[31, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='154] through a damping rate γ = 2π/τ, where τ = 35 fs is the relaxation time[27, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The numerical value is ex- tracted from the fit with the Drude model performed on the experimental results (See supplementary material S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' This value is in agreement with the literature[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2(c) and (d) compare the measurements with pre- dictions based on the analytical model shown as solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' We focus the comparison on the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2,4] eV but our conclusions are valid for the whole energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Indeed at low energy, the absolute value of the dielectric constant is large shrinking the curves variations if the whole energy range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='05,4] eV is plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The experi- mental sample is described by a distribution of metallic nanotube species: armchair (n,n), zigzag (n,0) and chi- ral (n,m), in the diameter range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='7,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2] nm determined from the Radial Breathing Modes study (See supplemen- tary material S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' To compare, we chose the (9,9) and (9,0) nanotubes, with a respective diameter of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='24 nm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='71 nm, which are the extrema that bound the di- ameter distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The chirality (6,6) whose diameter 4 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='81 nm is close to the center of the distribution, is chosen as a representative species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' This comparison does not include any chiral species since the analytical solu- tion for the interband transitions is not given explicitly for this type of m-SWCNT[18, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Results show that the comparison is only qualitative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' It under-estimates the real-part of the dielectric constant for energies larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='3 eV while it over-estimates the imaginary-part of the dielectric constant in the energy range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' This analytical model does not predict accurately the dielectric constant of a film of m-SWCNT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The optical properties of m-SWCNT can also be com- puted from a first-principles approach based on the Den- sity Functional Theory (DFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' (See supplementary ma- terial S6) This method relaxes some assumptions inher- ent to the analytical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Indeed, the electron dy- namics takes into account both the σ-electrons and the π-electrons and is not limited to the first three adjacent atoms in the hexagonal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Then, SWCNT are no more assumed to be infinitely thin perpendicular to its axis (1D-object), so that ε⊥(ω) ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Finally, DFT takes into account curvature effects, which lacks in the analytical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' In this paper, we compute both intraband and inter- band transitions to the complex dielectric constant for a large range of chiral indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Concerning interband transitions, both ε⊥(ω) and ε∥(ω) are characterized by a series of absorption lines in the high-energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' It is worth highlighting that ε⊥(ω) differs from unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Arm- chairs are the only species to be truly metallic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Zigzag and chiral m-SWCNT are semi-metallic, with a low en- ergy gap arising from curvature[11, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' This gap arises at very low energy (e < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='05 eV) below our measurement range, but the tail of their contribution could contribute to the dielectric constant in our measurement range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The intraband transition contributes to the dielectric constant as a Drude-like term[25] with a damping factor γ = 2π/τ (τ = 35 fs for all nanotube species) to take into account electronic relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The comparison between the experimental dielectric constant and DFT calculations is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2(e) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The chiral nanotube (12,3) of diameter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='09 nm was also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The experimental measurements fall in the range bounded by the two species (9,0) and (9,9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' It follows quite closely the predictions for the chirality (6,6) that is close to the center of the diameter distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Overall, a very good agreement is achieved between the DFT calculations and the experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The cut-off energy E⋆ allows a quantitative assess- ment of the theoretical predictions compared to the ex- perimental reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='3(a) displays E⋆ as a function of m-SWCNT diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Experimental results are repre- sented as a black shaded area, with a confidence inter- val of ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='03 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' DFT calculations for various chiralities inside the experimental diameter distribution are repre- sented by solid circles, while the value predicted by the DFT Analytics Measurement ■ ■ ■ ■ ■ ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='4 CNT diameter (nm) E★ (eV) DFT: (m,m) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=': (m,m) DFT: (m,0) DFT: 2m+n=3q A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=': (m,0) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=': 2m+n=3q a) b) ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='6 0 1 2 3 4 5 CNT diameter (nm) ℏωp (eV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' (a) Cut-off energy as a function of the m-SWCNT diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Filled circles are predictions from the DFT calcu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' They are delimited by the blue-shaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Filled squares are predictions from the analytical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' They are delimited by the red-shaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Armchair (m,m), zigzag (m,0) and chiral (2m+n=3q) species are specified by the color blue, red or black, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' (b) Plasma frequency as a function of the m-SWCNT diameters predicted from the DFT calculations (filled circles) and the analytical model (filled squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Results from DFT are fitted with a linear curve and results from the analytical model follow a hyperbola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' analytical model are represented by solid squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Ex- perimental values are clearly within the interval of DFT calculations and clearly outside the values predicted by the analytical model, its lowest value being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='17 eV too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' To understand more deeply the discrepancies between those two models, the Drude model can be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Based on eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' (2), the cut-off energy E⋆ reads (see sup- plementary material S7): E⋆ = ℏ � ω2p ε∥ b + 2ε⊥ b − γ2 (4) with the plasma frequency ωp, the loss rate γ and the background dielectric constant ε∥ b and ε⊥ b , respectively parallel and perpendicular to the nanotube axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Except for the loss rate that is considered to be the same in the two models, all other quantities differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The quan- tity ε∥ b differs marginally in the two theories by roughly 10-20 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' This difference can not explain the large dis- crepancy between the experimental value and the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' As a consequence, we focus the analysis on the two other quantities ωp and ε⊥ b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The plasma energy ℏωp is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='3(b) for a wide distribution of nanotube diameters, wider than ex- perimentally available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' DFT calculations are represented by solid circles while the analytical model values are dis- played as solid squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' They strikingly differ for small diameter nanotubes and seem to converge toward the largest diameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The nanotube diameter dependence differs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' For the analytical model, ℏωp varies as the in- verse of the nanotube diameter[18, 27], while it seems 5 rather to decrease linearly for the DFT results .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' This is independent of the nanotube type: armchair, zigzag and chiral SWCNT align on the same line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' This is a new result leading to qualitative and quantitative differences between the two theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The second parameter that differs in the two model’s predictions is ε⊥ b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The analyt- ical model assumes ε⊥ b = 1 whereas DFT predicts that m-SWCNTs are also polarizable objects in the transverse directions despite their small diameter compared to the light wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' For energy smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='5 eV, the DFT calculations predict a value ranging between 2 and 3, significantly different from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Last but not least, DFT calculations take into account the curvature-induced low-energy gap existing in non- armchair SWCNT while the analytical model does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' This effect is not captured by the Drude model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' As ob- served in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='3, it leads to higher E⋆ for zigzag and chi- ral compared to armchair nanotubes, independently of the diameter effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Consequently, it could not explain the discrepancy between the measurement and the pre- dictions of the analytical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' We reported the measurement of the complex dielectric constant of metallic carbon nanotubes films over a wide range of energies ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='05 to 4 eV) and the comparison with predictions from two theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' This allowed us to investigate intraband transitions in m-SWCNT and in particular to extract the fundamental quantities that are the cut-off frequency E⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='03 eV, below which a m-SWCNT film behaves like a metal and plasma fre- quencies for a large range of chiralities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' We highlight a re- markable agreement with DFT calculations and exclude prediction based on the analytical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' A similar con- clusion was reached for graphene[29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The assumptions performed in the analytical model[18] are too crude to describe correctly the optical properties of m-SWCNT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Our methodology strengthens the need for ab initio cal- culations to predict accurately the electronic and optical properties of SWCNT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Based on this enhanced knowl- edge of m-SWCNT optical properties, we envision the use of metallic carbon nanotubes as fundamental build- ing blocks for nanometric scale plasmonic devices in the near infrared range, in particular telecom relevant optical windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' ER and NI contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' They initiated and lead this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The samples were sorted by DB under the supervision of NI with insights from SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The reflectance measurements were performed by DB under the supervision of NI and JLB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The Kramers- Kronig analysis were performed by DB and ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' The DFT calculations were performed by PH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' ER and NI wrote a draft of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' All the authors discussed the results, agreed with the conclusions, and contributed to the final version of the letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Authors thanks M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Ramonda from CTM Montpellier for his help with AFM measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Reflectance measurements were performed on the technological platform IRRAMAN of the University of Montpellier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' ∗ emmanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='rousseau@umontpellier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='fr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' nico- las.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='izard@umontpellier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content='fr [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Butt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Khonina and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Kazanskiy, Fiber and Inte- grated Optics 40, 14 (2021) [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} +page_content=' Li, S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQf3i1y/content/2301.11662v1.pdf'} diff --git a/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf b/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..93c876d1f133f760cb6f9a5a0a4a8e7a06c6f7bc --- /dev/null +++ b/X9AyT4oBgHgl3EQfvflc/content/2301.00631v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:794bbf518752eee3f99b6c8db07c4f72aa3b86f07e55076c8b06ccead4aec77c +size 1053321 diff --git a/Y9E2T4oBgHgl3EQfvQga/content/tmp_files/2301.04088v1.pdf.txt b/Y9E2T4oBgHgl3EQfvQga/content/tmp_files/2301.04088v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..719567d01ab70768c51a7ad75d2d23fb9f267544 --- /dev/null +++ b/Y9E2T4oBgHgl3EQfvQga/content/tmp_files/2301.04088v1.pdf.txt @@ -0,0 +1,3316 @@ +1 +Community Detection with Known, Unknown, or +Partially Known Auxiliary Latent Variables +Mohammad Esmaeili and Aria Nosratinia, Fellow, IEEE +Abstract—Empirical observations suggest that in practice, +community membership does not completely explain the depen- +dency between the edges of an observation graph. The residual +dependence of the graph edges are modeled in this paper, to +first order, by auxiliary node latent variables that affect the +statistics of the graph edges but carry no information about the +communities of interest. We then study community detection in +graphs obeying the stochastic block model and censored block +model with auxiliary latent variables. We analyze the conditions +for exact recovery when these auxiliary latent variables are +unknown, representing unknown nuisance parameters or model +mismatch. We also analyze exact recovery when these secondary +latent variables have been either fully or partially revealed. +Finally, we propose a semidefinite programming algorithm for +recovering the desired labels when the secondary labels are either +known or unknown. We show that exact recovery is possible +by semidefinite programming down to the respective maximum +likelihood exact recovery threshold. +Index +Terms—Community +Detection, +Latent +Variables, +Stochastic Block Model (SBM), Censored Block Model (CBM), +Graph Inference, Exact Recovery, Semidefinite Programming +(SDP), Chernoff-Hellinger Divergence. +I. INTRODUCTION +Community detection refers to a clustering of the nodes +of a graph based on the observation of the edges. In many +applications, this involves identifying groups of nodes that are +more densely connected within the group than to nodes outside +the group. Community detection has many applications such as +finding like-minded people in social networks [1], exploration +of biomedical networks [2], improving link predictors and rec- +ommendation systems [3]–[5], and is also relevant to network +reconstruction problems [6]–[9]. Community detection has +been widely investigated in the literature from both theoretical +and algorithmic perspectives. Community detection is based +on graph models such as the stochastic block model and the +censored block model [10]–[16]. Several metrics are used +in this field to characterize the asymptotic behavior of the +residual errors as the size of the graph grows, including +correlated recovery, weak recovery, almost exact recovery, +and exact recovery [17]–[26]. Among the various detection +techniques one can name spectral methods, belief propagation, +and semidefinite programming [27]–[32]. +In the graph models that have so far been studied for +community detection, the graph edges are generated indepen- +dently conditioned on the community labels. A brief survey of +models that are most closely related to the present work will +M. Esmaeili and A. Nosratinia are with the Department of Electrical +and Computer Engineering, The University of Texas at Dallas, Richardson, +TX 75083-0688, USA, Email: esmaeili@utdallas.edu, aria@utdallas.edu. This +work was supported in part by the NSF grant CIF-2008684. +be presented shortly. However, in many practical community +detection problems, the community labels do not fully explain +the dependence between the graph edges. In other words, in +many graphs encountered in practice, the graph edges condi- +tioned on the desired community labels are not statistically +independent. This happens when the structure of the graph is +also influenced by factors other than the community of interest. +For example, one may consider political affiliation communi- +ties on a social network in a university campus, where the +social network graph is also influenced by other variables +that may be unrelated to the community label of interest, +such as membership in intramural and extramural activities. +The nature and magnitude of the dependence of the graph +on these secondary or auxiliary factors can have an effect on +the performance of the community detection algorithm for the +community label of interest. The present study models and +analyzes community detection in this scenario. +Toward that goal, this paper introduces secondary or auxil- +iary latent variables in the graph model that are not subject to +community detection themselves, but influence the structure of +the graph. More specifically, we propose and employ a more +general version of the stochastic block model and censored +block model in which edges are independent conditioned on +both the community labels and a set of secondary latent +variables. The secondary or auxiliary latent variables represent +a first-order model for the residual dependence of the edges +of the graph once the effect of the community labels has been +removed. Auxiliary variables are independent of community +memberships and may or may not be observable. The aux- +iliary latent variable model is distinct from side-information +model [33], [34] where the side information variables are +directly observed and carry information about the communi- +ties. Side information represents non-graph information about +communities, while auxiliary variables model the graph con- +nectivity patterns that are unrelated to the communities. +We investigate the exact recovery threshold for community +detection in the graphs with secondary latent variables. We +also analyze the effect on the performance of community +detection when this secondary latent variable is fully or +partially known. We also propose and investigate a semidef- +inite programming algorithm for community detection with +secondary latent variables. Our analysis shows that exact +recovery via semidefinite programming is possible down to +the respective maximum likelihood exact recovery threshold, +for both unknown or known secondary latent variables. +In addition to addressing a novel problem, this paper also +provides a novel proof for bounding the summation of the +minimums of Poisson-distributed values from above and below +via Chernoff-Hellinger divergence. Our result (Lemma 1) +arXiv:2301.04088v1 [cs.SI] 8 Jan 2023 + +2 +eliminates certain technical difficulties that existed in earlier +proofs, e.g., does not impose restrictions on the domain of +Poisson distributions. This result is extended (Lemma 2) for +the general censored block model. Also, the analysis of exact +recovery for a graph generated based on two latent variables +involves subtleties in extracting the maximum likelihood esti- +mator and analyzing its semidefinite programming relaxation, +which go beyond earlier works. +To put the model of this paper in perspective, we review +several community detection graph models whose nodes are +associated, beyond a scalar community detection label, with +some other variables too. The latent space model [35]–[37] +associates with each node a vector, often with small dimension, +containing variables that are latent in the model. The graph +edges are generated from a distribution that is parameterized +based on the distance between the latent vectors of pairs of +nodes, and the community is a scalar generated as a func- +tion of each latent vector. The overlapping stochastic block +model [11], [38] recovers multiple independent, identically +distributed, binary communities via observing a graph whose +edges are drawn independently conditioned on all the commu- +nity labels of the terminating nodes. An important distinction +of overlapped communities from the present work is that +all communities must be recovered in the overlapped model, +therefore the overlapped model has significant similarity with a +multi-community model. In the overlapped model, the multiple +communities posses a structure that can be exploited, com- +pared with a general multi-community model. Finally, there +exists some work on combining non-graph observation with +graph observations [33], [34]; these works have a superficial +resemblance to the subsection in this paper where the sec- +ondary latent variable is revealed. However, the graph and +the side information in [33], [34] are assumed independent +of each other conditioned on community labels, therefore the +revealed side information in [33], [34] has no direct influence +on the graph. Thus, [33], [34] model a different phenomenon +and also have a different mathematical structure, compared +with the present work. In the interest of brevity, our coverage +of various community detection models is limited, and the +interested reader is referred to more comprehensive coverage +available, e.g., in [11]. +Notation: I is the identity matrix and J the all-one matrix. +S ⪰ 0 indicates a positive semidefinite matrix and S ≥ 0 +denotes a matrix with non-negative entries. ||S|| is the spectral +norm and λ2(S) is the second smallest eigenvalue (for a sym- +metric matrix). [a, b] is a vector that is obtained by stacking +vectors a and b. ⟨·, ·⟩ is the inner product and ∗ is the element- +wise product. We abbreviate [n] ≜ {1, · · · , n}. P(·) indicates +the probability operator and P(·) a probability distribution +which is identified by the choice of its variables whenever +there is no confusion. Random variables with Bernoulli and +Binomial distributions are indicated by Bern(p) and Bin(n, p), +respectively, with n trails and success probability p. Also, +random variables with Poisson distribution are indicated by +Pλ(n) with n trails and parameter λ. +II. SYSTEM MODEL +We start by considering a two-latent variable model, and +assume the cardinality of both is finite. For notational conve- +nience throughout the paper, x, y are length-n vectors holding +latent variable values for the whole graph, while the latent +variables for any node v are represented with xν, yν. In our +model, we aim to discover x, therefore nodes that share the +same value for x are called a community. By micro-community, +we refer to the set of nodes in the graph that share the same +value for both latent variables x, y. The matrix P denotes prior +probabilities +Pi,j = P(xv = i, yv = j). +For convenience and for avoiding tensor calculations, we +further define: +p ≜ vec(P). +For both the two-latent variable stochastic block model and +two-latent variable censored block model, the graph edges are +Bernoulli distributed, conditioned on the latent variables of +the two nodes terminating the edge. The conditional Bernoulli +parameters for an arbitrary edge are organized in a symmetric +matrix ¯Q, whose rows and columns are ordered in a manner +compatible with vector p. In other words, assuming the latent +variable xv has mx outcomes, then the probability of an edge +between two nodes with latent variable pairs taking values +(i, j) and (i′, j′) is given by the element of ¯Q in row jmx +i +and column j′mx + i′. +We are interested in a regime where edge probabilities +diminish with the size of the graph n, in particular, in the +context of our model there exist a constant matrix Q such +that: +¯Q = log n +n +Q. +This assumption asymptotically guarantees a fully connected +graph. +Example 1. Consider a two-latent variable stochastic block +model with mx = 2 and my = 3. Then +P = +�P0,0 +P0,1 +P0,2 +P1,0 +P1,1 +P1,2 +� +, +p = +�P0,0 +P0,1 +P0,2 +P1,0 +P1,1 +P1,2 +� +, +¯Q = log n +n +� +������� +Q0,0 +Q0,1 +Q0,2 +Q0,3 +Q0,4 +Q0,5 +Q1,0 +Q1,1 +Q1,2 +Q1,3 +Q1,4 +Q1,5 +Q2,0 +Q2,1 +Q2,2 +Q2,3 +Q2,4 +Q2,5 +Q3,0 +Q3,1 +Q3,2 +Q3,3 +Q3,4 +Q3,5 +Q4,0 +Q4,1 +Q4,2 +Q4,3 +Q4,4 +Q4,5 +Q5,0 +Q5,1 +Q5,2 +Q5,3 +Q5,4 +Q5,5 +� +������� +. +In addition, we define the columns of weighted versions of +the matrix Q as +q(i,j) ≜ diag(p)Q ejmx+i , +where ek is the k-th canonical coordinate vector, and for +convenience our notation of q(i,j) emphasizes dependence on + +3 +the latent variable outcomes rather than matrix coordinates. +Thus, q(i,j) is the column of diag(p)Q. This vector represents +the relative frequency of edges connecting a node from the +micro-community (i, j) to all nodes of each micro-community +(including the same micro-community). Also, we define the +vector ˜q(i,j) of size mx with entries +˜q(i,j) +i′ +≜ +� +j′ +Pi′,j′Qj′mx+i′,jmx+i , +representing the relative frequency of edges, connecting a +node from the micro-community (i, j) to all nodes of micro- +communities with similar community latent variable. +For the two-latent variable censored block model, if an edge +exists between a pair of nodes, the sign of the edge (positive +or negative) is determined by a random variable drawn from a +Bernoulli distribution with a certain parameter. The Bernoulli +parameters for the positive sign of an edge are organized in a +symmetric matrix Ξ, whose rows and columns are also ordered +in a manner compatible with vector p. Finally, for the censored +block model, we define similarly +g(i,j) ≜ diag(p)(Ξ ∗ Q) ejmx+i , +h(i,j) ≜ diag(p)((1 − Ξ) ∗ Q) ejmx+i , +and +˜g(i,j) +i′ +≜ +� +j′ +Pi′,j′(Ξ ∗ Q)j′mx+i′,jmx+i , +˜h(i,j) +i′ +≜ +� +j′ +Pi′,j′((1 − Ξ) ∗ Q)j′mx+i′,jmx+i . +Remark 1. The censored block model in [28], [39] with +parameters a and ξ is a special case of the general censored +model represented in this paper with +Q = +�a +a +a +a +� +, +Ξ = +�1 − ξ +ξ +ξ +1 − ξ +� +. +III. EXACT RECOVERY UNDER OPTIMAL DETECTION +The main results of this part are represented in the context +of three scenarios, where the latent variable x is unknown +and the latent variable y is either known or unknown (for all +nodes in the graph) or partially known (for some nodes in +the graph). Figure 1 shows graph realizations of a two-latent +variable stochastic block model with mx = 2 and my = 2. In +each node, the community latent variable is indicated by the +color of the inner circle, and the auxiliary latent variable is +represented by the color of a ring around the inner circle. +The Chernoff-Hellinger divergence is due to Abbe [24] +and is defined for two non-negative vectors a, b of the same +dimension: +Div(a, b) ≜ max +t∈[0,1] +� +i +� +tai + (1 − t)bi − at +ib1−t +i +� +. +(1) +This is a generalization of the Hellinger divergence and the +Chernoff divergence [11], [24]. In a manner similar to [11] we +present a lemma that bounds a summation of the minimums +of Poisson-distributed values. +Lemma 1. Let a, b ∈ Rm ++, with a ̸= b, and two positive +scalars p, ˆp. For any Poisson multivariate distributions Pa(d) +and Pb(d), define +I(a, b) ≜ +� +d∈Zm ++ +min{Pa(d)p, Pb(d)ˆp}. +Then +I(a, b) ≤ max{p, ˆp}e−Div(a,b), +I(a, b) ≥ min{p, ˆp}e−Div(a,b) +m +� +i=1 +1 +e +� +at∗ +i b1−t∗ +i +�− 1 +2 , +where t∗ is the optimal parameter in the definition of Chernoff- +Hellinger divergence Div(a, b). +Proof. See Appendix A. +Let D be a random variable vector representing the number +of edges that connect the node v to each micro-community. +More specifically, D(i′,j′) is an element of the D indicating +the number of edges connecting the node v to the micro- +community (i′, j′). For each node v, the proposed detection +tests hypotheses +Hi : xv = i. +If v belongs to micro-community (i, j), then +D(i′,j′) ∼ Bin(nPi′,j′, ¯Qj′mx+i′,jmx+i). +In the regime where ¯Q = Q log n +n , the Binomial distribution can +be approximated by a Poisson distribution with the same mean, +denoted λ(i′,j′) +i,j +. Indeed, using Le Cam’s inequality, the total +variation distance between Bin(nPi′,j′, log n +n Qj′mx+i′,jmx+i) +and P(Pi′,j′Qj′mx+i′,jmx+i log n) asymptotically goes to +zero. Then +P(D = d|Hi, yv = j) = +� +i′ +� +j′ +Pλ(i′,j′) +i,j +(d(i′,j′)), +where λ(i′,j′) +i,j += Pi′,j′Qj′mx+i′,jmx+i log n. +Theorem 1. Under the two-latent variable stochastic block +model, all micro-communities are exactly recovered if and only +if +min +(i,j)̸=(k,l) Div(q(i,j), q(k,l)) > 1. +Proof. It follows from the exact recovery under the general +stochastic bock model or the general overlapping stochastic +block model. +Theorem 2. Under the two-latent variable stochastic block +model, when the latent variable y is revealed, exact recovery +of x is possible if and only if +γ1 ≜ min +j +min +i̸=k Div(q(i,j), q(k,j)) > 1. +Proof. See Appendix B. +Theorem 3. Under the two-latent variable stochastic block +model, when both latent variables are unknown, exact recovery +of x is possible if and only if +γ2 ≜ min +j +min +i̸=k Div +� +˜q(i,j), ˜q(k,j)� +> 1. + +4 +(a) +(b) +(c) +(d) +Fig. 1: For each node, (a) both latent variables are unknown, (b) knowing the statistics of the graph, the community latent +variable is recovered while the auxiliary latent variable is unknown, (c) the auxiliary latent variable is known while the first +one is unknown, (d) knowing the statistics of the graph, the community latent variable is recovered while the auxiliary latent +variable is known. +Proof. See Appendix C. +Now we present the following Lemma which is similar to +Lemma 1 and is crucial for the analysis of the censored block +model. +Lemma 2. Let a, b, ˆa,ˆb ∈ Rm ++, with a ̸= b or ˆa ̸= ˆb, and two +positive scalars p, ˆp. For any Poisson multivariate distributions +Pa(d), Pb(d), Pˆa(w), and Pˆb(w), define +I(a, b, ˆa,ˆb) ≜ +� +d,w∈Zm ++ +min{Pa(d)Pˆa(w)p, Pb(d)Pˆb(w)ˆp}. +Then +I(a, b, ˆa,ˆb) ≤ max{p, ˆp}e−Div([a,ˆa],[b,ˆb]), +I(a, b, ˆa,ˆb) ≥ min{p, ˆp}e−Div([a,ˆa],[b,ˆb]) +× +� +i +1 +e2 +� +(aiˆai)t∗(biˆbi)1−t∗�− 1 +2 , +where t∗ is the optimal parameter in the definition of Chernoff- +Hellinger divergence Div([a, ˆa], [b,ˆb]). +Proof. See Appendix D. +Let D and W be random vectors representing the pos- +itive and negative edges that connect the node v to each +micro-community, respectively. More specifically, D(i′,j′) and +W (i′,j′) are elements of D and W indicating the number +of positive and negative edges connecting the node v to the +micro-community (i′, j′), respectively. For each node v, the +proposed detection tests hypotheses +Hi : xv = i. +If v belongs to micro-community (i, j), then +D(i′,j′) ∼ Bin(nPi′,j′, (Ξ ∗ ¯Q)j′mx+i′,jmx+i), +W (i′,j′) ∼ Bin(nPi′,j′, ((1 − Ξ) ∗ ¯Q)j′mx+i′,jmx+i). +In the regime where ¯Q = Q log n +n , the Binomial distribution +can be approximated by a Poisson distribution with the same +mean. The distributions of D and W can be approximated +by multivariate Poisson distributions Pλi,j and Pˆλi,j with the +vector means λi,j and ˆλi,j, respectively. Therefore +P(D = d, W = w|Hi, yv = j) += P(D = d|Hi, yv = j)P(W = w|Hi, yv = j) += +� +i′ +� +j′ +Pλ(i′,j′) +i,j +(d(i′,j′))Pˆλ(i′,j′) +i,j +(w(i′,j′)), +where +λ(i′,j′) +i,j += Pi′,j′(Ξ ∗ Q)j′mx+i′,jmx+i log n, +ˆλ(i′,j′) +i,j += Pi′,j′((1 − Ξ) ∗ Q)j′mx+i′,jmx+i log n. +Theorem 4. Under two-latent variable censored block model, +all micro-communities are exactly recovered if and only if +min +(i,j)̸=(k,l) Div +� +[g(i,j), h(i,j)], [g(k,l), h(k,l)] +� +> 1. +Proof. See Appendix E. +Theorem 5. Under the two-latent variable censored block +model, when the latent variable y is revealed, exact recovery +of x is possible if and only if +γ3 ≜ min +j +min +i̸=k Div +� +[g(i,j), h(i,j)], [g(k,j), h(k,j)] +� +> 1. +Proof. See Appendix F. +Theorem 6. Under the two-latent variable censored block +model, when both latent variables are unknown, exact recovery +of x is possible if and only if +γ4 ≜ min +j +min +i̸=k Div +� +[˜g(i,j), ˜h(i,j)], [˜g(k,j), ˜h(k,j)] +� +> 1. +Proof. See Appendix G. +Corollary 1. Assume x and y are unknown latent variables +for all nodes. We randomly reveal the latent variable y for +(1−ϵ)n nodes, where ϵ ∈ (0, 1). This is equivalent to erasing +the latent variable y which is a known latent variable from a +node with erasure probability ϵ. Define +β1 ≜ − lim +n→∞ +log(1 − ϵ) +log n +, +β2 ≜ − lim +n→∞ +log ϵ +log n. +• Under the two-latent variable stochastic block model ex- +act recovery is asymptotically possible for latent variable +x if and only if +min +� +γ1 + β1, γ2 + β2 +� +> 1. + +5 +• Under the two-latent variable censored block model exact +recovery is asymptotically possible for latent variable x +if and only if +min +� +γ3 + β1, γ4 + β2 +� +> 1. +The results of this part generalize to M latent variables +without difficulty. +Remark 2. To prove the “if” part of all theorems in Sec- +tion III, a partial recovery algorithm is required before apply- +ing a MAP estimator. For that purpose, the partial recovery +algorithm in [11] is adopted and modified to match the +scenarios in this paper. Please see Appendix I. +IV. SEMIDEFINITE PROGRAMMING RESULTS +This section describes a semidefinite programming algo- +rithm for recovering the desired latent variable. The main +results of this part are represented in the context of two +scenarios, where the latent variable x is unknown and the latent +variable y is either known or unknown (for all nodes in the +graph). We consider x, y ∈ {±1}n such that xT 1 = 0. Thus, +the latent variable x represents two equal-sized communities. +The sample size of the latent variable y, represented by +ρ ≜ 1 +n|{v ∈ [n] : yv = 1}|, is an unknown quantity.1 +A. Two-latent variable stochastic block model +We highlight the specifics of a two-latent variable stochastic +block model for the purposes of upcoming calculations. The +probability of an edge drawn between two nodes v, u is +characterized by four constants, q0, q1, q2, q3 such that: +Aij ∼ +� +� +� +� +� +� +� +� +� +Bern(q0 +log n +n ) +if +xv = xu, yv = yu +Bern(q1 +log n +n ) +if +xv ̸= xu, yv = yu +Bern(q2 +log n +n ) +if +xv = xu, yv ̸= yu +Bern(q3 +log n +n ) +if +xv ̸= xu, yv ̸= yu +. +The corresponding matrix Q, as defined earlier, in this case +will be: +Q = +� +��� +q0 +q1 +q2 +q3 +q1 +q0 +q3 +q2 +q2 +q3 +q0 +q1 +q3 +q2 +q1 +q0 +� +��� . +(2) +1) Recovering x when y is known: In the first scenario, +given an observation of the graph A and y which corresponds +to the observed graph, the latent variable xv is recovered +exactly for each node v ∈ [n]. In this part, y is considered +as an observation which helps the estimator to recover the +desired latent variable x. Let W ≜ yyT and B ≜ W ∗ A. +Since x is chosen uniformly over {x ∈ {±1}n : xT 1 = 0}, +the maximum likelihood estimator gives the optimal solution. +For this configuration, the log-likelihood is +log P(A|x, y) = T1 +8 xT Bx + T2 +8 xT Ax + c, +1Note that semidefinite programming results in this section are obtained for +binary equal-sized communities, while the results of Section III were more +general. +where T1 ≜ log( q0q3 +q2q1 ) and T2 ≜ log( q0q2 +q1q3 ), as n → ∞ and +c is a constant. Considering the constraints, the maximum +likelihood estimator is, +ˆx =arg max +x +T1xT Bx + T2xT Ax +subject to +xi ∈ {±1}, +i ∈ [n] +xT 1 = 0, +(3) +which is a non-convex optimization problem. Let Z = xxT . +Reorganizing (3), +ˆZ =arg max +Z +⟨Z, T1B + T2A⟩ +subject to +Z = xxT +Zii = 1, +i ∈ [n] +⟨Z, J⟩ = 0. +(4) +By relaxing the rank-one constraint on Z, we obtain the +following semidefinite programming relaxation of (4): +ˆZ =arg max +Z +⟨Z, T1B + T2A⟩ +subject to +Z ⪰ 0 +Zii = 1, +i ∈ [n] +⟨Z, J⟩ = 0. +(5) +For convenience define +η1(q, ρ) ≜ ρ +2(√q0 − √q1)2 + 1 − ρ +2 +(√q2 − √q3)2, +where q ≜ [q0, q1, q2, q3]. +Theorem 7. Under the two-latent variable stochastic block +model with binary alphabet where the latent variable y has +been revealed, if +� +η1(q, ρ) > 1 +when +ρ ≤ 0.5 +η1(q, 1 − ρ) > 1 +when +ρ > 0.5 +then the semidefinite programming estimator is asymptotically +optimal, i.e., P( ˆZ = Z∗) ≥ 1 − o(1). Also, if +� +η1(q, ρ) < 1 +when +ρ ≤ 0.5 +η1(q, 1 − ρ) < 1 +when +ρ > 0.5 +then for any sequence of estimators ˆZn, P( ˆZn = Z∗) → 0. +Proof. See Appendix H. +2) Recovering x when y is unknown: Given an observation +of the graph A, the aim is to exactly recover x while both latent +variables x and y are unknown latent variables. It is assumed +that the estimator does not know anything about the auxiliary +latent variable y, which its prior distribution is uniform over +{y : y ∈ {±1}n}. Notice that x is drawn uniformly from +{x ∈ {±1}n : xT 1 = 0}. The log-likelihood of A given x +and y is +log P(A|x, y) =T1 +8 yT (A ∗ xxT )y + T2 +8 xT Ax + T3 +8 yT Ay + c, + +6 +where T1 ≜ log( q0q3 +q2q1 ), T2 ≜ log( q0q2 +q1q3 ), and T3 ≜ log( q0q1 +q2q3 ), +as n → ∞ and c is a constant. Then +logP(A|x) ∝ log +� +Y +P(A|x, y) +∝ log +� +Y +e +T1 +T3 yT (A∗xxT )y+ T2 +T3 xT Ax+yT Ay +=T1 + T2 +T3 +xT Ax + +� +i +� +j +Aij ++ log +� +Y +e +T1 +T3 yT (A∗xxT )y+yT Ay− T1 +T3 xT Ax−� +i +� +j Aij. +Applying the log-sum-exp approximation, the maximum like- +lihood estimator is +ˆx =arg max +x +xT Ax +subject to +xi ∈ {±1}, +i ∈ [n] +xT 1 = 0, +(6) +that is a non-convex optimization problem. Let Z = xxT . +Reorganizing (6) yields +ˆZ =arg max +Z +⟨Z, A⟩ +subject to +Z = xxT +Zii = 1, +i ∈ [n] +⟨Z, J⟩ = 0. +(7) +Relaxing the rank-one constraint on Z, we obtain the following +semidefinite programming relaxation of (7): +ˆZ =arg max +Z +⟨Z, A⟩ +subject to +Z ⪰ 0 +Zii = 1, +i ∈ [n] +⟨Z, J⟩ = 0. +(8) +For convenience define +η2(q, ρ) ≜ 1 +2 +�� +q0ρ + q2(1 − ρ) − +� +q1ρ + q3(1 − ρ) +�2 +. +Theorem 8. Under the two-latent variable stochastic block +model with binary alphabet, if +min {η2(q, ρ), η2(q, 1 − ρ)} > 1, +then the semidefinite programming estimator is asymptotically +optimal, i.e., P( ˆZ = Z∗) ≥ 1 − o(1). Also, if +min {η2(q, ρ), η2(q, 1 − ρ)} < 1, +then for any sequence of estimators ˆZn, P( ˆZn = Z∗) → 0. +Proof. See Appendix J. +Remark 3. The results of Theorems 7 and 8 are consistent +with Theorems 2 and 3, respectively. +Remark 4. The constraint xT 1 = 0 that has been considered +for this part results in a well-defined phase transition threshold +for exact recovery of latent variable x. In general, x may be a +random variable which is drawn uniformly from {x ∈ {±1}n : +xT 1 = (2ρx − 1)n}, where ρx ≜ +1 +n|{v ∈ [n] : xv = 1}|. +Then xT 1 = 0 is substituted by xT 1 = (2ρx − 1)n in +semidefinite programming relaxations (5) and (8). Also, due to +the robustness of semidefinite programming, an approximation +of ρx can be replaced for recovering the latent variable +x. Investigating the constraint xT 1 = (2ρx − 1)n and the +robustness of semidefinite programming are beyond the scope +of this paper. +B. Two-latent variable censored block model +We highlight the specifics of a two-latent variable censored +block model for the purposes of upcoming calculations. Let +P(k; q0, ξ) be a discrete probability density function with +parameters q0 > 0 and ξ ∈ [0, 1] as, +P(k; q0, ξ) ≜ξq0 +log n +n +δ[k − 1] + (1 − ξ)q0 +log n +n +δ[k + 1] ++ +� +1 − q0 +log n +n +� +δ[k], +where δ is Dirac delta function. The probability of an edge +drawn between two nodes v, u is characterized by constants +q0, q1, q2, q3 and ξ such that: +Aij ∼ +� +� +� +� +� +� +� +� +� +P(k; q0, 1 − ξ) +if +xv = xu, yv = yu +P(k; q1, ξ) +if +xv ̸= xu, yv = yu +P(k; q2, ξ) +if +xv = xu, yv ̸= yu +P(k; q3, ξ) +if +xv ̸= xu, yv ̸= yu +. +The corresponding matrix Q, as defined earlier, is the same +as (2). Also, in this case, the corresponding matrix Ξ will be: +Ξ = +� +��� +(1 − ξ) +ξ +ξ +ξ +ξ +(1 − ξ) +ξ +ξ +ξ +ξ +(1 − ξ) +ξ +ξ +ξ +ξ +(1 − ξ) +� +��� . +(9) +1) Recovering x when y is known: Given an observation of +the graph A and y which corresponds to the observed graph, +the latent variable xv is recovered exactly for each node v ∈ +[n]. In this part, y is considered as an observation which helps +the estimator to recover the desired latent variable x. Let +R ≜ TA + T(A ∗ W) + T1(A ∗ A ∗ W) + T2(A ∗ A), +where T ≜ log +� 1−ξ +ξ +� +and W ≜ yyT . Since x is chosen +uniformly over {x ∈ {±1}n : xT 1 = 0}, the maximum +likelihood estimator gives the optimal solution. Similar to +Section IV-A1, it can be shown that the semidefinite pro- +gramming relaxation of maximum likelihood estimator for this +configuration is +ˆZ =arg max +Z +⟨Z, R⟩ +subject to +Z ⪰ 0 +Zii = 1, +i ∈ [n] +⟨Z, J⟩ = 0. +(10) +For convenience define +g ≜ [(1 − ξ)q0, ξq1, ξq2, ξq3], +h ≜ [ξq0, (1 − ξ)q1, (1 − ξ)q2, (1 − ξ)q3]. + +7 +Theorem 9. Under the two-latent variable censored block +model with binary alphabet where the latent variable y has +been revealed, if +� +η1(g, ρ) + η1(h, ρ) > 1 +when +ρ ≤ 0.5 +η1(g, 1 − ρ) + η1(h, 1 − ρ) > 1 +when +ρ > 0.5 +then the semidefinite programming estimator is asymptotically +optimal, i.e., P( ˆZ = Z∗) ≥ 1 − o(1). Also, if +� +η1(g, ρ) + η1(h, ρ) < 1 +when +ρ ≤ 0.5 +η1(g, 1 − ρ) + η1(h, 1 − ρ) < 1 +when +ρ > 0.5 +then for any sequence of estimators ˆZn, P( ˆZn = Z∗) → 0. +Proof. See Appendix K. +2) Recovering x when y is unknown: Given an observation +of the graph A, the aim is to exactly recover x while both latent +variables x and y are unknown. It is assumed that the estimator +does not know anything about the auxiliary latent variable y, +which its prior distribution is uniform over {y : y ∈ {±1}n}. +Notice that x is drawn uniformly from {x ∈ {±1}n : xT 1 = +0}. Similar to Section IV-A2, it can be shown that for this +configuration the semidefinite programming relaxation of the +maximum likelihood estimator is +ˆZ =arg max +Z +⟨Z, TA + T2(A ∗ A)⟩ +subject to +Z ⪰ 0 +Zii = 1, +i ∈ [n] +⟨Z, J⟩ = 0. +(11) +Theorem 10. Under the two-latent variable censored block +model with binary alphabet, if +min +� +η2(g, ρ) + η2(h, ρ), η2(g, 1 − ρ) + η2(h, 1 − ρ) +� +> 1, +then the semidefinite programming estimator is asymptotically +optimal, i.e., P( ˆZ = Z∗) ≥ 1 − o(1). Also, if +min +� +η2(g, ρ) + η2(h, ρ), η2(g, 1 − ρ) + η2(h, 1 − ρ) +� +< 1, +then for any sequence of estimators ˆZn, P( ˆZn = Z∗) → 0. +Proof. See Appendix L. +Remark 5. The results of Theorems 9 and 10 are consistent +with Theorems 5 and 6, respectively. +V. DISCUSSION & NUMERICAL RESULTS +It is illuminating to review the flow of the development of +the achievability results througout this paper: +1) Calculate the Lagrangian of the corresponding optimiza- +tion +2) Extract the dual optimal solution based on the Lagrange +multipliers +3) Show that ˆZ = Z∗ is primal optimal solution +4) Show that ˆZ = Z∗ is unique +5) Extract the conditions under which the dual optimal +solution holds +The converses follow the following sequence: +Fig. 2: Exact recovery region of x in the context of Eq. (2), +with q2 = 3, q1 = q3 = 1. +Fig. 3: Exact recovery region of x in the context of Eq. (2), +with q1 = q2 = q3 = 1. +1) Extract the maximum likelihood estimator +2) Extract the conditions under which the maximum like- +lihood estimator fails +To give a pictorial view of some results of the paper, we +plot some results in the context of the two-latent variable +stochastic block model represented by (2) and two-latent +variable censored block model represented by (2) and (9). For +ease of notation, we define +γ1 ≜ min{η1(q, ρ), η1(q, 1 − ρ)}, +γ2 ≜ min{η2(q, ρ), η2(q, 1 − ρ)}, +γ3 ≜ min{η1(g, ρ) + η1(h, ρ), η1(g, 1 − ρ) + η1(h, 1 − ρ)}, +γ4 ≜ min{η2(g, ρ) + η2(h, ρ), η2(g, 1 − ρ) + η2(h, 1 − ρ)}. +For the two-latent variable stochastic block model, Figures 2 +and 3 show the exact recovery region for recovering the latent + +810 +qo12140.5y191.5 +2 +1Y2'p=0.520=d +p=0 +,p=0.4 +.4 +.52.5=0.3 +.30 +46810 +qo12140.51.5 +2 +Y1'p=0.52p=0 +p=0 +,=0.4 +.4 +.52.5 O +=0.3 +.30 +468 +Fig. 4: Exact recovery region of x in the context of Eq. (2) +and Eq. (9), with ξ = 0.1, q2 = 3, and q1 = q3 = 1. +Fig. 5: Exact recovery region of x in the context of Eq. (2) +and Eq. (9), with ξ = 0.1, and q1 = q2 = q3 = 1. +variable x when the secondary latent variable y is either known +or unknown. The curves in these figures are based on the +obtained results in Theorem 7 and Theorem 8. These figures +encompass several curves plotted for different values of q0, q1, +q2, q3 in (2), and ρ. At each figure, we consider fixed values +for q1, q2, q3 and vary the values of q0 and ρ. A comparison +between the curves in Figures 2 and 3 clarifies the role of +the revealed latent variable y for recovering the desired latent +variable x. +For the two-latent variable censored block model, Figures 4 +and 5 show the exact recovery region for recovering the latent +variable x when the secondary latent variable y is either known +or unknown. The curves in these figures are based on the +obtained results in Theorem 9 and Theorem 10. These figures +consist of several curves plotted for different values of q0, q1, +y +n +BSBM +BCBM +q0 +AEP +q0 +ξ +AEP +Known +100 +7 +3.8 × 10−2 +4 +0.1 +6.7 × 10−2 +Known +200 +7 +2.4 × 10−2 +4 +0.1 +4.9 × 10−2 +Known +300 +7 +1.9 × 10−2 +4 +0.1 +3.6 × 10−2 +Known +400 +7 +1.5 × 10−2 +4 +0.1 +2.5 × 10−2 +Known +500 +7 +1.1 × 10−2 +4 +0.1 +1.6 × 10−2 +Known +100 +9 +8.1 × 10−5 +6 +0.1 +4.6 × 10−5 +Known +200 +9 +5.9 × 10−5 +6 +0.1 +3.2 × 10−5 +Known +300 +9 +4.2 × 10−5 +6 +0.1 +2.4 × 10−5 +Known +400 +9 +2.8 × 10−5 +6 +0.1 +1.7 × 10−5 +Known +500 +9 +1.8 × 10−5 +6 +0.1 +1.2 × 10−5 +Unknown +100 +8 +5.7 × 10−2 +5 +0.1 +5.6 × 10−2 +Unknown +200 +8 +4.1 × 10−2 +5 +0.1 +3.9 × 10−2 +Unknown +300 +8 +2.7 × 10−2 +5 +0.1 +2.5 × 10−2 +Unknown +400 +8 +1.8 × 10−2 +5 +0.1 +1.6 × 10−2 +Unknown +500 +8 +1.3 × 10−2 +5 +0.1 +1.1 × 10−2 +Unknown +100 +10 +6.2 × 10−5 +7 +0.1 +6.3 × 10−5 +Unknown +200 +10 +4.4 × 10−5 +7 +0.1 +4.0 × 10−5 +Unknown +300 +10 +3.3 × 10−5 +7 +0.1 +2.3 × 10−5 +Unknown +400 +10 +2.3 × 10−5 +7 +0.1 +1.7 × 10−5 +Unknown +500 +10 +1.4 × 10−5 +7 +0.1 +1.3 × 10−5 +TABLE I: Semidefinite programming optimization of (8) and +(10), with q2 = 3, q1 = q3 = 1, and ρ = 0.5. +q2, q3 in (2) and ρ, while ξ = 0.1 in (9). At each figure, +we consider fixed values for ξ, q1, q2, q3 and vary the values +of q0 and ρ. A comparison between the curves in Figures 4 +and 5 clarifies the role of the revealed latent variable y for +recovering the desired latent variable x. +To gain an understanding of the scope of our asymptotic +results, under the conditions of Figures 2 and 4, we performed +several simulations on 104 graph realizations with various +graph sizes obtained from the proposed models in Section II. +The obtained average error probability (AEP) is around 10−5 +in the regimes just inside the region of exact recovery, and +around 10−2 in the regimes just outside the region of exact +recovery. The details of these simulations are represented in +Tables I and II. At each simulation, we consider fixed values +for q1, q2, q3 and vary the values of q0, ρ, and n. +VI. CONCLUSION +This paper presents and analyzes a new generalization of the +stochastic and censored block models in which, in addition +to the latent variable representing community labels, there +exists another (secondary) latent variables that are not part of +community detection. These secondary latent variables may be +known, unknown, or partially known. This model represents +community detection problems where the community labels +alone does not explain all the dependencies between the graph +edges. +We investigate the exact recovery threshold for these models +under maximum likelihood detection, and also analyze a +semidefinite programming algorithm for recovering the desired +latent variable under the two-latent variable stochastic block +model and the two-latent variable censored block model for +both scenarios. + +810 +qo12140.54 +1.5 +34'p=0.52.5 +2p=0 +p=0 +,p=0.4 +Exact Re +.4 +.5covery Regicn3=0 +p=0.3 +.30 +46810 +qo12140.54 +1.5 +3—4' p= 0.52.5 +2p=0 += 0 +p=0.4 +Exact Re +.4 +.5covery Regicn3p=0.3 +.30 +469 +y +n +BSBM +BCBM +q0 +AEP +q0 +ξ +AEP +Known +100 +10 +7.6 × 10−2 +7 +0.1 +4.1 × 10−2 +Known +200 +10 +5.1 × 10−2 +7 +0.1 +3.1 × 10−2 +Known +300 +10 +3.0 × 10−2 +7 +0.1 +2.3 × 10−2 +Known +400 +10 +2.1 × 10−2 +7 +0.1 +1.8 × 10−2 +Known +500 +10 +1.3 × 10−2 +7 +0.1 +1.3 × 10−2 +Known +100 +12 +6.7 × 10−5 +9 +0.1 +3.9 × 10−5 +Known +200 +12 +5.1 × 10−5 +9 +0.1 +2.5 × 10−5 +Known +300 +12 +3.6 × 10−5 +9 +0.1 +1.8 × 10−5 +Known +400 +12 +2.5 × 10−5 +9 +0.1 +1.2 × 10−5 +Known +500 +12 +1.6 × 10−5 +9 +0.1 +1.0 × 10−5 +Unknown +100 +11 +4.3 × 10−2 +8 +0.1 +4.2 × 10−2 +Unknown +200 +11 +3.3 × 10−2 +8 +0.1 +2.9 × 10−2 +Unknown +300 +11 +2.4 × 10−2 +8 +0.1 +2.0 × 10−2 +Unknown +400 +11 +1.7 × 10−2 +8 +0.1 +1.3 × 10−2 +Unknown +500 +11 +1.2 × 10−2 +8 +0.1 +1.0 × 10−2 +Unknown +100 +13 +4.2 × 10−5 +10 +0.1 +4.8 × 10−5 +Unknown +200 +13 +2.6 × 10−5 +10 +0.1 +3.3 × 10−5 +Unknown +300 +13 +1.7 × 10−5 +10 +0.1 +2.2 × 10−5 +Unknown +400 +13 +1.3 × 10−5 +10 +0.1 +1.5 × 10−5 +Unknown +500 +13 +1.1 × 10−5 +10 +0.1 +1.0 × 10−5 +TABLE II: Semidefinite programming optimization of (8) and +(10), with q2 = 3, q1 = q3 = 1, and ρ = 0.3. +APPENDIX A +PROOF OF LEMMA 1 +Define +f1(t) ≜ +m +� +i=1 +� bi +ai +�(t−1)di +e(t−1)(ai−bi), +f2(t) ≜ +m +� +i=1 +� bi +ai +�tdi +et(ai−bi). +For any t ∈ [0, 1], +� +d∈Zm ++ +min{Pa(d)p, Pb(d)ˆp} +≤ max{p, ˆp} +� +d∈Zm ++ +min{Pa(d), Pb(d)} += max{p, ˆp} exp +� +− +� +i +� +tai + (1 − t)bi − at +ib1−t +i +�� +× +� +d∈Zm ++ +� +i +(at +ib1−t +i +)di +di! +e−at +ib1−t +i +min{f1(t), f2(t)}. +Both f1(t) and f2(t) are monotonic and +f2(t) +f1(t) is a positive +constant (does not depend on t), thus min{f1, f2} is also +monotonic in t. Since f1(1) = f2(0) = 1, for all t we have: +min{f1(t), f2(t)} ≤ 1. +Notice that +� +d∈Zm ++ +� +i +(at +ib1−t +i +)di +di! +e−at +ib1−t +i += 1. +Then +I(a, b) ≤ max{p, ˆp}e− �m +i=1 +� +tai+(1−t)bi−at +ib1−t +i +� +. +(12) +For the value of t that maximizes the right-hand side of +inequality (12), we have +� +d∈Zm ++ +min{Pa(d)p, Pb(d)ˆp} ≤ max{p, ˆp}e−Div(a,b). +Notice that t∗ satisfies +m +� +i=1 +� bi +ai +�at∗ +i b1−t∗ +i +eai−bi = 1. +Then at the optimal t∗, +� +d∈Zm ++ +min{Pa(d)p, Pb(d)ˆp} +≥ min{p, ˆp} +� +d∈Zm ++ +min{Pa(d), Pb(d)} +(a) +≥ min{p, ˆp}e−Div(a,b) � +i +(at∗ +i b1−t∗ +i +)at∗ +i b1−t∗ +i +at∗ +i b1−t∗ +i +! +e−at∗ +i b1−t∗ +i +(b) +≥ min{p, ˆp}e−Div(a,b) � +i +1 +e +� +at∗ +i b1−t∗ +i +�− 1 +2 , +where (a) holds because +� +d∈Zm ++ +min{Pa(d), Pb(d)} ≥ min{Pa(d∗), Pb(d∗)}, +where d∗ is defined by d∗ +i ≜ at∗ +i b1−t∗ +i +, and (b) is due to +Stirling’s approximation n! ≤ nn+ 1 +2 e−n+1 for any n ≥ 1. +APPENDIX B +PROOF OF THEOREM 2 +We aim to recover xv when yv is known. Given a realization +of D and yv, our goal is to minimize the error probability by +selecting the most likely hypothesis, i.e., +argmax +i +P{Hi|D = d, yv}, +or equivalently, since d, yv are known observations, +argmax +i +P(d|Hi, yv)P{Hi, yv}, +which is the maximum a posteriori (MAP) detector, which we +rewrite: +argmax +i +P(d|Hi, yv)Pi,yv. +(13) +Solving (13) requires mx − 1 pairwise comparisons of the +hypotheses. From this viewpoint, if +P(d|Hi, yv)Pi,yv ≤ P(d|Hk, yv)Pk,yv, +(14) +then a pairwise comparison will choose Hk over Hi. Now +assume the correct hypothesis is Hi, and denote by Bik the +region of D for which (14) is satisfied, i.e., Hi has a worse +metric compared with Hk. Also denote by Bi the region for D +where the overall MAP decoder is in error. The dependence of +error regions Bik and Bi on yv is implicit. Then the probability +of error +Pe = +� +i +P{D ∈ Bi|Hi, yv}Pi,yv. +(15) + +10 +Since Bi ⊂ ∪kBik, +Pe ≤ +� +i +� +k̸=i +P{D ∈ Bik|Hi, yv}Pi,yv. +From the earlier Poisson assumption P(d|Hi, yv) = Pλi,yv (d) +it follows that: +min{Pλi,yv (d)Pi,yv, Pλk,yv (d)Pk,yv} = +� +Pλi,yv (d) Pi,yv +when D ∈ Bik +Pλk,yv (d) Pk,yv +when D ∈ Bc +ik +. +Therefore, substituting into the union bound: +Pe ≤ +� +d +� +i +� +k>i +min{Pλi,yv (d)Pi,yv, Pλk,yv (d)Pk,yv}. +(16) +For bounding the error probability (16), it suffices to find an +upper bound for +� +d +min{Pλi,yv (d)Pi,yv, Pλk,yv (d)Pk,yv}. +(17) +It follows from Lemma 1 that +Pe ≤ +� +i +� +k>i +max{Pi,yv, Pk,yv}e−Div(λi,yv ,λk,yv ) += +� +i +� +k>i +n−Div +� +q(i,yv),q(k,yv)� ++o(1). +(18) +We now bound the error probability of decoding rule (13) +from below. Since +� +k̸=i +P{D ∈ Bik|Hi, yk} ≤ (mx − 1)P{D ∈ Bi|Hi, yv}, (19) +substituting (19) into (15) yields +Pe ≥ +1 +mx − 1 +� +i +� +k̸=i +P{D ∈ Bik|Hi, yv}Pi,yv = +1 +mx − 1 +× +� +i +� +k>i +� +d +min +� +Pλi,yv (d)Pi,yv, Pλk,yv (d)Pk,yv +� +. +Then it suffices to find a lower bound for (17) to bound the +error probability from below. It follows from Lemma 1 that +Pe ≥ +� +i +� +k>i +c′ min{Pi,yv, Pk,yv}(log n)− m +2 e−Div(λi,yv ,λk,yv ) += +� +i +� +k>i +n−Div(q(i,yv),q(k,yv))+o(1), +(20) +where c′ is a constant and m is the number of elements in +vector d, i.e., the product of alphabet sizes of xv and yv. The +lower and upper bounds (18) and (20) imply that the true +hypothesis is recovered correctly if Div(q(i,yv), q(k,yv)) > 1, +for a given yv and any i ̸= k. This means that a known latent +variable restricts the number of pairwise comparisons. Then +under the two-latent variable stochastic block model in which +the latent variable y is known, and the latent variable x is +unknown, exact recovery is possible for x if and only if +min +j +min +i̸=k Div(q(i,j), q(k,j)) > 1. +(21) +APPENDIX C +PROOF OF THEOREM 3 +We aim to recover xv when yv is unknown, given a +realization of D for node v. For this setting the MAP detector +is +argmax +i +P{Hi|D = d}, +or equivalently, +argmax +i +� +yv +� +l +P +� � +j +d(l,j)|Hi, yv +� +Pi,yv. +(22) +Solving (22) requires mx − 1 pairwise comparisons. In these +comparisons, if +� +yv +� +l +P +� � +j +d(l,j)|Hi, yv +� +Pi,yv +< +� +yv +� +l +P +� � +j +d(l,j)|Hk, yv +� +Pk,yv, +(23) +then we conclude hypothesis Hi is ruled out, i.e., xv ̸= i, +because another hypothesis Hk has a better metric. Denote by +Bik the region of D for which Hi has a worse metric compared +with Hk, i.e., the region for D in which (23) is satisfied. Also +denote by Bi the region for D where the overall MAP decoder +is in error. The error probability of MAP decoder (22) is given +by +Pe = +� +i +� +yv +P(D ∈ Bi|Hi, yv}Pi,yv. +(24) +Since Bi ⊂ ∪kBik, via the union bound, +� +yv +P{D ∈ Bi|Hi, yv}Pi,yv +≤ +� +yv +� +k̸=i +P{D ∈ Bik|Hi, yv}Pi,yv. +(25) +Using the Poisson approximation and the additive property of +Poisson distribution: +I(d, i, yv) ≜ +� +l +P +� � +j +d(l,j)|Hi, yv +� += +� +l +P� +j λ(l,j) +i,yv +� � +j +d(l,j) +� +. +Therefore, +min +� +I(d, i, yv)Pi,yv,I(d, k, yv)Pk,yv +� += +� +I(d, i, yv)Pi,yv +when D ∈ Bik +I(d, k, yv)Pk,yv +when D ∈ Bc +ik +. +Substituting (25) into (24) yields +Pe ≤ +� +i +� +k̸=i +� +yv +P{D ∈ Bik|Hi, yv}Pi,yv += +� +i +� +k>i +� +yv +� +d +min +� +I(d, i, yv)Pi,yv, I(d, k, yv)Pk,yv +� +. +(26) + +11 +For bounding the error probability (26) from above, it suffices +to find an upper bound for +� +d∈Zm ++ +min +� +I(d, i, yv)Pi,yv, I(d, k, yv)Pk,yv +� +. +(27) +Applying Lemma 1 yields +Pe ≤ +� +i +� +k>i +� +yv +n−Div +� +˜q(i,yv),˜q(k,yv)� ++o(1). +(28) +We now bound the error probability of decoding rule (22) from +below. Notice that +� +k̸=i +P{D ∈ Bik|Hi, yv} ≤ (mx − 1)P{D ∈ Bi|Hi, yv}. +(29) +Substituting (29) into (24) yields +Pe ≥ +1 +mx − 1 +� +i +� +k̸=i +� +yv +P{D ∈ Bik|Hi, yv}Pi,yv = +1 +mx − 1 +× +� +i +� +k>i +� +yv +� +d +min +� +I(d, i, yv)Pi,yv, I(d, k, yv)Pk,yv +� +. +Then it suffices to find a lower bound for (27). Applying +Lemma 1 yields +Pe ≥ +� +i +� +k>i +� +yv +n−Div +� +˜q(i,yv),˜q(k,yv)� ++o(1). +(30) +The lower and upper bounds (28) and (30) imply that the true +hypothesis is recovered correctly if Div(˜q(i,yv), ˜q(k,yv)) > 1 +for any i ̸= k and any yv. Then under two-latent variable +stochastic block model in which both latent variables x, y are +unknown, exact recovery is solvable for x if and only if +min +j +min +i̸=k Div +� +˜q(i,j), ˜q(k,j)� +> 1. +(31) +APPENDIX D +PROOF OF LEMMA 2 +Define +f1(t) ≜ +�Pa(d)Pˆa(w) +Pb(d)Pˆb(w) +�1−t +, +f2(t) ≜ +� Pb(d)Pˆb(w) +Pa(d)Pˆa(w) +�t +, +f(t) ≜ Pa(d)tPb(d)1−tPˆa(w)tPˆb(w)1−t. +For any t ∈ [0, 1], +� +d,w∈Zm ++ +min{Pa(d)Pˆa(w)p, Pb(d)Pˆb(w)ˆp} +≤ max{p, ˆp} +� +d,w∈Zm ++ +min{Pa(d)Pˆa(w), Pb(d)Pˆb(w)} +≤ max{p, ˆp} exp +� +− +� +i +� +tai + (1 − t)bi − at +ib1−t +i +�� +× exp +� +− +� +i +� +tˆai + (1 − t)ˆbi − ˆat +iˆb1−t +i +�� +, +(32) +where the last inequality holds because min{f1(t), f2(t)} ≤ 1, +and +� +d,w∈Zm ++ +� +i +(at +ib1−t +i +)di +di! +e−at +ib1−t +i +(ˆat +iˆb1−t +i +)wi +wi! +e−ˆat +iˆb1−t +i += 1. +For the value of t that minimizes the upper bound of (32), we +have +I(a, b, ˆa,ˆb) ≤ max{p, ˆp}e−Div([a,ˆa],[b,ˆb]). +Notice that t∗ satisfies +m +� +i=1 +� bi +ai +�at∗ +i b1−t∗ +i +�ˆbi +ˆai +�ˆat∗ +i ˆb1−t∗ +i +eai−bi+ˆai−ˆbi = 1. +Then at the optimal t∗, +� +d,w∈Zm ++ +min{Pa(d)Pˆa(w)p, Pb(d)Pˆb(w)ˆp} +≥ min{p, ˆp} +� +d,w∈Zm ++ +min{Pa(d)Pˆa(w), Pb(d)Pˆb(w)} +(a) +≥ min{p, ˆp}e−Div([a,ˆa],[b,ˆb]) +× +� +i +(at∗ +i b1−t∗ +i +)at∗ +i b1−t∗ +i +at∗ +i b1−t∗ +i +! +e−at∗ +i b1−t∗ +i +× +� +i +(ˆat∗ +i ˆb1−t∗ +i +)ˆat∗ +i ˆb1−t∗ +i +ˆat∗ +i ˆb1−t∗ +i +! +e−ˆat∗ +i ˆb1−t∗ +i +(b) +≥ min{p, ˆp}e−Div([a,ˆa],[b,ˆb]) � +i +1 +e2 +� +(aiˆai)t∗(biˆbi)1−t∗�− 1 +2 , +where (a) holds because +� +d,w∈Zm ++ +min{Pa(d)Pˆa(w), Pb(d)Pˆb(w)} +≥ min{Pa(d∗)Pˆa(w∗), Pb(d∗)Pˆb(w∗)}, +where d∗ is defined by d∗ +i ≜ at∗ +i b1−t∗ +i +and w∗ is defined by +w∗ +i ≜ ˆat∗ +i ˆb1−t∗ +i +, and (b) is due to Stirling’s approximation +n! ≤ nn+ 1 +2 e−n+1 for any n ≥ 1. +APPENDIX E +PROOF OF THEOREM 4 +We aim to recover both xv and yv for node v, given a +realization of D and a realization of W. Our goal is to +minimize the error probability by selecting the most likely +hypothesis, i.e., +argmax +i,j +P{Hi,j|D = d, W = w}, +where +Hi,j : xv = i, yv = j. +The maximum a posteriori (MAP) detector is rewrite as +argmax +i,j +P(d, w|Hi,j)Pi,j. +(33) + +12 +Solving (33) requires mxmy − 1 pairwise comparisons of the +hypotheses. From this viewpoint, if +P(d, w|Hi,j)Pi,j ≤ P(d, w|Hk,l)Pk,l, +then a pairwise comparison will choose Hk,l over Hi,j. Now +assume the correct hypothesis is Hi,j. Similar to the proof of +Theorems 2 and 3, it can be shown that the probability of error +for recovering the true hypothesis is bounded from above and +below by controlling +� +d,w +min{Pλi,j(d)Pˆλi,j(w)Pi,j, Pλk,l(d)Pˆλk,l(w)Pk,l}. +It follows from Lemma 2 that +Pe ≤ +� +i,k>i +� +j,l>j +max{Pi,j, Pk,l}e−Div([λi,j,ˆλi,j],[λk,l,ˆλk,l]) += +� +i,k>i +� +j,l>j +n−Div +� +[q(i,j),g(i,j)],[q(k,l),g(k,l)] +� ++o(1), +(34) +and +Pe ≥ +� +i,k>i +� +j,l>j +c′ min{Pi,j, Pk,l} +(log n)m +e−Div([λi,j,ˆλi,j],[λk,l,ˆλk,l]) += +� +i,k>i +� +j,l>j +n−Div +� +[g(i,j),h(i,j)],[g(k,l),h(k,l)] +� ++o(1), +(35) +where c′ is a constant and m is the number of elements in vec- +tor d, i.e., the product of alphabet sizes of xv and yv. The lower +and upper bounds (34) and (35) imply that the true hypothesis +is recovered correctly if Div +� +[g(i,j), h(i,j)], [g(k,l), h(k,l)] +� +> 1, +for any (i, j) ̸= (k, l). This means that under the two- +latent variable censored block model all micro-communities +are exactly recovered if and only if +min +(i,j)̸=(k,l) Div +� +[g(i,j), h(i,j)], [g(k,l), h(k,l)] +� +> 1. +APPENDIX F +PROOF OF THEOREM 5 +We aim to recover xv when yv is known. Given a realization +of D, a realization of W, and yv, our goal is to minimize the +error probability by selecting the most likely hypothesis, i.e., +argmax +i +P{Hi|D = d, W = w, yv}, +or equivalently, +argmax +i +P(d|Hi, yv)P(w|Hi, yv)Pi,yv, +(36) +which is the MAP detector. Solving (36) requires mx − 1 +pairwise comparisons of the hypotheses. Similar to the proof +of Theorem 2, it can be shown that the error probability of +finding true hypothesis is bounded from above and below by +controlling +� +d,w +min{Pλi,yv (d)Pˆλi,yv (w)Pi,yv, Pλk,yv (d)Pˆλk,yv (w)Pk,yv}. +It follows from Lemma 2 that +Pe ≤ +� +i +� +k>i +max{Pi,yv, Pk,yv}e−Div([λi,yv ,ˆλi,yv ],[λk,yv ,ˆλk,l]) += +� +i +� +k>i +n−Div +� +[g(i,yv),h(i,yv)],[g(k,yv),h(k,yv)] +� ++o(1). (37) +and +Pe ≥ +� +i +� +k>i +c +(log n) +m +2 e−Div([λi,yv ,ˆλi,yv ],[λk,yv ,ˆλk,yv ]) += +� +i +� +k>i +n−Div +� +[g(i,yv),h(i,yv)],[g(k,yv),h(k,yv)] +� ++o(1), (38) +where c ≜ c′ min{Pi,yv, Pk,yv} is a constant and m is +the number of elements in vector d. The lower and upper +bounds (37) and (38) imply that the true hypothesis is recov- +ered correctly if Div +� +[g(i,yv), h(i,yv)], [g(k,yv), h(k,yv)] +� +> 1, +for a given yv and any i ̸= k. This means that a known latent +variable restricts the number of pairwise comparisons. Then +under the two-latent variable censored block model in which +the latent variable y is known, and the latent variable x is +unknown, exact recovery is possible for x if and only if +min +j +min +i̸=k Div +� +[g(i,j), h(i,j)], [g(k,j), h(k,j)] +� +> 1. +APPENDIX G +PROOF OF THEOREM 6 +We aim to recover xv when yv is unknown, given a +realization of D and a realization of W for node v. For this +setting the MAP detector is +argmax +i +P{Hi|D = d, W = w}. +For convenience define +I(d, w, i, yv) ≜ +� +l +P +� � +j +d(l,j), +� +j +w(l,j)|Hi, yv +� +, +where � +j w(l,j) and � +j d(l,j) are independent given Hi and +yv. Then the MAP detector rewrite as +argmax +i +� +yv +I(d, w, i, yv)Pi,yv. +(39) +Solving (39) requires mx − 1 pairwise comparisons. In these +comparisons, if +� +yv +I(d, w, i, yv)Pi,yv < +� +yv +I(d, w, k, yv)Pk,yv, +then we conclude hypothesis Hi is ruled out, i.e., xv ̸= i, +because another hypothesis Hk has a better metric. Notice that +using the Poisson approximation and the additive property of +Poisson distribution, I(d, w, i, yv) can be reorganized as +I(d, w, i, yv) = +� +l +P� +j λ(l,j) +i,yv +� � +j +d(l,j) +� +× +� +l +P� +j λ(l,j) +i,yv +� � +j +w(l,j) +� +. + +13 +Similar to the proof of Theorem 3, it can be shown that the +error probability of recovering the true hypothesis is bounded +from above and below by controlling +� +d,w∈Zm ++ +min +� +I(d, w, i, yv)Pi,yv, I(d, w, k, yv)Pk,yv +� +. +Applying Lemma 2 yields +Pe ≤ +� +i +� +k>i +� +yv +n−Div +� +[˜g(i,yv),˜h(i,yv)],[˜g(k,yv),˜h(k,yv)] +� ++o(1), +(40) +and +Pe ≥ +� +i +� +k>i +� +yv +n−Div +� +[˜g(i,yv),˜h(i,yv)],[˜g(k,yv),˜h(k,yv)] +� ++o(1). +(41) +The +lower +and +upper +bounds +(40) +and +(41) +imply +that +the +true +hypothesis +is +recovered +correctly +if +Div +� +[˜g(i,yv), ˜h(i,yv)], [˜g(k,yv), ˜h(k,yv)] +� +> 1 for any i ̸= k +and any yv. Then under two-latent variable censored block +model in which both latent variables x, y are unknown, exact +recovery is solvable for x if and only if +min +j +min +i̸=k Div +� +[˜g(i,j), ˜h(i,j)], [˜g(k,j), ˜h(k,j)] +� +> 1. +APPENDIX H +PROOF OF THEOREM 7 +We begin by stating sufficient conditions for the optimum +solution of (5) matching the true labels x∗. +Lemma 3. For the optimization problem (5), consider the +Lagrange multipliers +λ∗, +D∗ = diag(d∗ +i ), +S∗. +If we have +S∗ = D∗ + λ∗J − T1B − T2A, +S∗ ⪰ 0, +λ2(S∗) > 0, +S∗x∗ = 0, +then (λ∗, D∗, S∗) is the dual optimal solution and ˆZ = x∗x∗T +is the unique primal optimal solution of (5). +Proof. Let D = diag(di), λ ∈ R, and S ⪰ 0 denote the +Lagrangian of (5). For any Z that satisfies the constraints in +(5), we have +T1⟨B, Z⟩ + T2⟨A, Z⟩ +(a) +≤L(Z, S∗, D∗, λ∗) = ⟨D∗, I⟩ +(b) +=⟨S∗ − λ∗J + T1B + T2A, Z∗⟩ +(c) +=T1⟨B, Z∗⟩ + T2⟨A, Z∗⟩, +where (a) holds because ⟨S∗, Z⟩ ≥ 0, (b) holds because Zii = +1 for all i ∈ [n] and S∗ = D∗+λ∗J−T1B−T2A, and (c) holds +because S∗x∗ = 0 and x∗T 1 = 0. Therefore, Z∗ = x∗x∗T is +an optimal solution of (5). Now, assume ˜Z is another optimal +solution. Then +⟨S∗, ˜Z⟩ =⟨D∗ + λ∗J − T1B − T2A, ˜Z⟩ +(a) +=⟨D∗ + λ∗J − T1B − T2A, Z∗⟩ = ⟨S∗, Z∗⟩ = 0, +where (a) holds because ⟨T1B+T2A, Z∗⟩ = ⟨T1B+T2A, ˜Z⟩, +Z∗ +ii = ˜Zii = 1 for all i ∈ [n], and ⟨J, Z∗⟩ = ⟨J, ˜Z⟩ = 0. Since +˜Z ⪰ 0, and S∗ ⪰ 0 while its second smallest eigenvalue +λ2(S∗) is positive (since S∗ˆx∗ = 0), ˜Z must be a multiple +of Z∗. Also, since ˜Zii = Z∗ +ii = 1 for all i ∈ [n], we have +˜Z = Z∗. +We now show that S∗ = D∗ + λ∗J − T1B − T2A satisfies +other conditions in Lemma 3 with probability 1 − o(1). Let +d∗ +i = T1 +n +� +j=1 +Bijx∗ +jx∗ +i + T2 +n +� +j=1 +Aijx∗ +jx∗ +i . +(42) +Then D∗x∗ = T1Bx∗ + T2Ax∗ and based on the definition +of S∗ in Lemma 3, S∗ satisfies the condition S∗x∗ = 0. It +remains to show that S∗ ⪰ 0 and λ2(S∗) > 0 with probability +1 − o(1). In other words, we need to show that +P +� +inf +v⊥x∗,∥v∥=1vT S∗v > 0 +� +≥ 1 − o(1), +(43) +where v is a n × 1 vector. Then for any v such that vT x∗ = 0 +and ∥v∥ = 1, +vT S∗v =vT D∗v + λ∗vT Jv − T1vT (B − E[B])v +− T2vT (A − E[A])v − T1vT E[B]v − T2vT E[A]v +≥ min +i +d∗ +i + λ∗vT Jv − T1∥B − E[B]∥ +− T2∥A − E[A]∥ − T1vT E[B]v − T2vT E[A]v. +Notice that +T1vT E[B]v + T2vT E[A]v =1 +4[T1c1 + T2c2]vT Wv ++ 1 +4[T1c3 + T2c4]vT (Z ∗ W)v ++ 1 +4[T1c1 + T2c2]vT Jv +− (T1 + T2)q0 +log n +n +, +where +c1 ≜ log n +n +(q0 − q2 + q1 − q3), +c2 ≜ log n +n +(q0 + q2 + q1 + q3), +c3 ≜ log n +n +(q0 − q2 − q1 + q3), +c4 ≜ log n +n +(q0 + q2 − q1 + q3). +Lemma 4. For any c > 0, there exists c′, c′′ > 0 such that +for any n ≥ 1, ∥A − E[A]∥ ≤ c′′√log n and ∥B − E[B]∥ ≤ +c′√log n with probability at least 1 − n−c. +Proof. The proof is similar to the proofs [39, Thoerem 9] and +[32, Thoerem 5]. + +14 +Lemma 5. With probability at least 1 − n− 1 +2 , +vT (Z ∗ W)v ≤ +� +log n, +vT Wv ≤ +� +log n + (2ρ − 1)2vT Jv + 2|2ρ − 1| +� +n log n. +Proof. Since −|vi| ≤ viyi ≤ |vi|, by applying the Chernoff +bound we have +P(vT y − E[vT y] ≥ +� +log n) ≤ n− 1 +2 . +Since E[vT y] = (2ρ − 1)vT 1 and |vT 1| ≤ ∥v∥2∥1∥2 = √n, +with probability converging to one, +(vT y)2 ≤ log n + (2ρ − 1)2vT Jv + 2|vT 1||2ρ − 1| +� +log n +≤ log n + (2ρ − 1)2vT Jv + 2|2ρ − 1| +� +n log n. +Similarly, since E[� +i xiyivi] = 0 and −|vi| ≤ xiyivi ≤ |vi|, +applying the Chernoff bound yields vT (Z ∗ W)v ≤ √log n +with probability converging to one. +Lemma 6. For δ = +log n +log log n, +P +� +min +i∈[n] d∗ +i ≥ δ +� +≥ 1 − n1−η1(q,ρ)+o(1) − n1−η1(q,1−ρ)+o(1). +Proof. The proof is achieved by applying the Chernoff bound +and taking the union bound. +Notice that ρ ≤ 0.5 implies η1(q, ρ) ≤ η1(q, 1−ρ) and ρ > +0.5 implies η1(q, ρ) ≥ η1(q, 1 − ρ). Then mini d∗ +i ≥ +log n +log log n +if +� +η1(q, ρ) > 1 +when +ρ ≤ 0.5 +η1(q, 1 − ρ) > 1 +when +ρ > 0.5 . +(44) +Let λ∗ ≥ +1 +4[T1c1 + T2c2](2ρ − 1)2. Therefore, applying +Lemmas 4, 5, and 6, we get that if (44) holds, then +vT S∗v ≥ +log n +log log n − (T1c′ + T2c′′) +� +log n ++ (T1 + T2)q0 +log n +n +> 0, +and the first part of Theorem 7 follows. +To prove the second part, since x∗ has a uniform distribution +over {x ∈ {±1}n : xT 1 = 0}, maximum likelihood estimator +minimizes the error probability among all estimators. Then we +need to find when the maximum likelihood estimator fails. Let +e(i, H) ≜ � +j∈H Aij(T1yiyj + T2). Define the events +F1 ≜ +� +min +i∈C∗ +1 +(e(i, C∗ +1) − e(i, C∗ +2)) ≤ −2 +� +, +F2 ≜ +� +min +i∈C∗ +2 +(e(i, C∗ +2) − e(i, C∗ +1)) ≤ −2 +� +, +where C∗ +1 = {v ∈ [n] : x∗ +v = 1} and C∗ +2 = {v ∈ [n] : x∗ +v = +−1}. Then P(ML fails) ≥ P(F1∩F2). Thus, it suffices to show +that with high probability P(F1) → 1 and P(F2) → 1. Here +we just prove that P(F1) → 1, while P(F2) → 1 is proved +similarly. By symmetry, we can condition on C∗ +1 being the +first n +2 nodes. Let T denote the set of first ⌊ +n +log2 n⌋ nodes of +C∗ +1. Then +min +i∈C∗ +1 +(e(i, C∗ +1) − e(i, C∗ +2)) ≤ min +i∈T (e(i, C∗ +1) − e(i, C∗ +2)) +≤ min +i∈T (e(i, C∗ +1 \ T ) − e(i, C∗ +2)) ++ max +i∈T e(i, T ). +Define the events +E1 ≜ +� +max +i∈T e(i, T ) ≤ δ − 2 +� +, +E2 ≜ +� +min +i∈T (e(i, C∗ +1 \ T ) − e(i, C∗ +2)) ≤ −δ +� +. +It suffices to show that P(E1) → 1 and P(E2) → 1, to have +P(F1) → 1. For any i ∈ T , +e(i, T ) = (T2 + T1)X1 + (T2 − T1)X2, +where +X1 +∼ +Binom(|T |, q0 log n/n) +and +X2 +∼ +Binom(|T |, q2 log n/n). +Lemma 7. +[28, Lemma 5] When S ∼ Bin(n, p), for any +r ≥ 1, +P(S ≥ rnp) ≤ +�e +r +�rnp +e−np. +From Lemma 7, +P +� +X1 ≥ +δ − 2 +2(T1 + T2) +� +≤ +� (δ − 2) log n +4(T1 + T2)eq0 +� +2−δ +2(T1+T2) +≤ n−2+o(1), +P +� +X2 ≥ +δ − 2 +2|T2 − T1| +� +≤ +� (δ − 2) log n +4|T2 − T1|eq2 +� +2−δ +2|T2−T1| +≤ n−2+o(1). +Since |T2 − T1| > 0 and T1 + T2 > 0, +P(e(i, T ) ≥ δ − 2) +≤ P((T1 + T2)X1 + |T2 − T1|X2 ≥ δ − 2) ≤ n−2+o(1). +Using the union bound yields P(E1) ≥ 1 − n−1+o(1). There- +fore, P(E1) → 1 with high probability. +Lemma 8. [34, Lemma 15] Let {S1, . . . , Sm} be a sequence +of i.i.d. random variables, where m − n = o(n). Then for any +µ ∈ R and ν ≥ 0 we have +P +� m +� +i=1 +Si ≥ µ − ν +� +≥ min +t>0 e−tµ−|t|νM(t) +� +1 − +σ2 +ˆ +Z +ν2 +� +, +where M(t) is the moment generating function of Z += +�m +i=1 Si and ˆZ is a random variable distributed according +to etzP(z) +EZ[etz] with variance σ2 +ˆ +Z. +Lemma 9. Let e(i, H) ≜ � +j∈H Aij(T1yiyj + T2). Define +E′ +2 ≜ +� +e(i, C∗ +1 \ T ) − e(i, C∗ +2) ≤ −δ +� +. +Then +P(E′ +2) ≥ n−η1(q,ρ)+o(1) + n−η1(q,1−ρ)+o(1). + +15 +Proof. The proof is achieved by applying Lemma 8 and the +Chernoff bound. +Applying Lemma 9 yields +P(E2) = 1 − +� +i∈T +[1 − P(E′ +2)] +≥ 1 − +� +1 − n−η1(q,1−ρ)+o(1) − n−η1(q,ρ)+o(1)�|T | +≥ 1 − e−n1−η1(q,1−ρ)+o(1)−n1−η1(q,ρ)+o(1). +Recall that ρ ≤ 0.5 implies η1(q, ρ) ≤ η1(q, 1 − ρ) and +ρ > 0.5 implies η1(q, ρ) ≥ η1(q, 1 − ρ). When ρ ≤ 0.5, +if η1(q, ρ) < 1 then P(E2) → 1. When ρ ≥ 0.5, if +η1(q, 1 − ρ) < 1 then P(E2) → 1 and the second part of +Theorem 7 follows. +APPENDIX I +PARTIAL RECOVERY ALGORITHM +In this paper, the partial recovery algorithm in [11] is +employed with few changes to make it compatible for each +Scenario. For the two-latent variable stochastic block model +we can directly use the partial recovery algorithm in [11]: +A. The two-latent variable stochastic block model with known +auxiliary latent variable y: +1) Cluster nodes according to the value of the auxiliary +latent variable y, call them auxiliary clusters. +2) Extract submatrices of P and ¯Q representing each value +of y, call them P (k) and ¯Q(k). +3) Separately in each auxiliary cluster, use respective sub- +matrices P (k) and ¯Q(k) to construct a partial recovery +estimator of communities x, and find the community +estimate for all members of each cluster. +B. The two-latent variable stochastic block model with un- +known latent variable y: +1) Use matrices P and ¯Q to construct a partial recovery +estimator of all micro-communities. +2) Cluster nodes with the same community variable repre- +senting each value of x. +For the two-latent variable censored block model, we need a +new variant of the partial recovery algorithm in [11]. In the +new variant, the vertex comparison algorithm in [11] is used +twice for each pair of nodes. First, the algorithm is employed +using the eigenvalues of diag(p)(Ξ ∗ Q). For this case, if the +two nodes belong to the same community, the output of the +algorithm is 1; otherwise it returns 0. Then, the algorithm is +employed using the eigenvalues of diag(p)((1 − Ξ) ∗ Q). For +this case, if the two nodes belong to the same community, +the output of the algorithm is 0; otherwise it returns 1. If +the outputs are not equal, we are able to determine reliably +whether the two nodes belong to the same community. If the +outputs are equal, another pair of nodes are selected to repeat +the partial recovery algorithm. +C. The two-latent variable censored block model with known +latent variable y: +1) Cluster nodes according to the value of the auxiliary +latent variable y, call them auxiliary clusters. +2) Extract submatrices of P, ¯Q, and Ξ representing each +value of y, call them P (k) and ¯Q(k), and Ξ(k). +3) Separately in each auxiliary cluster, use respective sub- +matrices P (k), ¯Q(k), and Ξ(k) to construct a partial +recovery estimator of communities x, and find the com- +munity estimate for all members of each cluster. +D. The two-latent variable censored block model with un- +known latent variable y: +1) Use matrices P, ¯Q, and Ξ to construct a partial recovery +estimator of all micro-communities. +2) Cluster nodes with the same community variable repre- +senting each value of x. +Remark 6. When y is known, for each auxiliary latent +variable y, definitions 4 and 5 in [11] are restated based on the +new matrices P (k), ¯Q(k), and Ξ(k). Using these new matrices, +the vertex comparison algorithm, the vertex classification +algorithm, the unreliable graph classification algorithm, +and the reliable graph classification algorithm in [11] are +exploited separately. When y is unknown, these definitions and +algorithms are followed from matrices P, ¯Q, and Ξ. +APPENDIX J +PROOF OF THEOREM 8 +We begin by deriving sufficient conditions for the semidefi- +nite programming estimator (8) to produce the true labels x∗. +Lemma 10. The sufficient conditions of Lemma 3 apply to +semidefinite programming (8) by replacing +S∗ = D∗ + λ∗J − A. +Proof. The proof is similar to the proof of Lemma 3. +It suffices to show that S∗ = D∗ + λ∗J − A satisfies other +conditions in Lemma 10 with probability 1 − o(1). Let +d∗ +i = +n +� +j=1 +Aijx∗ +jx∗ +i . +Then D∗x∗ = Ax∗ and based on the definition of S∗ in +Lemma 10, S∗ satisfies the condition S∗x∗ = 0. It remains to +show that S∗ ⪰ 0 and λ2(S∗) > 0 with probability 1 − o(1), +i.e., (43) holds. For any v such that vT x∗ = 0 and ∥v∥ = 1, +vT S∗v =vT D∗v + λ∗vT Jv − vT (A − E[A])v − vT E[A]v +≥ min +i +d∗ +i + λ∗vT Jv − ∥A − E[A]∥ − vT E[A]v. +Notice that +−vT E[A]v = − 1 +4[c1vT Wv − c2vT Jv − c3vT (Z ∗ W)v] ++ q0 +log n +n +. + +16 +Lemma 11. For δ = +log n +log log n, +P +� +min +i +d∗ +i ≥ δ +� +≥ 1 − n1−η2(q,ρ)+o(1) − n1−η2(q,1−ρ)+o(1). +Proof. The proof is achieved by applying the Chernoff bound +and the union bound. +Using Lemma 11, mini d∗ +i +≥ +log n +log log n with probability +converging to one, if min{η2(q, ρ), η2(q, 1 − ρ)} > 1. Let +λ∗ ≥ 1 +4[c1(2ρ−1)2 +c2]. Applying Lemmas 4, 5, and 11, we +get that when min{η2(q, ρ), η2(q, 1 − ρ)} > 1, +vT S∗v ≥ +log n +log log n − c′� +log n + q0 +log n +n +> 0, +and the first part of Theorem 8 follows. +To prove the second part, it suffices to find when the +maximum likelihood detector fails. The events F1, F2, E1, +E2, and E′ +2 are the same as we defined them in the proof of +Theorem 7. Also, the definitions for C∗ +1, C∗ +2, and T remain +valid for this part. Then P(ML fails) ≥ P(F1 ∩ F2). Here +we just prove that P(F1) → 1, while P(F2) → 1 is proved +similarly. By symmetry, we can condition on C∗ +1 being the +first n +2 nodes. Then +min +i∈C∗ +1 +(e(i, C∗ +1) − e(i, C∗ +2)) ≤ min +i∈T (e(i, C∗ +1) − e(i, C∗ +2)) +≤ min +i∈T (e(i, C∗ +1 \ T ) − e(i, C∗ +2)) ++ max +i∈T e(i, T ), +where e(i, H) +≜ +� +j∈H Aij. For i +∈ +T , e(i, T ) += +X1 + X2, where X1 ∼ Binom(|T |, q0 log n/n) and X2 ∼ +Binom(|T |, q2 log n/n). It follows from Lemma 7 that +P +� +X1 ≥ δ +2 − 1 +� +≤ +�log n +2eq0 +�δ +2 − 1 +��1− δ +2 +≤ n−2+o(1), +P +� +X2 ≥ δ +2 − 1 +� +≤ +�log n +2eq2 +�δ +2 − 1 +��1− δ +2 +≤ n−2+o(1). +Then P(e(i, T ) ≥ δ − 2) ≤ n−2+o(1). Using the union bound, +P(E1) ≥ 1 − n−1+o(1). Therefore, P(E1) → 1 with high +probability. +Lemma 12. When e(i, H) ≜ � +j∈H Aij, +P(E′ +2) ≥ n−η2(q,ρ)+o(1) + n−η2(q,1−ρ)+o(1). +Proof. The proof is achieved by applying Lemma 8 and the +Chernoff bound. +Applying Lemma 12 yields +P(E2) = 1 − +� +i∈T +[1 − P(E′ +2)] +≥ 1 − +� +1 − n−η2(q,ρ)+o(1) − n−η2(q,1−ρ)+o(1)�|T | +≥ 1 − e−n1−η2(q,ρ)+o(1)−n1−η2(q,1−ρ)+o(1). +Therefore, if min{η2(q, ρ), η2(q, 1−ρ)} < 1 then P(E2) → 1 +and the second part of Theorem 8 follows. +APPENDIX K +PROOF OF THEOREM 9 +The proof is similar to the proof of Theorem 7. Here we just +mention the proof outlines and important Lemmas for brevity. +The following Lemma declares the sufficient conditions for +the optimum solution of (10) matching the true labels x∗. +Lemma 13. For the optimization problem (10), consider the +Lagrange multipliers +λ∗, +D∗ = diag(d∗ +i ), +S∗. +If we have +S∗ = D∗ + λ∗J − R, +S∗ ⪰ 0, +λ2(S∗) > 0, +S∗x∗ = 0, +then (λ∗, D∗, S∗) is the dual optimal solution and ˆZ = x∗x∗T +is the unique primal optimal solution of (10). +Proof. The proof is similar to the proof of Lemma 3. +Let +d∗ +i =T +n +� +j=1 +Aijx∗ +jx∗ +i + T +n +� +j=1 +Aijyiyjx∗ +jx∗ +i ++ T1 +n +� +j=1 +A2 +ijyiyjx∗ +jx∗ +i + T2 +n +� +j=1 +A2 +ijx∗ +jx∗ +i . +Then D∗x∗ = TA + T(A ∗ W) + T1(A ∗ A ∗ W) + T2(A ∗ A) +and based on the definition of S∗ in Lemma 13, S∗ satisfies +the condition S∗x∗ = 0. +Lemma 14. For δ = +log n +log log n, +P +� +min +i∈[n] d∗ +i ≥ δ +� +≥1 − n1−η1(g,ρ)−η1(h,ρ)+o(1) +− n1−η1(g,1−ρ)−η1(h,1−ρ)+o(1). +Proof. The proof is achieved by applying the Chernoff bound +and taking the union bound. +Similar to the proof of Theorem 7, using Lemma 14, it +can be shown that S∗ ⪰ 0 and λ2(S∗) > 0 with probability +1 − o(1) if +� +η1(g, ρ) + η1(h, ρ) > 1 +when +ρ ≤ 0.5 +η1(g, 1 − ρ) + η1(h, 1 − ρ) > 1 +when +ρ > 0.5 . +To prove the second part, we start to find when the maxi- +mum likelihood estimator fails. To this end, let +e(i, H) ≜ +� +j∈H +Aij(Tyiyj + T) + A2 +ij(T1yiyj + T2). +The definition of events F1, F2, E1, and E2 in the proof +of Theorem 7 are used to show that with high probability +P(F1) → 1 and P(F2) → 1. Also, the definitions for C∗ +1, C∗ +2, +and T remain valid for this part. We prove that P(F1) → 1, +while P(F2) → 1 is proved similarly. To show that P(F1) → 1, + +17 +we must have P(E1) → 1 and P(E2) → 1. It can be shown +that P(E1) ≥ 1 − n−1+o(1) without difficulty. +Lemma 15. Let E′ +2 ≜ +� +e(i, C∗ +1 \ T ) − e(i, C∗ +2) ≤ −δ +� +. Then +P(E′ +2) ≥n−η1(g,ρ)−η1(h,ρ)+o(1) ++ n−η1(g,1−ρ)−η1(h,1−ρ)+o(1). +Proof. The proof is achieved by applying Lemma 8 and the +Chernoff bound. +Applying Lemma 15 yields +P(E2) = 1 − +� +i∈T +[1 − P(E′ +2)] +≥ 1 − e−n−η1(g,ρ)−η1(h,ρ)+o(1)−n−η1(g,1−ρ)−η1(h,1−ρ)+o(1). +Recall that ρ ≤ 0.5 implies +η1(g, ρ) + η1(h, ρ) ≤ η1(g, 1 − ρ) − η1(h, 1 − ρ), +and ρ > 0.5 implies +η1(g, ρ) + η1(h, ρ) ≥ η1(g, 1 − ρ) + η1(h, 1 − ρ). +When ρ ≤ 0.5, if η1(g, ρ) + η1(h, ρ) < 1, then P(E2) → 1. +When ρ ≥ 0.5, if η1(g, 1 − ρ) + η1(h, 1 − ρ) < 1, then +P(E2) → 1 and the second part of Theorem 9 follows. +APPENDIX L +PROOF OF THEOREM 10 +The proof is similar to the proof of Theorem 8. Here we just +mention the proof outlines and important Lemmas for brevity. +The following Lemma declares the sufficient conditions for +the optimum solution of (11) matching the true labels x∗. +Lemma 16. For the optimization problem (11), consider the +Lagrange multipliers +λ∗, +D∗ = diag(d∗ +i ), +S∗. +If we have +S∗ = D∗ + λ∗J − TA − T2(A ∗ A), +S∗ ⪰ 0, +λ2(S∗) > 0, +S∗x∗ = 0, +then (λ∗, D∗, S∗) is the dual optimal solution and ˆZ = x∗x∗T +is the unique primal optimal solution of (11). +Proof. The proof is similar to the proof of Lemma 3. +Let +d∗ +i =T +n +� +j=1 +Aijx∗ +jx∗ +i + T2 +n +� +j=1 +A2 +ijx∗ +jx∗ +i . +Then D∗x∗ = TA + T2(A ∗ A) and based on the definition of +S∗ in Lemma 16, S∗ satisfies the condition S∗x∗ = 0. +Lemma 17. For δ = +log n +log log n, +P +� +min +i∈[n] d∗ +i ≥ δ +� +≥1 − n1−η2(g,ρ)−η2(h,ρ)+o(1) +− n1−η2(g,1−ρ)−η2(h,1−ρ)+o(1). +Proof. The proof is achieved by applying the Chernoff bound +and taking the union bound. +Similar to the proof of Theorem 8, using Lemma 17, it +can be shown that S∗ ⪰ 0 and λ2(S∗) > 0 with probability +1 − o(1) if +min +� +η2(g, ρ) + η2(h, ρ), η2(g, 1 − ρ) + η2(h, 1 − ρ) +� +> 1. +To prove the second part, we start to find when the maxi- +mum likelihood estimator fails. To this end, let +e(i, H) ≜ +� +j∈H +TAij + T2A2 +ij. +The definition of events F1, F2, E1, and E2 in Theorem 7 +are used to show that with high probability P(F1) → 1 and +P(F2) → 1. Also, the definitions for C∗ +1, C∗ +2, and T remain +valid for this part. We prove that P(F1) → 1, while P(F2) → 1 +is proved similarly. To show that P(F1) → 1, we must have +P(E1) → 1 and P(E2) → 1. It can be shown that P(E1) ≥ +1 − n−1+o(1) without difficulty. +Lemma 18. Let E′ +2 ≜ +� +e(i, C∗ +1 \ T ) − e(i, C∗ +2) ≤ −δ +� +. Then +P(E′ +2) ≥n−η2(g,ρ)−η2(h,ρ)+o(1) ++ n−η2(g,1−ρ)−η2(h,1−ρ)+o(1). +Proof. The proof is achieved by applying Lemma 8 and the +Chernoff bound. +Applying Lemma 18 yields +P(E2) = 1 − +� +i∈T +[1 − P(E′ +2)] +≥ 1 − e−n−η2(g,ρ)−η2(h,ρ)+o(1)−n−η2(g,1−ρ)−η2(h,1−ρ)+o(1). +If +min +� +η2(g, ρ) + η2(h, ρ), η2(g, 1 − ρ) + η2(h, 1 − ρ) +� +< 1, +then P(E2) → 1 and the second part of Theorem 10 follows. +REFERENCES +[1] M. Girvan and M. E. J. Newman, “Community structure in social and +biological networks,” Proceedings of the National Academy of Sciences, +vol. 99, no. 12, pp. 7821–7826, 2002. +[2] J. Chen and B. Yuan, “Detecting functional modules in the yeast protein– +protein interaction network,” Bioinformatics, vol. 22, no. 18, pp. 2283– +2290, 2006. +[3] K. Berahmand, E. Nasiri, S. Forouzandeh, and Y. Li, “A preference +random walk algorithm for link prediction through mutual influence +nodes in complex networks,” Journal of King Saud University-Computer +and Information Sciences, 2021. +[4] K. Berahmand, E. Nasiri, M. Rostami, and S. 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Xu, “Achieving exact cluster recovery thresh- +old via semidefinite programming: Extensions,” IEEE Transactions on +Information Theory, vol. 62, no. 10, pp. 5918–5937, 2016. + diff --git a/Y9E2T4oBgHgl3EQfvQga/content/tmp_files/load_file.txt b/Y9E2T4oBgHgl3EQfvQga/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..63dd60edee7b4afba20ac3758c5aae65eea43ec4 --- /dev/null +++ b/Y9E2T4oBgHgl3EQfvQga/content/tmp_files/load_file.txt @@ -0,0 +1,1049 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf,len=1048 +page_content='1 Community Detection with Known, Unknown, or Partially Known Auxiliary Latent Variables Mohammad Esmaeili and Aria Nosratinia, Fellow, IEEE Abstract—Empirical observations suggest that in practice, community membership does not completely explain the depen- dency between the edges of an observation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The residual dependence of the graph edges are modeled in this paper, to first order, by auxiliary node latent variables that affect the statistics of the graph edges but carry no information about the communities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We then study community detection in graphs obeying the stochastic block model and censored block model with auxiliary latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We analyze the conditions for exact recovery when these auxiliary latent variables are unknown, representing unknown nuisance parameters or model mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We also analyze exact recovery when these secondary latent variables have been either fully or partially revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Finally, we propose a semidefinite programming algorithm for recovering the desired labels when the secondary labels are either known or unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We show that exact recovery is possible by semidefinite programming down to the respective maximum likelihood exact recovery threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Index Terms—Community Detection, Latent Variables, Stochastic Block Model (SBM), Censored Block Model (CBM), Graph Inference, Exact Recovery, Semidefinite Programming (SDP), Chernoff-Hellinger Divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' INTRODUCTION Community detection refers to a clustering of the nodes of a graph based on the observation of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In many applications, this involves identifying groups of nodes that are more densely connected within the group than to nodes outside the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Community detection has many applications such as finding like-minded people in social networks [1], exploration of biomedical networks [2], improving link predictors and rec- ommendation systems [3]–[5], and is also relevant to network reconstruction problems [6]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Community detection has been widely investigated in the literature from both theoretical and algorithmic perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Community detection is based on graph models such as the stochastic block model and the censored block model [10]–[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Several metrics are used in this field to characterize the asymptotic behavior of the residual errors as the size of the graph grows, including correlated recovery, weak recovery, almost exact recovery, and exact recovery [17]–[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Among the various detection techniques one can name spectral methods, belief propagation, and semidefinite programming [27]–[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In the graph models that have so far been studied for community detection, the graph edges are generated indepen- dently conditioned on the community labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' A brief survey of models that are most closely related to the present work will M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Esmaeili and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Nosratinia are with the Department of Electrical and Computer Engineering, The University of Texas at Dallas, Richardson, TX 75083-0688, USA, Email: esmaeili@utdallas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='edu, aria@utdallas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' This work was supported in part by the NSF grant CIF-2008684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' be presented shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' However, in many practical community detection problems, the community labels do not fully explain the dependence between the graph edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In other words, in many graphs encountered in practice, the graph edges condi- tioned on the desired community labels are not statistically independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' This happens when the structure of the graph is also influenced by factors other than the community of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For example, one may consider political affiliation communi- ties on a social network in a university campus, where the social network graph is also influenced by other variables that may be unrelated to the community label of interest, such as membership in intramural and extramural activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The nature and magnitude of the dependence of the graph on these secondary or auxiliary factors can have an effect on the performance of the community detection algorithm for the community label of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The present study models and analyzes community detection in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Toward that goal, this paper introduces secondary or auxil- iary latent variables in the graph model that are not subject to community detection themselves, but influence the structure of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' More specifically, we propose and employ a more general version of the stochastic block model and censored block model in which edges are independent conditioned on both the community labels and a set of secondary latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The secondary or auxiliary latent variables represent a first-order model for the residual dependence of the edges of the graph once the effect of the community labels has been removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Auxiliary variables are independent of community memberships and may or may not be observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The aux- iliary latent variable model is distinct from side-information model [33], [34] where the side information variables are directly observed and carry information about the communi- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Side information represents non-graph information about communities, while auxiliary variables model the graph con- nectivity patterns that are unrelated to the communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We investigate the exact recovery threshold for community detection in the graphs with secondary latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We also analyze the effect on the performance of community detection when this secondary latent variable is fully or partially known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We also propose and investigate a semidef- inite programming algorithm for community detection with secondary latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Our analysis shows that exact recovery via semidefinite programming is possible down to the respective maximum likelihood exact recovery threshold, for both unknown or known secondary latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In addition to addressing a novel problem, this paper also provides a novel proof for bounding the summation of the minimums of Poisson-distributed values from above and below via Chernoff-Hellinger divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Our result (Lemma 1) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='04088v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='SI] 8 Jan 2023 2 eliminates certain technical difficulties that existed in earlier proofs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', does not impose restrictions on the domain of Poisson distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' This result is extended (Lemma 2) for the general censored block model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also, the analysis of exact recovery for a graph generated based on two latent variables involves subtleties in extracting the maximum likelihood esti- mator and analyzing its semidefinite programming relaxation, which go beyond earlier works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' To put the model of this paper in perspective, we review several community detection graph models whose nodes are associated, beyond a scalar community detection label, with some other variables too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The latent space model [35]–[37] associates with each node a vector, often with small dimension, containing variables that are latent in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The graph edges are generated from a distribution that is parameterized based on the distance between the latent vectors of pairs of nodes, and the community is a scalar generated as a func- tion of each latent vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The overlapping stochastic block model [11], [38] recovers multiple independent, identically distributed, binary communities via observing a graph whose edges are drawn independently conditioned on all the commu- nity labels of the terminating nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' An important distinction of overlapped communities from the present work is that all communities must be recovered in the overlapped model, therefore the overlapped model has significant similarity with a multi-community model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In the overlapped model, the multiple communities posses a structure that can be exploited, com- pared with a general multi-community model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Finally, there exists some work on combining non-graph observation with graph observations [33], [34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' these works have a superficial resemblance to the subsection in this paper where the sec- ondary latent variable is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' However, the graph and the side information in [33], [34] are assumed independent of each other conditioned on community labels, therefore the revealed side information in [33], [34] has no direct influence on the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Thus, [33], [34] model a different phenomenon and also have a different mathematical structure, compared with the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In the interest of brevity, our coverage of various community detection models is limited, and the interested reader is referred to more comprehensive coverage available, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Notation: I is the identity matrix and J the all-one matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' S ⪰ 0 indicates a positive semidefinite matrix and S ≥ 0 denotes a matrix with non-negative entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' ||S|| is the spectral norm and λ2(S) is the second smallest eigenvalue (for a sym- metric matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' [a, b] is a vector that is obtained by stacking vectors a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' ⟨·, ·⟩ is the inner product and ∗ is the element- wise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We abbreviate [n] ≜ {1, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' P(·) indicates the probability operator and P(·) a probability distribution which is identified by the choice of its variables whenever there is no confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Random variables with Bernoulli and Binomial distributions are indicated by Bern(p) and Bin(n, p), respectively, with n trails and success probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also, random variables with Poisson distribution are indicated by Pλ(n) with n trails and parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' SYSTEM MODEL We start by considering a two-latent variable model, and assume the cardinality of both is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For notational conve- nience throughout the paper, x, y are length-n vectors holding latent variable values for the whole graph, while the latent variables for any node v are represented with xν, yν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In our model, we aim to discover x, therefore nodes that share the same value for x are called a community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' By micro-community, we refer to the set of nodes in the graph that share the same value for both latent variables x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The matrix P denotes prior probabilities Pi,j = P(xv = i, yv = j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For convenience and for avoiding tensor calculations, we further define: p ≜ vec(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For both the two-latent variable stochastic block model and two-latent variable censored block model, the graph edges are Bernoulli distributed, conditioned on the latent variables of the two nodes terminating the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The conditional Bernoulli parameters for an arbitrary edge are organized in a symmetric matrix ¯Q, whose rows and columns are ordered in a manner compatible with vector p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In other words, assuming the latent variable xv has mx outcomes, then the probability of an edge between two nodes with latent variable pairs taking values (i, j) and (i′, j′) is given by the element of ¯Q in row jmx +i and column j′mx + i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We are interested in a regime where edge probabilities diminish with the size of the graph n, in particular, in the context of our model there exist a constant matrix Q such that: ¯Q = log n n Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' This assumption asymptotically guarantees a fully connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Consider a two-latent variable stochastic block model with mx = 2 and my = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then P = �P0,0 P0,1 P0,2 P1,0 P1,1 P1,2 � , p = �P0,0 P0,1 P0,2 P1,0 P1,1 P1,2 � , ¯Q = log n n � ������� Q0,0 Q0,1 Q0,2 Q0,3 Q0,4 Q0,5 Q1,0 Q1,1 Q1,2 Q1,3 Q1,4 Q1,5 Q2,0 Q2,1 Q2,2 Q2,3 Q2,4 Q2,5 Q3,0 Q3,1 Q3,2 Q3,3 Q3,4 Q3,5 Q4,0 Q4,1 Q4,2 Q4,3 Q4,4 Q4,5 Q5,0 Q5,1 Q5,2 Q5,3 Q5,4 Q5,5 � ������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In addition, we define the columns of weighted versions of the matrix Q as q(i,j) ≜ diag(p)Q ejmx+i , where ek is the k-th canonical coordinate vector, and for convenience our notation of q(i,j) emphasizes dependence on 3 the latent variable outcomes rather than matrix coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Thus, q(i,j) is the column of diag(p)Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' This vector represents the relative frequency of edges connecting a node from the micro-community (i, j) to all nodes of each micro-community (including the same micro-community).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also, we define the vector ˜q(i,j) of size mx with entries ˜q(i,j) i′ ≜ � j′ Pi′,j′Qj′mx+i′,jmx+i , representing the relative frequency of edges, connecting a node from the micro-community (i, j) to all nodes of micro- communities with similar community latent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For the two-latent variable censored block model, if an edge exists between a pair of nodes, the sign of the edge (positive or negative) is determined by a random variable drawn from a Bernoulli distribution with a certain parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The Bernoulli parameters for the positive sign of an edge are organized in a symmetric matrix Ξ, whose rows and columns are also ordered in a manner compatible with vector p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Finally, for the censored block model, we define similarly g(i,j) ≜ diag(p)(Ξ ∗ Q) ejmx+i , h(i,j) ≜ diag(p)((1 − Ξ) ∗ Q) ejmx+i , and ˜g(i,j) i′ ≜ � j′ Pi′,j′(Ξ ∗ Q)j′mx+i′,jmx+i , ˜h(i,j) i′ ≜ � j′ Pi′,j′((1 − Ξ) ∗ Q)j′mx+i′,jmx+i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The censored block model in [28], [39] with parameters a and ξ is a special case of the general censored model represented in this paper with Q = �a a a a � , Ξ = �1 − ξ ξ ξ 1 − ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' EXACT RECOVERY UNDER OPTIMAL DETECTION The main results of this part are represented in the context of three scenarios, where the latent variable x is unknown and the latent variable y is either known or unknown (for all nodes in the graph) or partially known (for some nodes in the graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Figure 1 shows graph realizations of a two-latent variable stochastic block model with mx = 2 and my = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In each node, the community latent variable is indicated by the color of the inner circle, and the auxiliary latent variable is represented by the color of a ring around the inner circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The Chernoff-Hellinger divergence is due to Abbe [24] and is defined for two non-negative vectors a, b of the same dimension: Div(a, b) ≜ max t∈[0,1] � i � tai + (1 − t)bi − at ib1−t i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (1) This is a generalization of the Hellinger divergence and the Chernoff divergence [11], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In a manner similar to [11] we present a lemma that bounds a summation of the minimums of Poisson-distributed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let a, b ∈ Rm +, with a ̸= b, and two positive scalars p, ˆp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For any Poisson multivariate distributions Pa(d) and Pb(d), define I(a, b) ≜ � d∈Zm + min{Pa(d)p, Pb(d)ˆp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then I(a, b) ≤ max{p, ˆp}e−Div(a,b), I(a, b) ≥ min{p, ˆp}e−Div(a,b) m � i=1 1 e � at∗ i b1−t∗ i �− 1 2 , where t∗ is the optimal parameter in the definition of Chernoff- Hellinger divergence Div(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let D be a random variable vector representing the number of edges that connect the node v to each micro-community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' More specifically, D(i′,j′) is an element of the D indicating the number of edges connecting the node v to the micro- community (i′, j′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For each node v, the proposed detection tests hypotheses Hi : xv = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' If v belongs to micro-community (i, j), then D(i′,j′) ∼ Bin(nPi′,j′, ¯Qj′mx+i′,jmx+i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In the regime where ¯Q = Q log n n , the Binomial distribution can be approximated by a Poisson distribution with the same mean, denoted λ(i′,j′) i,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Indeed, using Le Cam’s inequality, the total variation distance between Bin(nPi′,j′, log n n Qj′mx+i′,jmx+i) and P(Pi′,j′Qj′mx+i′,jmx+i log n) asymptotically goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then P(D = d|Hi, yv = j) = � i′ � j′ Pλ(i′,j′) i,j (d(i′,j′)), where λ(i′,j′) i,j = Pi′,j′Qj′mx+i′,jmx+i log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Under the two-latent variable stochastic block model, all micro-communities are exactly recovered if and only if min (i,j)̸=(k,l) Div(q(i,j), q(k,l)) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' It follows from the exact recovery under the general stochastic bock model or the general overlapping stochastic block model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Under the two-latent variable stochastic block model, when the latent variable y is revealed, exact recovery of x is possible if and only if γ1 ≜ min j min i̸=k Div(q(i,j), q(k,j)) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Under the two-latent variable stochastic block model, when both latent variables are unknown, exact recovery of x is possible if and only if γ2 ≜ min j min i̸=k Div � ˜q(i,j), ˜q(k,j)� > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 4 (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 1: For each node, (a) both latent variables are unknown, (b) knowing the statistics of the graph, the community latent variable is recovered while the auxiliary latent variable is unknown, (c) the auxiliary latent variable is known while the first one is unknown, (d) knowing the statistics of the graph, the community latent variable is recovered while the auxiliary latent variable is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Now we present the following Lemma which is similar to Lemma 1 and is crucial for the analysis of the censored block model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let a, b, ˆa,ˆb ∈ Rm +, with a ̸= b or ˆa ̸= ˆb, and two positive scalars p, ˆp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For any Poisson multivariate distributions Pa(d), Pb(d), Pˆa(w), and Pˆb(w), define I(a, b, ˆa,ˆb) ≜ � d,w∈Zm + min{Pa(d)Pˆa(w)p, Pb(d)Pˆb(w)ˆp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then I(a, b, ˆa,ˆb) ≤ max{p, ˆp}e−Div([a,ˆa],[b,ˆb]), I(a, b, ˆa,ˆb) ≥ min{p, ˆp}e−Div([a,ˆa],[b,ˆb]) × � i 1 e2 � (aiˆai)t∗(biˆbi)1−t∗�− 1 2 , where t∗ is the optimal parameter in the definition of Chernoff- Hellinger divergence Div([a, ˆa], [b,ˆb]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let D and W be random vectors representing the pos- itive and negative edges that connect the node v to each micro-community, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' More specifically, D(i′,j′) and W (i′,j′) are elements of D and W indicating the number of positive and negative edges connecting the node v to the micro-community (i′, j′), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For each node v, the proposed detection tests hypotheses Hi : xv = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' If v belongs to micro-community (i, j), then D(i′,j′) ∼ Bin(nPi′,j′, (Ξ ∗ ¯Q)j′mx+i′,jmx+i), W (i′,j′) ∼ Bin(nPi′,j′, ((1 − Ξ) ∗ ¯Q)j′mx+i′,jmx+i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In the regime where ¯Q = Q log n n , the Binomial distribution can be approximated by a Poisson distribution with the same mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The distributions of D and W can be approximated by multivariate Poisson distributions Pλi,j and Pˆλi,j with the vector means λi,j and ˆλi,j, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Therefore P(D = d, W = w|Hi, yv = j) = P(D = d|Hi, yv = j)P(W = w|Hi, yv = j) = � i′ � j′ Pλ(i′,j′) i,j (d(i′,j′))Pˆλ(i′,j′) i,j (w(i′,j′)), where λ(i′,j′) i,j = Pi′,j′(Ξ ∗ Q)j′mx+i′,jmx+i log n, ˆλ(i′,j′) i,j = Pi′,j′((1 − Ξ) ∗ Q)j′mx+i′,jmx+i log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Under two-latent variable censored block model, all micro-communities are exactly recovered if and only if min (i,j)̸=(k,l) Div � [g(i,j), h(i,j)], [g(k,l), h(k,l)] � > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' See Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Under the two-latent variable censored block model, when the latent variable y is revealed, exact recovery of x is possible if and only if γ3 ≜ min j min i̸=k Div � [g(i,j), h(i,j)], [g(k,j), h(k,j)] � > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' See Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Under the two-latent variable censored block model, when both latent variables are unknown, exact recovery of x is possible if and only if γ4 ≜ min j min i̸=k Div � [˜g(i,j), ˜h(i,j)], [˜g(k,j), ˜h(k,j)] � > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' See Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Assume x and y are unknown latent variables for all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We randomly reveal the latent variable y for (1−ϵ)n nodes, where ϵ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' This is equivalent to erasing the latent variable y which is a known latent variable from a node with erasure probability ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Define β1 ≜ − lim n→∞ log(1 − ϵ) log n , β2 ≜ − lim n→∞ log ϵ log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Under the two-latent variable stochastic block model ex- act recovery is asymptotically possible for latent variable x if and only if min � γ1 + β1, γ2 + β2 � > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 5 Under the two-latent variable censored block model exact recovery is asymptotically possible for latent variable x if and only if min � γ3 + β1, γ4 + β2 � > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The results of this part generalize to M latent variables without difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' To prove the “if” part of all theorems in Sec- tion III, a partial recovery algorithm is required before apply- ing a MAP estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For that purpose, the partial recovery algorithm in [11] is adopted and modified to match the scenarios in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Please see Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' SEMIDEFINITE PROGRAMMING RESULTS This section describes a semidefinite programming algo- rithm for recovering the desired latent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The main results of this part are represented in the context of two scenarios, where the latent variable x is unknown and the latent variable y is either known or unknown (for all nodes in the graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We consider x, y ∈ {±1}n such that xT 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Thus, the latent variable x represents two equal-sized communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The sample size of the latent variable y, represented by ρ ≜ 1 n|{v ∈ [n] : yv = 1}|, is an unknown quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Two-latent variable stochastic block model We highlight the specifics of a two-latent variable stochastic block model for the purposes of upcoming calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The probability of an edge drawn between two nodes v, u is characterized by four constants, q0, q1, q2, q3 such that: Aij ∼ � � � � � � � � � Bern(q0 log n n ) if xv = xu, yv = yu Bern(q1 log n n ) if xv ̸= xu, yv = yu Bern(q2 log n n ) if xv = xu, yv ̸= yu Bern(q3 log n n ) if xv ̸= xu, yv ̸= yu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The corresponding matrix Q, as defined earlier, in this case will be: Q = � ��� q0 q1 q2 q3 q1 q0 q3 q2 q2 q3 q0 q1 q3 q2 q1 q0 � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (2) 1) Recovering x when y is known: In the first scenario, given an observation of the graph A and y which corresponds to the observed graph, the latent variable xv is recovered exactly for each node v ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In this part, y is considered as an observation which helps the estimator to recover the desired latent variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let W ≜ yyT and B ≜ W ∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Since x is chosen uniformly over {x ∈ {±1}n : xT 1 = 0}, the maximum likelihood estimator gives the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For this configuration, the log-likelihood is log P(A|x, y) = T1 8 xT Bx + T2 8 xT Ax + c, 1Note that semidefinite programming results in this section are obtained for binary equal-sized communities, while the results of Section III were more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' where T1 ≜ log( q0q3 q2q1 ) and T2 ≜ log( q0q2 q1q3 ), as n → ∞ and c is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Considering the constraints, the maximum likelihood estimator is, ˆx =arg max x T1xT Bx + T2xT Ax subject to xi ∈ {±1}, i ∈ [n] xT 1 = 0, (3) which is a non-convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let Z = xxT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Reorganizing (3), ˆZ =arg max Z ⟨Z, T1B + T2A⟩ subject to Z = xxT Zii = 1, i ∈ [n] ⟨Z, J⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (4) By relaxing the rank-one constraint on Z, we obtain the following semidefinite programming relaxation of (4): ˆZ =arg max Z ⟨Z, T1B + T2A⟩ subject to Z ⪰ 0 Zii = 1, i ∈ [n] ⟨Z, J⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (5) For convenience define η1(q, ρ) ≜ ρ 2(√q0 − √q1)2 + 1 − ρ 2 (√q2 − √q3)2, where q ≜ [q0, q1, q2, q3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Under the two-latent variable stochastic block model with binary alphabet where the latent variable y has been revealed, if � η1(q, ρ) > 1 when ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 η1(q, 1 − ρ) > 1 when ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 then the semidefinite programming estimator is asymptotically optimal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', P( ˆZ = Z∗) ≥ 1 − o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also, if � η1(q, ρ) < 1 when ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 η1(q, 1 − ρ) < 1 when ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 then for any sequence of estimators ˆZn, P( ˆZn = Z∗) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' See Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 2) Recovering x when y is unknown: Given an observation of the graph A, the aim is to exactly recover x while both latent variables x and y are unknown latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' It is assumed that the estimator does not know anything about the auxiliary latent variable y, which its prior distribution is uniform over {y : y ∈ {±1}n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Notice that x is drawn uniformly from {x ∈ {±1}n : xT 1 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The log-likelihood of A given x and y is log P(A|x, y) =T1 8 yT (A ∗ xxT )y + T2 8 xT Ax + T3 8 yT Ay + c, 6 where T1 ≜ log( q0q3 q2q1 ), T2 ≜ log( q0q2 q1q3 ), and T3 ≜ log( q0q1 q2q3 ), as n → ∞ and c is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then logP(A|x) ∝ log � Y P(A|x, y) ∝ log � Y e T1 T3 yT (A∗xxT )y+ T2 T3 xT Ax+yT Ay =T1 + T2 T3 xT Ax + � i � j Aij + log � Y e T1 T3 yT (A∗xxT )y+yT Ay− T1 T3 xT Ax−� i � j Aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Applying the log-sum-exp approximation, the maximum like- lihood estimator is ˆx =arg max x xT Ax subject to xi ∈ {±1}, i ∈ [n] xT 1 = 0, (6) that is a non-convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let Z = xxT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Reorganizing (6) yields ˆZ =arg max Z ⟨Z, A⟩ subject to Z = xxT Zii = 1, i ∈ [n] ⟨Z, J⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (7) Relaxing the rank-one constraint on Z, we obtain the following semidefinite programming relaxation of (7): ˆZ =arg max Z ⟨Z, A⟩ subject to Z ⪰ 0 Zii = 1, i ∈ [n] ⟨Z, J⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (8) For convenience define η2(q, ρ) ≜ 1 2 �� q0ρ + q2(1 − ρ) − � q1ρ + q3(1 − ρ) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Under the two-latent variable stochastic block model with binary alphabet, if min {η2(q, ρ), η2(q, 1 − ρ)} > 1, then the semidefinite programming estimator is asymptotically optimal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', P( ˆZ = Z∗) ≥ 1 − o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also, if min {η2(q, ρ), η2(q, 1 − ρ)} < 1, then for any sequence of estimators ˆZn, P( ˆZn = Z∗) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' See Appendix J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The results of Theorems 7 and 8 are consistent with Theorems 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The constraint xT 1 = 0 that has been considered for this part results in a well-defined phase transition threshold for exact recovery of latent variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In general, x may be a random variable which is drawn uniformly from {x ∈ {±1}n : xT 1 = (2ρx − 1)n}, where ρx ≜ 1 n|{v ∈ [n] : xv = 1}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then xT 1 = 0 is substituted by xT 1 = (2ρx − 1)n in semidefinite programming relaxations (5) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also, due to the robustness of semidefinite programming, an approximation of ρx can be replaced for recovering the latent variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Investigating the constraint xT 1 = (2ρx − 1)n and the robustness of semidefinite programming are beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Two-latent variable censored block model We highlight the specifics of a two-latent variable censored block model for the purposes of upcoming calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let P(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' q0, ξ) be a discrete probability density function with parameters q0 > 0 and ξ ∈ [0, 1] as, P(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' q0, ξ) ≜ξq0 log n n δ[k − 1] + (1 − ξ)q0 log n n δ[k + 1] + � 1 − q0 log n n � δ[k], where δ is Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The probability of an edge drawn between two nodes v, u is characterized by constants q0, q1, q2, q3 and ξ such that: Aij ∼ � � � � � � � � � P(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' q0, 1 − ξ) if xv = xu, yv = yu P(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' q1, ξ) if xv ̸= xu, yv = yu P(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' q2, ξ) if xv = xu, yv ̸= yu P(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' q3, ξ) if xv ̸= xu, yv ̸= yu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The corresponding matrix Q, as defined earlier, is the same as (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also, in this case, the corresponding matrix Ξ will be: Ξ = � ��� (1 − ξ) ξ ξ ξ ξ (1 − ξ) ξ ξ ξ ξ (1 − ξ) ξ ξ ξ ξ (1 − ξ) � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (9) 1) Recovering x when y is known: Given an observation of the graph A and y which corresponds to the observed graph, the latent variable xv is recovered exactly for each node v ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In this part, y is considered as an observation which helps the estimator to recover the desired latent variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let R ≜ TA + T(A ∗ W) + T1(A ∗ A ∗ W) + T2(A ∗ A), where T ≜ log � 1−ξ ξ � and W ≜ yyT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Since x is chosen uniformly over {x ∈ {±1}n : xT 1 = 0}, the maximum likelihood estimator gives the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Similar to Section IV-A1, it can be shown that the semidefinite pro- gramming relaxation of maximum likelihood estimator for this configuration is ˆZ =arg max Z ⟨Z, R⟩ subject to Z ⪰ 0 Zii = 1, i ∈ [n] ⟨Z, J⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (10) For convenience define g ≜ [(1 − ξ)q0, ξq1, ξq2, ξq3], h ≜ [ξq0, (1 − ξ)q1, (1 − ξ)q2, (1 − ξ)q3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 7 Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Under the two-latent variable censored block model with binary alphabet where the latent variable y has been revealed, if � η1(g, ρ) + η1(h, ρ) > 1 when ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 η1(g, 1 − ρ) + η1(h, 1 − ρ) > 1 when ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 then the semidefinite programming estimator is asymptotically optimal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', P( ˆZ = Z∗) ≥ 1 − o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also, if � η1(g, ρ) + η1(h, ρ) < 1 when ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 η1(g, 1 − ρ) + η1(h, 1 − ρ) < 1 when ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 then for any sequence of estimators ˆZn, P( ˆZn = Z∗) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' See Appendix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 2) Recovering x when y is unknown: Given an observation of the graph A, the aim is to exactly recover x while both latent variables x and y are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' It is assumed that the estimator does not know anything about the auxiliary latent variable y, which its prior distribution is uniform over {y : y ∈ {±1}n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Notice that x is drawn uniformly from {x ∈ {±1}n : xT 1 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Similar to Section IV-A2, it can be shown that for this configuration the semidefinite programming relaxation of the maximum likelihood estimator is ˆZ =arg max Z ⟨Z, TA + T2(A ∗ A)⟩ subject to Z ⪰ 0 Zii = 1, i ∈ [n] ⟨Z, J⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (11) Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Under the two-latent variable censored block model with binary alphabet, if min � η2(g, ρ) + η2(h, ρ), η2(g, 1 − ρ) + η2(h, 1 − ρ) � > 1, then the semidefinite programming estimator is asymptotically optimal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', P( ˆZ = Z∗) ≥ 1 − o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also, if min � η2(g, ρ) + η2(h, ρ), η2(g, 1 − ρ) + η2(h, 1 − ρ) � < 1, then for any sequence of estimators ˆZn, P( ˆZn = Z∗) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' See Appendix L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The results of Theorems 9 and 10 are consistent with Theorems 5 and 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' DISCUSSION & NUMERICAL RESULTS It is illuminating to review the flow of the development of the achievability results througout this paper: 1) Calculate the Lagrangian of the corresponding optimiza- tion 2) Extract the dual optimal solution based on the Lagrange multipliers 3) Show that ˆZ = Z∗ is primal optimal solution 4) Show that ˆZ = Z∗ is unique 5) Extract the conditions under which the dual optimal solution holds The converses follow the following sequence: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 2: Exact recovery region of x in the context of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (2), with q2 = 3, q1 = q3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 3: Exact recovery region of x in the context of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (2), with q1 = q2 = q3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 1) Extract the maximum likelihood estimator 2) Extract the conditions under which the maximum like- lihood estimator fails To give a pictorial view of some results of the paper, we plot some results in the context of the two-latent variable stochastic block model represented by (2) and two-latent variable censored block model represented by (2) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For ease of notation, we define γ1 ≜ min{η1(q, ρ), η1(q, 1 − ρ)}, γ2 ≜ min{η2(q, ρ), η2(q, 1 − ρ)}, γ3 ≜ min{η1(g, ρ) + η1(h, ρ), η1(g, 1 − ρ) + η1(h, 1 − ρ)}, γ4 ≜ min{η2(g, ρ) + η2(h, ρ), η2(g, 1 − ρ) + η2(h, 1 − ρ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For the two-latent variable stochastic block model, Figures 2 and 3 show the exact recovery region for recovering the latent 810 qo12140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5y191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content="5 2 1Y2'p=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='520=d p=0 ,p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='30 46810 qo12140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content="5 2 Y1'p=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='52p=0 p=0 ,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 O =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='30 468 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 4: Exact recovery region of x in the context of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (2) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (9), with ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1, q2 = 3, and q1 = q3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 5: Exact recovery region of x in the context of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (2) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (9), with ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1, and q1 = q2 = q3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' variable x when the secondary latent variable y is either known or unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The curves in these figures are based on the obtained results in Theorem 7 and Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' These figures encompass several curves plotted for different values of q0, q1, q2, q3 in (2), and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' At each figure, we consider fixed values for q1, q2, q3 and vary the values of q0 and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' A comparison between the curves in Figures 2 and 3 clarifies the role of the revealed latent variable y for recovering the desired latent variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For the two-latent variable censored block model, Figures 4 and 5 show the exact recovery region for recovering the latent variable x when the secondary latent variable y is either known or unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The curves in these figures are based on the obtained results in Theorem 9 and Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' These figures consist of several curves plotted for different values of q0, q1, y n BSBM BCBM q0 AEP q0 ξ AEP Known 100 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='8 × 10−2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='7 × 10−2 Known 200 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='4 × 10−2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='9 × 10−2 Known 300 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='9 × 10−2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='6 × 10−2 Known 400 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 × 10−2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 × 10−2 Known 500 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 × 10−2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='6 × 10−2 Known 100 9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 × 10−5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='6 × 10−5 Known 200 9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='9 × 10−5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='2 × 10−5 Known 300 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='2 × 10−5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='4 × 10−5 Known 400 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='8 × 10−5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='7 × 10−5 Known 500 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='8 × 10−5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='2 × 10−5 Unknown 100 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='7 × 10−2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='6 × 10−2 Unknown 200 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 × 10−2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='9 × 10−2 Unknown 300 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='7 × 10−2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 × 10−2 Unknown 400 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='8 × 10−2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='6 × 10−2 Unknown 500 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 × 10−2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 × 10−2 Unknown 100 10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='2 × 10−5 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 × 10−5 Unknown 200 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='4 × 10−5 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='0 × 10−5 Unknown 300 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 × 10−5 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 × 10−5 Unknown 400 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 × 10−5 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='7 × 10−5 Unknown 500 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='4 × 10−5 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 × 10−5 TABLE I: Semidefinite programming optimization of (8) and (10), with q2 = 3, q1 = q3 = 1, and ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' q2, q3 in (2) and ρ, while ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' At each figure, we consider fixed values for ξ, q1, q2, q3 and vary the values of q0 and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' A comparison between the curves in Figures 4 and 5 clarifies the role of the revealed latent variable y for recovering the desired latent variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' To gain an understanding of the scope of our asymptotic results, under the conditions of Figures 2 and 4, we performed several simulations on 104 graph realizations with various graph sizes obtained from the proposed models in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The obtained average error probability (AEP) is around 10−5 in the regimes just inside the region of exact recovery, and around 10−2 in the regimes just outside the region of exact recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The details of these simulations are represented in Tables I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' At each simulation, we consider fixed values for q1, q2, q3 and vary the values of q0, ρ, and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' CONCLUSION This paper presents and analyzes a new generalization of the stochastic and censored block models in which, in addition to the latent variable representing community labels, there exists another (secondary) latent variables that are not part of community detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' These secondary latent variables may be known, unknown, or partially known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' This model represents community detection problems where the community labels alone does not explain all the dependencies between the graph edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We investigate the exact recovery threshold for these models under maximum likelihood detection, and also analyze a semidefinite programming algorithm for recovering the desired latent variable under the two-latent variable stochastic block model and the two-latent variable censored block model for both scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 810 qo12140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content="5 34'p=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 2p=0 p=0 ,p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='4 Exact Re .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5covery Regicn3=0 p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='30 46810 qo12140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content="5 3—4' p= 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 2p=0 = 0 p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='4 Exact Re .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5covery Regicn3p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='30 469 y n BSBM BCBM q0 AEP q0 ξ AEP Known 100 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='6 × 10−2 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 × 10−2 Known 200 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 × 10−2 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 × 10−2 Known 300 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='0 × 10−2 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 × 10−2 Known 400 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 × 10−2 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='8 × 10−2 Known 500 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 × 10−2 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 × 10−2 Known 100 12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='7 × 10−5 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='9 × 10−5 Known 200 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 × 10−5 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 × 10−5 Known 300 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='6 × 10−5 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='8 × 10−5 Known 400 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 × 10−5 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='2 × 10−5 Known 500 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='6 × 10−5 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='0 × 10−5 Unknown 100 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 × 10−2 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='2 × 10−2 Unknown 200 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 × 10−2 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='9 × 10−2 Unknown 300 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='4 × 10−2 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='0 × 10−2 Unknown 400 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='7 × 10−2 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 × 10−2 Unknown 500 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='2 × 10−2 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='0 × 10−2 Unknown 100 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='2 × 10−5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='8 × 10−5 Unknown 200 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='6 × 10−5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 × 10−5 Unknown 300 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='7 × 10−5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='2 × 10−5 Unknown 400 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3 × 10−5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 × 10−5 Unknown 500 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 × 10−5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='0 × 10−5 TABLE II: Semidefinite programming optimization of (8) and (10), with q2 = 3, q1 = q3 = 1, and ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' APPENDIX A PROOF OF LEMMA 1 Define f1(t) ≜ m � i=1 � bi ai �(t−1)di e(t−1)(ai−bi), f2(t) ≜ m � i=1 � bi ai �tdi et(ai−bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For any t ∈ [0, 1], � d∈Zm + min{Pa(d)p, Pb(d)ˆp} ≤ max{p, ˆp} � d∈Zm + min{Pa(d), Pb(d)} = max{p, ˆp} exp � − � i � tai + (1 − t)bi − at ib1−t i �� × � d∈Zm + � i (at ib1−t i )di di!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' e−at ib1−t i min{f1(t), f2(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Both f1(t) and f2(t) are monotonic and f2(t) f1(t) is a positive constant (does not depend on t), thus min{f1, f2} is also monotonic in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Since f1(1) = f2(0) = 1, for all t we have: min{f1(t), f2(t)} ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Notice that � d∈Zm + � i (at ib1−t i )di di!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' e−at ib1−t i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then I(a, b) ≤ max{p, ˆp}e− �m i=1 � tai+(1−t)bi−at ib1−t i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (12) For the value of t that maximizes the right-hand side of inequality (12), we have � d∈Zm + min{Pa(d)p, Pb(d)ˆp} ≤ max{p, ˆp}e−Div(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Notice that t∗ satisfies m � i=1 � bi ai �at∗ i b1−t∗ i eai−bi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then at the optimal t∗, � d∈Zm + min{Pa(d)p, Pb(d)ˆp} ≥ min{p, ˆp} � d∈Zm + min{Pa(d), Pb(d)} (a) ≥ min{p, ˆp}e−Div(a,b) � i (at∗ i b1−t∗ i )at∗ i b1−t∗ i at∗ i b1−t∗ i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' e−at∗ i b1−t∗ i (b) ≥ min{p, ˆp}e−Div(a,b) � i 1 e � at∗ i b1−t∗ i �− 1 2 , where (a) holds because � d∈Zm + min{Pa(d), Pb(d)} ≥ min{Pa(d∗), Pb(d∗)}, where d∗ is defined by d∗ i ≜ at∗ i b1−t∗ i , and (b) is due to Stirling’s approximation n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' ≤ nn+ 1 2 e−n+1 for any n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' APPENDIX B PROOF OF THEOREM 2 We aim to recover xv when yv is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Given a realization of D and yv, our goal is to minimize the error probability by selecting the most likely hypothesis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', argmax i P{Hi|D = d, yv}, or equivalently, since d, yv are known observations, argmax i P(d|Hi, yv)P{Hi, yv}, which is the maximum a posteriori (MAP) detector, which we rewrite: argmax i P(d|Hi, yv)Pi,yv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (13) Solving (13) requires mx − 1 pairwise comparisons of the hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' From this viewpoint, if P(d|Hi, yv)Pi,yv ≤ P(d|Hk, yv)Pk,yv, (14) then a pairwise comparison will choose Hk over Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Now assume the correct hypothesis is Hi, and denote by Bik the region of D for which (14) is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', Hi has a worse metric compared with Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also denote by Bi the region for D where the overall MAP decoder is in error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The dependence of error regions Bik and Bi on yv is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then the probability of error Pe = � i P{D ∈ Bi|Hi, yv}Pi,yv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (15) 10 Since Bi ⊂ ∪kBik, Pe ≤ � i � k̸=i P{D ∈ Bik|Hi, yv}Pi,yv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' From the earlier Poisson assumption P(d|Hi, yv) = Pλi,yv (d) it follows that: min{Pλi,yv (d)Pi,yv, Pλk,yv (d)Pk,yv} = � Pλi,yv (d) Pi,yv when D ∈ Bik Pλk,yv (d) Pk,yv when D ∈ Bc ik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Therefore, substituting into the union bound: Pe ≤ � d � i � k>i min{Pλi,yv (d)Pi,yv, Pλk,yv (d)Pk,yv}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (16) For bounding the error probability (16), it suffices to find an upper bound for � d min{Pλi,yv (d)Pi,yv, Pλk,yv (d)Pk,yv}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (17) It follows from Lemma 1 that Pe ≤ � i � k>i max{Pi,yv, Pk,yv}e−Div(λi,yv ,λk,yv ) = � i � k>i n−Div � q(i,yv),q(k,yv)� +o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (18) We now bound the error probability of decoding rule (13) from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Since � k̸=i P{D ∈ Bik|Hi, yk} ≤ (mx − 1)P{D ∈ Bi|Hi, yv}, (19) substituting (19) into (15) yields Pe ≥ 1 mx − 1 � i � k̸=i P{D ∈ Bik|Hi, yv}Pi,yv = 1 mx − 1 × � i � k>i � d min � Pλi,yv (d)Pi,yv, Pλk,yv (d)Pk,yv � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then it suffices to find a lower bound for (17) to bound the error probability from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' It follows from Lemma 1 that Pe ≥ � i � k>i c′ min{Pi,yv, Pk,yv}(log n)− m 2 e−Div(λi,yv ,λk,yv ) = � i � k>i n−Div(q(i,yv),q(k,yv))+o(1), (20) where c′ is a constant and m is the number of elements in vector d, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', the product of alphabet sizes of xv and yv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The lower and upper bounds (18) and (20) imply that the true hypothesis is recovered correctly if Div(q(i,yv), q(k,yv)) > 1, for a given yv and any i ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' This means that a known latent variable restricts the number of pairwise comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then under the two-latent variable stochastic block model in which the latent variable y is known, and the latent variable x is unknown, exact recovery is possible for x if and only if min j min i̸=k Div(q(i,j), q(k,j)) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (21) APPENDIX C PROOF OF THEOREM 3 We aim to recover xv when yv is unknown, given a realization of D for node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For this setting the MAP detector is argmax i P{Hi|D = d}, or equivalently, argmax i � yv � l P � � j d(l,j)|Hi, yv � Pi,yv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (22) Solving (22) requires mx − 1 pairwise comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In these comparisons, if � yv � l P � � j d(l,j)|Hi, yv � Pi,yv < � yv � l P � � j d(l,j)|Hk, yv � Pk,yv, (23) then we conclude hypothesis Hi is ruled out, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', xv ̸= i, because another hypothesis Hk has a better metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Denote by Bik the region of D for which Hi has a worse metric compared with Hk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', the region for D in which (23) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also denote by Bi the region for D where the overall MAP decoder is in error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The error probability of MAP decoder (22) is given by Pe = � i � yv P(D ∈ Bi|Hi, yv}Pi,yv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (24) Since Bi ⊂ ∪kBik, via the union bound, � yv P{D ∈ Bi|Hi, yv}Pi,yv ≤ � yv � k̸=i P{D ∈ Bik|Hi, yv}Pi,yv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (25) Using the Poisson approximation and the additive property of Poisson distribution: I(d, i, yv) ≜ � l P � � j d(l,j)|Hi, yv � = � l P� j λ(l,j) i,yv � � j d(l,j) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Therefore, min � I(d, i, yv)Pi,yv,I(d, k, yv)Pk,yv � = � I(d, i, yv)Pi,yv when D ∈ Bik I(d, k, yv)Pk,yv when D ∈ Bc ik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Substituting (25) into (24) yields Pe ≤ � i � k̸=i � yv P{D ∈ Bik|Hi, yv}Pi,yv = � i � k>i � yv � d min � I(d, i, yv)Pi,yv, I(d, k, yv)Pk,yv � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (26) 11 For bounding the error probability (26) from above, it suffices to find an upper bound for � d∈Zm + min � I(d, i, yv)Pi,yv, I(d, k, yv)Pk,yv � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (27) Applying Lemma 1 yields Pe ≤ � i � k>i � yv n−Div � ˜q(i,yv),˜q(k,yv)� +o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (28) We now bound the error probability of decoding rule (22) from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Notice that � k̸=i P{D ∈ Bik|Hi, yv} ≤ (mx − 1)P{D ∈ Bi|Hi, yv}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (29) Substituting (29) into (24) yields Pe ≥ 1 mx − 1 � i � k̸=i � yv P{D ∈ Bik|Hi, yv}Pi,yv = 1 mx − 1 × � i � k>i � yv � d min � I(d, i, yv)Pi,yv, I(d, k, yv)Pk,yv � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then it suffices to find a lower bound for (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Applying Lemma 1 yields Pe ≥ � i � k>i � yv n−Div � ˜q(i,yv),˜q(k,yv)� +o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (30) The lower and upper bounds (28) and (30) imply that the true hypothesis is recovered correctly if Div(˜q(i,yv), ˜q(k,yv)) > 1 for any i ̸= k and any yv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then under two-latent variable stochastic block model in which both latent variables x, y are unknown, exact recovery is solvable for x if and only if min j min i̸=k Div � ˜q(i,j), ˜q(k,j)� > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (31) APPENDIX D PROOF OF LEMMA 2 Define f1(t) ≜ �Pa(d)Pˆa(w) Pb(d)Pˆb(w) �1−t , f2(t) ≜ � Pb(d)Pˆb(w) Pa(d)Pˆa(w) �t , f(t) ≜ Pa(d)tPb(d)1−tPˆa(w)tPˆb(w)1−t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For any t ∈ [0, 1], � d,w∈Zm + min{Pa(d)Pˆa(w)p, Pb(d)Pˆb(w)ˆp} ≤ max{p, ˆp} � d,w∈Zm + min{Pa(d)Pˆa(w), Pb(d)Pˆb(w)} ≤ max{p, ˆp} exp � − � i � tai + (1 − t)bi − at ib1−t i �� × exp � − � i � tˆai + (1 − t)ˆbi − ˆat iˆb1−t i �� , (32) where the last inequality holds because min{f1(t), f2(t)} ≤ 1, and � d,w∈Zm + � i (at ib1−t i )di di!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' e−at ib1−t i (ˆat iˆb1−t i )wi wi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' e−ˆat iˆb1−t i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For the value of t that minimizes the upper bound of (32), we have I(a, b, ˆa,ˆb) ≤ max{p, ˆp}e−Div([a,ˆa],[b,ˆb]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Notice that t∗ satisfies m � i=1 � bi ai �at∗ i b1−t∗ i �ˆbi ˆai �ˆat∗ i ˆb1−t∗ i eai−bi+ˆai−ˆbi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then at the optimal t∗, � d,w∈Zm + min{Pa(d)Pˆa(w)p, Pb(d)Pˆb(w)ˆp} ≥ min{p, ˆp} � d,w∈Zm + min{Pa(d)Pˆa(w), Pb(d)Pˆb(w)} (a) ≥ min{p, ˆp}e−Div([a,ˆa],[b,ˆb]) × � i (at∗ i b1−t∗ i )at∗ i b1−t∗ i at∗ i b1−t∗ i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' e−at∗ i b1−t∗ i × � i (ˆat∗ i ˆb1−t∗ i )ˆat∗ i ˆb1−t∗ i ˆat∗ i ˆb1−t∗ i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' e−ˆat∗ i ˆb1−t∗ i (b) ≥ min{p, ˆp}e−Div([a,ˆa],[b,ˆb]) � i 1 e2 � (aiˆai)t∗(biˆbi)1−t∗�− 1 2 , where (a) holds because � d,w∈Zm + min{Pa(d)Pˆa(w), Pb(d)Pˆb(w)} ≥ min{Pa(d∗)Pˆa(w∗), Pb(d∗)Pˆb(w∗)}, where d∗ is defined by d∗ i ≜ at∗ i b1−t∗ i and w∗ is defined by w∗ i ≜ ˆat∗ i ˆb1−t∗ i , and (b) is due to Stirling’s approximation n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' ≤ nn+ 1 2 e−n+1 for any n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' APPENDIX E PROOF OF THEOREM 4 We aim to recover both xv and yv for node v, given a realization of D and a realization of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Our goal is to minimize the error probability by selecting the most likely hypothesis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', argmax i,j P{Hi,j|D = d, W = w}, where Hi,j : xv = i, yv = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The maximum a posteriori (MAP) detector is rewrite as argmax i,j P(d, w|Hi,j)Pi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (33) 12 Solving (33) requires mxmy − 1 pairwise comparisons of the hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' From this viewpoint, if P(d, w|Hi,j)Pi,j ≤ P(d, w|Hk,l)Pk,l, then a pairwise comparison will choose Hk,l over Hi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Now assume the correct hypothesis is Hi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Similar to the proof of Theorems 2 and 3, it can be shown that the probability of error for recovering the true hypothesis is bounded from above and below by controlling � d,w min{Pλi,j(d)Pˆλi,j(w)Pi,j, Pλk,l(d)Pˆλk,l(w)Pk,l}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' It follows from Lemma 2 that Pe ≤ � i,k>i � j,l>j max{Pi,j, Pk,l}e−Div([λi,j,ˆλi,j],[λk,l,ˆλk,l]) = � i,k>i � j,l>j n−Div � [q(i,j),g(i,j)],[q(k,l),g(k,l)] � +o(1), (34) and Pe ≥ � i,k>i � j,l>j c′ min{Pi,j, Pk,l} (log n)m e−Div([λi,j,ˆλi,j],[λk,l,ˆλk,l]) = � i,k>i � j,l>j n−Div � [g(i,j),h(i,j)],[g(k,l),h(k,l)] � +o(1), (35) where c′ is a constant and m is the number of elements in vec- tor d, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', the product of alphabet sizes of xv and yv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The lower and upper bounds (34) and (35) imply that the true hypothesis is recovered correctly if Div � [g(i,j), h(i,j)], [g(k,l), h(k,l)] � > 1, for any (i, j) ̸= (k, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' This means that under the two- latent variable censored block model all micro-communities are exactly recovered if and only if min (i,j)̸=(k,l) Div � [g(i,j), h(i,j)], [g(k,l), h(k,l)] � > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' APPENDIX F PROOF OF THEOREM 5 We aim to recover xv when yv is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Given a realization of D, a realization of W, and yv, our goal is to minimize the error probability by selecting the most likely hypothesis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', argmax i P{Hi|D = d, W = w, yv}, or equivalently, argmax i P(d|Hi, yv)P(w|Hi, yv)Pi,yv, (36) which is the MAP detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Solving (36) requires mx − 1 pairwise comparisons of the hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Similar to the proof of Theorem 2, it can be shown that the error probability of finding true hypothesis is bounded from above and below by controlling � d,w min{Pλi,yv (d)Pˆλi,yv (w)Pi,yv, Pλk,yv (d)Pˆλk,yv (w)Pk,yv}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' It follows from Lemma 2 that Pe ≤ � i � k>i max{Pi,yv, Pk,yv}e−Div([λi,yv ,ˆλi,yv ],[λk,yv ,ˆλk,l]) = � i � k>i n−Div � [g(i,yv),h(i,yv)],[g(k,yv),h(k,yv)] � +o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (37) and Pe ≥ � i � k>i c (log n) m 2 e−Div([λi,yv ,ˆλi,yv ],[λk,yv ,ˆλk,yv ]) = � i � k>i n−Div � [g(i,yv),h(i,yv)],[g(k,yv),h(k,yv)] � +o(1), (38) where c ≜ c′ min{Pi,yv, Pk,yv} is a constant and m is the number of elements in vector d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The lower and upper bounds (37) and (38) imply that the true hypothesis is recov- ered correctly if Div � [g(i,yv), h(i,yv)], [g(k,yv), h(k,yv)] � > 1, for a given yv and any i ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' This means that a known latent variable restricts the number of pairwise comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then under the two-latent variable censored block model in which the latent variable y is known, and the latent variable x is unknown, exact recovery is possible for x if and only if min j min i̸=k Div � [g(i,j), h(i,j)], [g(k,j), h(k,j)] � > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' APPENDIX G PROOF OF THEOREM 6 We aim to recover xv when yv is unknown, given a realization of D and a realization of W for node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For this setting the MAP detector is argmax i P{Hi|D = d, W = w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For convenience define I(d, w, i, yv) ≜ � l P � � j d(l,j), � j w(l,j)|Hi, yv � , where � j w(l,j) and � j d(l,j) are independent given Hi and yv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then the MAP detector rewrite as argmax i � yv I(d, w, i, yv)Pi,yv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (39) Solving (39) requires mx − 1 pairwise comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In these comparisons, if � yv I(d, w, i, yv)Pi,yv < � yv I(d, w, k, yv)Pk,yv, then we conclude hypothesis Hi is ruled out, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', xv ̸= i, because another hypothesis Hk has a better metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Notice that using the Poisson approximation and the additive property of Poisson distribution, I(d, w, i, yv) can be reorganized as I(d, w, i, yv) = � l P� j λ(l,j) i,yv � � j d(l,j) � × � l P� j λ(l,j) i,yv � � j w(l,j) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 13 Similar to the proof of Theorem 3, it can be shown that the error probability of recovering the true hypothesis is bounded from above and below by controlling � d,w∈Zm + min � I(d, w, i, yv)Pi,yv, I(d, w, k, yv)Pk,yv � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Applying Lemma 2 yields Pe ≤ � i � k>i � yv n−Div � [˜g(i,yv),˜h(i,yv)],[˜g(k,yv),˜h(k,yv)] � +o(1), (40) and Pe ≥ � i � k>i � yv n−Div � [˜g(i,yv),˜h(i,yv)],[˜g(k,yv),˜h(k,yv)] � +o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (41) The lower and upper bounds (40) and (41) imply that the true hypothesis is recovered correctly if Div � [˜g(i,yv), ˜h(i,yv)], [˜g(k,yv), ˜h(k,yv)] � > 1 for any i ̸= k and any yv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then under two-latent variable censored block model in which both latent variables x, y are unknown, exact recovery is solvable for x if and only if min j min i̸=k Div � [˜g(i,j), ˜h(i,j)], [˜g(k,j), ˜h(k,j)] � > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' APPENDIX H PROOF OF THEOREM 7 We begin by stating sufficient conditions for the optimum solution of (5) matching the true labels x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For the optimization problem (5), consider the Lagrange multipliers λ∗, D∗ = diag(d∗ i ), S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' If we have S∗ = D∗ + λ∗J − T1B − T2A, S∗ ⪰ 0, λ2(S∗) > 0, S∗x∗ = 0, then (λ∗, D∗, S∗) is the dual optimal solution and ˆZ = x∗x∗T is the unique primal optimal solution of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let D = diag(di), λ ∈ R, and S ⪰ 0 denote the Lagrangian of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For any Z that satisfies the constraints in (5), we have T1⟨B, Z⟩ + T2⟨A, Z⟩ (a) ≤L(Z, S∗, D∗, λ∗) = ⟨D∗, I⟩ (b) =⟨S∗ − λ∗J + T1B + T2A, Z∗⟩ (c) =T1⟨B, Z∗⟩ + T2⟨A, Z∗⟩, where (a) holds because ⟨S∗, Z⟩ ≥ 0, (b) holds because Zii = 1 for all i ∈ [n] and S∗ = D∗+λ∗J−T1B−T2A, and (c) holds because S∗x∗ = 0 and x∗T 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Therefore, Z∗ = x∗x∗T is an optimal solution of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Now, assume ˜Z is another optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then ⟨S∗, ˜Z⟩ =⟨D∗ + λ∗J − T1B − T2A, ˜Z⟩ (a) =⟨D∗ + λ∗J − T1B − T2A, Z∗⟩ = ⟨S∗, Z∗⟩ = 0, where (a) holds because ⟨T1B+T2A, Z∗⟩ = ⟨T1B+T2A, ˜Z⟩, Z∗ ii = ˜Zii = 1 for all i ∈ [n], and ⟨J, Z∗⟩ = ⟨J, ˜Z⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Since ˜Z ⪰ 0, and S∗ ⪰ 0 while its second smallest eigenvalue λ2(S∗) is positive (since S∗ˆx∗ = 0), ˜Z must be a multiple of Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also, since ˜Zii = Z∗ ii = 1 for all i ∈ [n], we have ˜Z = Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We now show that S∗ = D∗ + λ∗J − T1B − T2A satisfies other conditions in Lemma 3 with probability 1 − o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let d∗ i = T1 n � j=1 Bijx∗ jx∗ i + T2 n � j=1 Aijx∗ jx∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (42) Then D∗x∗ = T1Bx∗ + T2Ax∗ and based on the definition of S∗ in Lemma 3, S∗ satisfies the condition S∗x∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' It remains to show that S∗ ⪰ 0 and λ2(S∗) > 0 with probability 1 − o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In other words, we need to show that P � inf v⊥x∗,∥v∥=1vT S∗v > 0 � ≥ 1 − o(1), (43) where v is a n × 1 vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then for any v such that vT x∗ = 0 and ∥v∥ = 1, vT S∗v =vT D∗v + λ∗vT Jv − T1vT (B − E[B])v − T2vT (A − E[A])v − T1vT E[B]v − T2vT E[A]v ≥ min i d∗ i + λ∗vT Jv − T1∥B − E[B]∥ − T2∥A − E[A]∥ − T1vT E[B]v − T2vT E[A]v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Notice that T1vT E[B]v + T2vT E[A]v =1 4[T1c1 + T2c2]vT Wv + 1 4[T1c3 + T2c4]vT (Z ∗ W)v + 1 4[T1c1 + T2c2]vT Jv − (T1 + T2)q0 log n n , where c1 ≜ log n n (q0 − q2 + q1 − q3), c2 ≜ log n n (q0 + q2 + q1 + q3), c3 ≜ log n n (q0 − q2 − q1 + q3), c4 ≜ log n n (q0 + q2 − q1 + q3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For any c > 0, there exists c′, c′′ > 0 such that for any n ≥ 1, ∥A − E[A]∥ ≤ c′′√log n and ∥B − E[B]∥ ≤ c′√log n with probability at least 1 − n−c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The proof is similar to the proofs [39, Thoerem 9] and [32, Thoerem 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 14 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' With probability at least 1 − n− 1 2 , vT (Z ∗ W)v ≤ � log n, vT Wv ≤ � log n + (2ρ − 1)2vT Jv + 2|2ρ − 1| � n log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Since −|vi| ≤ viyi ≤ |vi|, by applying the Chernoff bound we have P(vT y − E[vT y] ≥ � log n) ≤ n− 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Since E[vT y] = (2ρ − 1)vT 1 and |vT 1| ≤ ∥v∥2∥1∥2 = √n, with probability converging to one, (vT y)2 ≤ log n + (2ρ − 1)2vT Jv + 2|vT 1||2ρ − 1| � log n ≤ log n + (2ρ − 1)2vT Jv + 2|2ρ − 1| � n log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Similarly, since E[� i xiyivi] = 0 and −|vi| ≤ xiyivi ≤ |vi|, applying the Chernoff bound yields vT (Z ∗ W)v ≤ √log n with probability converging to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For δ = log n log log n, P � min i∈[n] d∗ i ≥ δ � ≥ 1 − n1−η1(q,ρ)+o(1) − n1−η1(q,1−ρ)+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The proof is achieved by applying the Chernoff bound and taking the union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Notice that ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 implies η1(q, ρ) ≤ η1(q, 1−ρ) and ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 implies η1(q, ρ) ≥ η1(q, 1 − ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then mini d∗ i ≥ log n log log n if � η1(q, ρ) > 1 when ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 η1(q, 1 − ρ) > 1 when ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' (44) Let λ∗ ≥ 1 4[T1c1 + T2c2](2ρ − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Therefore, applying Lemmas 4, 5, and 6, we get that if (44) holds, then vT S∗v ≥ log n log log n − (T1c′ + T2c′′) � log n + (T1 + T2)q0 log n n > 0, and the first part of Theorem 7 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' To prove the second part, since x∗ has a uniform distribution over {x ∈ {±1}n : xT 1 = 0}, maximum likelihood estimator minimizes the error probability among all estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then we need to find when the maximum likelihood estimator fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let e(i, H) ≜ � j∈H Aij(T1yiyj + T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Define the events F1 ≜ � min i∈C∗ 1 (e(i, C∗ 1) − e(i, C∗ 2)) ≤ −2 � , F2 ≜ � min i∈C∗ 2 (e(i, C∗ 2) − e(i, C∗ 1)) ≤ −2 � , where C∗ 1 = {v ∈ [n] : x∗ v = 1} and C∗ 2 = {v ∈ [n] : x∗ v = −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then P(ML fails) ≥ P(F1∩F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Thus, it suffices to show that with high probability P(F1) → 1 and P(F2) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Here we just prove that P(F1) → 1, while P(F2) → 1 is proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' By symmetry, we can condition on C∗ 1 being the first n 2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let T denote the set of first ⌊ n log2 n⌋ nodes of C∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then min i∈C∗ 1 (e(i, C∗ 1) − e(i, C∗ 2)) ≤ min i∈T (e(i, C∗ 1) − e(i, C∗ 2)) ≤ min i∈T (e(i, C∗ 1 \\ T ) − e(i, C∗ 2)) + max i∈T e(i, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Define the events E1 ≜ � max i∈T e(i, T ) ≤ δ − 2 � , E2 ≜ � min i∈T (e(i, C∗ 1 \\ T ) − e(i, C∗ 2)) ≤ −δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' It suffices to show that P(E1) → 1 and P(E2) → 1, to have P(F1) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For any i ∈ T , e(i, T ) = (T2 + T1)X1 + (T2 − T1)X2, where X1 ∼ Binom(|T |, q0 log n/n) and X2 ∼ Binom(|T |, q2 log n/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' [28, Lemma 5] When S ∼ Bin(n, p), for any r ≥ 1, P(S ≥ rnp) ≤ �e r �rnp e−np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' From Lemma 7, P � X1 ≥ δ − 2 2(T1 + T2) � ≤ � (δ − 2) log n 4(T1 + T2)eq0 � 2−δ 2(T1+T2) ≤ n−2+o(1), P � X2 ≥ δ − 2 2|T2 − T1| � ≤ � (δ − 2) log n 4|T2 − T1|eq2 � 2−δ 2|T2−T1| ≤ n−2+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Since |T2 − T1| > 0 and T1 + T2 > 0, P(e(i, T ) ≥ δ − 2) ≤ P((T1 + T2)X1 + |T2 − T1|X2 ≥ δ − 2) ≤ n−2+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Using the union bound yields P(E1) ≥ 1 − n−1+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' There- fore, P(E1) → 1 with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' [34, Lemma 15] Let {S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' , Sm} be a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' random variables, where m − n = o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then for any µ ∈ R and ν ≥ 0 we have P � m � i=1 Si ≥ µ − ν � ≥ min t>0 e−tµ−|t|νM(t) � 1 − σ2 ˆ Z ν2 � , where M(t) is the moment generating function of Z = �m i=1 Si and ˆZ is a random variable distributed according to etzP(z) EZ[etz] with variance σ2 ˆ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let e(i, H) ≜ � j∈H Aij(T1yiyj + T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Define E′ 2 ≜ � e(i, C∗ 1 \\ T ) − e(i, C∗ 2) ≤ −δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then P(E′ 2) ≥ n−η1(q,ρ)+o(1) + n−η1(q,1−ρ)+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The proof is achieved by applying Lemma 8 and the Chernoff bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Applying Lemma 9 yields P(E2) = 1 − � i∈T [1 − P(E′ 2)] ≥ 1 − � 1 − n−η1(q,1−ρ)+o(1) − n−η1(q,ρ)+o(1)�|T | ≥ 1 − e−n1−η1(q,1−ρ)+o(1)−n1−η1(q,ρ)+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Recall that ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 implies η1(q, ρ) ≤ η1(q, 1 − ρ) and ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 implies η1(q, ρ) ≥ η1(q, 1 − ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' When ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5, if η1(q, ρ) < 1 then P(E2) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' When ρ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5, if η1(q, 1 − ρ) < 1 then P(E2) → 1 and the second part of Theorem 7 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' APPENDIX I PARTIAL RECOVERY ALGORITHM In this paper, the partial recovery algorithm in [11] is employed with few changes to make it compatible for each Scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For the two-latent variable stochastic block model we can directly use the partial recovery algorithm in [11]: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The two-latent variable stochastic block model with known auxiliary latent variable y: 1) Cluster nodes according to the value of the auxiliary latent variable y, call them auxiliary clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 2) Extract submatrices of P and ¯Q representing each value of y, call them P (k) and ¯Q(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 3) Separately in each auxiliary cluster, use respective sub- matrices P (k) and ¯Q(k) to construct a partial recovery estimator of communities x, and find the community estimate for all members of each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The two-latent variable stochastic block model with un- known latent variable y: 1) Use matrices P and ¯Q to construct a partial recovery estimator of all micro-communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 2) Cluster nodes with the same community variable repre- senting each value of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For the two-latent variable censored block model, we need a new variant of the partial recovery algorithm in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' In the new variant, the vertex comparison algorithm in [11] is used twice for each pair of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' First, the algorithm is employed using the eigenvalues of diag(p)(Ξ ∗ Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For this case, if the two nodes belong to the same community, the output of the algorithm is 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' otherwise it returns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then, the algorithm is employed using the eigenvalues of diag(p)((1 − Ξ) ∗ Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For this case, if the two nodes belong to the same community, the output of the algorithm is 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' otherwise it returns 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' If the outputs are not equal, we are able to determine reliably whether the two nodes belong to the same community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' If the outputs are equal, another pair of nodes are selected to repeat the partial recovery algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The two-latent variable censored block model with known latent variable y: 1) Cluster nodes according to the value of the auxiliary latent variable y, call them auxiliary clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 2) Extract submatrices of P, ¯Q, and Ξ representing each value of y, call them P (k) and ¯Q(k), and Ξ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 3) Separately in each auxiliary cluster, use respective sub- matrices P (k), ¯Q(k), and Ξ(k) to construct a partial recovery estimator of communities x, and find the com- munity estimate for all members of each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The two-latent variable censored block model with un- known latent variable y: 1) Use matrices P, ¯Q, and Ξ to construct a partial recovery estimator of all micro-communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 2) Cluster nodes with the same community variable repre- senting each value of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' When y is known, for each auxiliary latent variable y, definitions 4 and 5 in [11] are restated based on the new matrices P (k), ¯Q(k), and Ξ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Using these new matrices, the vertex comparison algorithm, the vertex classification algorithm, the unreliable graph classification algorithm, and the reliable graph classification algorithm in [11] are exploited separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' When y is unknown, these definitions and algorithms are followed from matrices P, ¯Q, and Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' APPENDIX J PROOF OF THEOREM 8 We begin by deriving sufficient conditions for the semidefi- nite programming estimator (8) to produce the true labels x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The sufficient conditions of Lemma 3 apply to semidefinite programming (8) by replacing S∗ = D∗ + λ∗J − A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The proof is similar to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' It suffices to show that S∗ = D∗ + λ∗J − A satisfies other conditions in Lemma 10 with probability 1 − o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let d∗ i = n � j=1 Aijx∗ jx∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then D∗x∗ = Ax∗ and based on the definition of S∗ in Lemma 10, S∗ satisfies the condition S∗x∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' It remains to show that S∗ ⪰ 0 and λ2(S∗) > 0 with probability 1 − o(1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=', (43) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For any v such that vT x∗ = 0 and ∥v∥ = 1, vT S∗v =vT D∗v + λ∗vT Jv − vT (A − E[A])v − vT E[A]v ≥ min i d∗ i + λ∗vT Jv − ∥A − E[A]∥ − vT E[A]v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Notice that −vT E[A]v = − 1 4[c1vT Wv − c2vT Jv − c3vT (Z ∗ W)v] + q0 log n n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' 16 Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For δ = log n log log n, P � min i d∗ i ≥ δ � ≥ 1 − n1−η2(q,ρ)+o(1) − n1−η2(q,1−ρ)+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The proof is achieved by applying the Chernoff bound and the union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Using Lemma 11, mini d∗ i ≥ log n log log n with probability converging to one, if min{η2(q, ρ), η2(q, 1 − ρ)} > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let λ∗ ≥ 1 4[c1(2ρ−1)2 +c2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Applying Lemmas 4, 5, and 11, we get that when min{η2(q, ρ), η2(q, 1 − ρ)} > 1, vT S∗v ≥ log n log log n − c′� log n + q0 log n n > 0, and the first part of Theorem 8 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' To prove the second part, it suffices to find when the maximum likelihood detector fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The events F1, F2, E1, E2, and E′ 2 are the same as we defined them in the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also, the definitions for C∗ 1, C∗ 2, and T remain valid for this part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then P(ML fails) ≥ P(F1 ∩ F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Here we just prove that P(F1) → 1, while P(F2) → 1 is proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' By symmetry, we can condition on C∗ 1 being the first n 2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then min i∈C∗ 1 (e(i, C∗ 1) − e(i, C∗ 2)) ≤ min i∈T (e(i, C∗ 1) − e(i, C∗ 2)) ≤ min i∈T (e(i, C∗ 1 \\ T ) − e(i, C∗ 2)) + max i∈T e(i, T ), where e(i, H) ≜ � j∈H Aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For i ∈ T , e(i, T ) = X1 + X2, where X1 ∼ Binom(|T |, q0 log n/n) and X2 ∼ Binom(|T |, q2 log n/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' It follows from Lemma 7 that P � X1 ≥ δ 2 − 1 � ≤ �log n 2eq0 �δ 2 − 1 ��1− δ 2 ≤ n−2+o(1), P � X2 ≥ δ 2 − 1 � ≤ �log n 2eq2 �δ 2 − 1 ��1− δ 2 ≤ n−2+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then P(e(i, T ) ≥ δ − 2) ≤ n−2+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Using the union bound, P(E1) ≥ 1 − n−1+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Therefore, P(E1) → 1 with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' When e(i, H) ≜ � j∈H Aij, P(E′ 2) ≥ n−η2(q,ρ)+o(1) + n−η2(q,1−ρ)+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The proof is achieved by applying Lemma 8 and the Chernoff bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Applying Lemma 12 yields P(E2) = 1 − � i∈T [1 − P(E′ 2)] ≥ 1 − � 1 − n−η2(q,ρ)+o(1) − n−η2(q,1−ρ)+o(1)�|T | ≥ 1 − e−n1−η2(q,ρ)+o(1)−n1−η2(q,1−ρ)+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Therefore, if min{η2(q, ρ), η2(q, 1−ρ)} < 1 then P(E2) → 1 and the second part of Theorem 8 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' APPENDIX K PROOF OF THEOREM 9 The proof is similar to the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Here we just mention the proof outlines and important Lemmas for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The following Lemma declares the sufficient conditions for the optimum solution of (10) matching the true labels x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For the optimization problem (10), consider the Lagrange multipliers λ∗, D∗ = diag(d∗ i ), S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' If we have S∗ = D∗ + λ∗J − R, S∗ ⪰ 0, λ2(S∗) > 0, S∗x∗ = 0, then (λ∗, D∗, S∗) is the dual optimal solution and ˆZ = x∗x∗T is the unique primal optimal solution of (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The proof is similar to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let d∗ i =T n � j=1 Aijx∗ jx∗ i + T n � j=1 Aijyiyjx∗ jx∗ i + T1 n � j=1 A2 ijyiyjx∗ jx∗ i + T2 n � j=1 A2 ijx∗ jx∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then D∗x∗ = TA + T(A ∗ W) + T1(A ∗ A ∗ W) + T2(A ∗ A) and based on the definition of S∗ in Lemma 13, S∗ satisfies the condition S∗x∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For δ = log n log log n, P � min i∈[n] d∗ i ≥ δ � ≥1 − n1−η1(g,ρ)−η1(h,ρ)+o(1) − n1−η1(g,1−ρ)−η1(h,1−ρ)+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The proof is achieved by applying the Chernoff bound and taking the union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Similar to the proof of Theorem 7, using Lemma 14, it can be shown that S∗ ⪰ 0 and λ2(S∗) > 0 with probability 1 − o(1) if � η1(g, ρ) + η1(h, ρ) > 1 when ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 η1(g, 1 − ρ) + η1(h, 1 − ρ) > 1 when ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' To prove the second part, we start to find when the maxi- mum likelihood estimator fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' To this end, let e(i, H) ≜ � j∈H Aij(Tyiyj + T) + A2 ij(T1yiyj + T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The definition of events F1, F2, E1, and E2 in the proof of Theorem 7 are used to show that with high probability P(F1) → 1 and P(F2) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also, the definitions for C∗ 1, C∗ 2, and T remain valid for this part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We prove that P(F1) → 1, while P(F2) → 1 is proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' To show that P(F1) → 1, 17 we must have P(E1) → 1 and P(E2) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' It can be shown that P(E1) ≥ 1 − n−1+o(1) without difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let E′ 2 ≜ � e(i, C∗ 1 \\ T ) − e(i, C∗ 2) ≤ −δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then P(E′ 2) ≥n−η1(g,ρ)−η1(h,ρ)+o(1) + n−η1(g,1−ρ)−η1(h,1−ρ)+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The proof is achieved by applying Lemma 8 and the Chernoff bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Applying Lemma 15 yields P(E2) = 1 − � i∈T [1 − P(E′ 2)] ≥ 1 − e−n−η1(g,ρ)−η1(h,ρ)+o(1)−n−η1(g,1−ρ)−η1(h,1−ρ)+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Recall that ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 implies η1(g, ρ) + η1(h, ρ) ≤ η1(g, 1 − ρ) − η1(h, 1 − ρ), and ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5 implies η1(g, ρ) + η1(h, ρ) ≥ η1(g, 1 − ρ) + η1(h, 1 − ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' When ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5, if η1(g, ρ) + η1(h, ρ) < 1, then P(E2) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' When ρ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content='5, if η1(g, 1 − ρ) + η1(h, 1 − ρ) < 1, then P(E2) → 1 and the second part of Theorem 9 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' APPENDIX L PROOF OF THEOREM 10 The proof is similar to the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Here we just mention the proof outlines and important Lemmas for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The following Lemma declares the sufficient conditions for the optimum solution of (11) matching the true labels x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For the optimization problem (11), consider the Lagrange multipliers λ∗, D∗ = diag(d∗ i ), S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' If we have S∗ = D∗ + λ∗J − TA − T2(A ∗ A), S∗ ⪰ 0, λ2(S∗) > 0, S∗x∗ = 0, then (λ∗, D∗, S∗) is the dual optimal solution and ˆZ = x∗x∗T is the unique primal optimal solution of (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The proof is similar to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let d∗ i =T n � j=1 Aijx∗ jx∗ i + T2 n � j=1 A2 ijx∗ jx∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then D∗x∗ = TA + T2(A ∗ A) and based on the definition of S∗ in Lemma 16, S∗ satisfies the condition S∗x∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' For δ = log n log log n, P � min i∈[n] d∗ i ≥ δ � ≥1 − n1−η2(g,ρ)−η2(h,ρ)+o(1) − n1−η2(g,1−ρ)−η2(h,1−ρ)+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The proof is achieved by applying the Chernoff bound and taking the union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Similar to the proof of Theorem 8, using Lemma 17, it can be shown that S∗ ⪰ 0 and λ2(S∗) > 0 with probability 1 − o(1) if min � η2(g, ρ) + η2(h, ρ), η2(g, 1 − ρ) + η2(h, 1 − ρ) � > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' To prove the second part, we start to find when the maxi- mum likelihood estimator fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' To this end, let e(i, H) ≜ � j∈H TAij + T2A2 ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The definition of events F1, F2, E1, and E2 in Theorem 7 are used to show that with high probability P(F1) → 1 and P(F2) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Also, the definitions for C∗ 1, C∗ 2, and T remain valid for this part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' We prove that P(F1) → 1, while P(F2) → 1 is proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' To show that P(F1) → 1, we must have P(E1) → 1 and P(E2) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' It can be shown that P(E1) ≥ 1 − n−1+o(1) without difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Let E′ 2 ≜ � e(i, C∗ 1 \\ T ) − e(i, C∗ 2) ≤ −δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Then P(E′ 2) ≥n−η2(g,ρ)−η2(h,ρ)+o(1) + n−η2(g,1−ρ)−η2(h,1−ρ)+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' The proof is achieved by applying Lemma 8 and the Chernoff bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfvQga/content/2301.04088v1.pdf'} +page_content=' Applying Lemma 18 yields P(E2) = 1 − � i∈T [1 − P(E′ 2)] ≥ 1 − e−n−η2(g,ρ)−η2(h,ρ)+o(1)−n−η2(g,1−ρ)−η2(h,1−ρ)+o(1).' metadata={'source': 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A. Hollands,1★ S. Stopkowicz,2,3,4 M.-P. Kitsaras,3 F. Hampe,3 S. Blaschke,3 and J.J. Hermes5 +1 Department of Physics and Astronomy, University of Sheffield, Sheffield, S3 7RH, UK +2 Fachrichtung Chemie, Universität des Saarlandes, D-66123 Saarbrücken, Germany +3 Department Chemie, Johannes Gutenberg-Universität Mainz, Duesbergweg 10-14, D-55128 Mainz, Germany +4 Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway +5 Department of Astronomy & Institute for Astrophysical Research, Boston University, 725 Commonwealth Ave., Boston, MA 02215, USA +Accepted 2023 January 11. Received 2023 January 11; in original form 2022 November 28 +ABSTRACT +Magnetic white dwarfs with field strengths below 10 MG are easy to recognise since the Zeeman splitting of spectral lines appears +proportional to the magnetic field strength. For fields ≳ 100 MG, however, transition wavelengths become chaotic, requiring +quantum-chemical predictions of wavelengths and oscillator strengths with a non-perturbative treatment of the magnetic field. +While highly accurate calculations have previously been performed for hydrogen and helium, the variational techniques employed +become computationally intractable for systems with more than three to four electrons. Modern computational techniques, such as +finite-field coupled-cluster theory, allow the calculation of many-electron systems in arbitrarily strong magnetic fields. Because +around 25 percent of white dwarfs have metal lines in their spectra, and some of those are also magnetic, the possibility +arises for some metals to be observed in very strong magnetic fields, resulting in unrecognisable spectra. We have identified +SDSS J114333.48+661531.83 as a magnetic DZ white dwarf, with a spectrum exhibiting many unusually shaped lines at unknown +wavelengths. Using atomic data calculated from computational finite-field coupled-cluster methods, we have identified some of +these lines arising from Na, Mg, and Ca. Surprisingly, we find a relatively low field strength of 30 MG, where the large number +of overlapping lines from different elements make the spectrum challenging to interpret at a much lower field strength than for +DAs and DBs. Finally we model the field structure of SDSS J1143+6615 finding the data are consistent with an offset dipole. +Key words: white dwarfs – stars: magnetic field – atomic data +1 INTRODUCTION +The first magnetic white dwarf was discovered by Kemp et al. (1970), +through the detection of circularly polarised light from GJ 742. Since +then, many hundreds of magnetic white dwarfs have been discovered +(Kawka et al. 2007; Kepler et al. 2013), with observed fields strengths +spanning a few 10 kG up to about 1000 MG. For fields ranging be- +tween a few 100 kG to a few 10 MG, magnetic DA white dwarfs (i.e. +those with spectra dominated by hydrogen absorption lines) are easy +to identify in intensity spectra and their field strengths are simple to +measure, as many hydrogen lines split into three components, where +the degree of splitting is proportional to field strength. For smaller +fields, where such splitting is unresolved, spectropolarimetry can be +used instead (Bagnulo & Landstreet 2018, 2019, 2021; Landstreet +& Bagnulo 2019). However, due to reduced throughput, spectropo- +larimetry is limited to only the brightest white dwarfs. +For higher fields, particularly those beyond 100 MG, identification +is often still straightforward, though measuring the field strength is +no-longer trivial. The diamagnetic term in the Hamiltonian of the +hydrogen atom (Wickramasinghe & Ferrario 2000) (resulting in the +quadratic Zeeman effect due to its 𝐵2 dependence), quickly exceeds +the interaction strength of the paramagnetic term (linear Zeeman +★ E-mail: m.hollands@sheffield.ac.uk +effect), and eventually even the electrostatic potential. This results +in large shifts in wavelength, which ostensibly appear chaotic in +their field strength dependence. Due to the 𝑛4 dependence on the +quadratic Zeeman effect (where 𝑛 is the principle quantum number, +Wickramasinghe & Ferrario 2000), the shifts are first observed in the +higher order Balmer lines, but beyond a few 10 MG also causes the +wavelengths of the H𝛼 components to become chaotic. Because the +size of the diamagnetic term in the Hamiltonian becomes comparable +to the other terms, and overall the magnetic field is no longer a +small perturbation to the system, the energies (and hence transition +wavelengths), cannot be determined using perturbation theory, and +instead must be determined numerically. +For hydrogen, the first detailed atomistic calculations were per- +formed in the 1980s (Roesner et al. 1984; Forster et al. 1984; Henry +& O’Connell 1985; Wunner 1987). The results of these calcula- +tions quickly found application to assignment of lines in strongly +magnetic white dwarf spectra (Greenstein et al. 1985; Angel et al. +1985; Schmidt et al. 1986). More recent calculations have refined +the atomic data for hydrogen in strong fields (Schimeczek & Wunner +2014b; Schimeczek & Wunner 2014a). +Even at these early stages, however, the magnetic white dwarf +GD 229 was found to defy assignment of hydrogen spectral lines, +leading to speculation that it may instead have a helium dominated +atmosphere (Green & Liebert 1981; Schmidt et al. 1990, 1996). This +© 2022 The Authors +arXiv:2301.04665v1 [astro-ph.SR] 11 Jan 2023 + +2 +M. A. Hollands et al. +hypothesis was proved correct when the first calculations of He i by +Jordan et al. (1998) were matched to lines in the spectrum of GD 229, +implying a surface field varying between 300–700 MG. The calcu- +lations themselves relied on finite-field full configuration interaction +(ff-FCI) theory, a variational technique providing near-exact solu- +tions to the time-independent electronic Schrödinger equation. Such +a description is needed due to electron-electron repulsion term in the +Hamiltonian. Similar calculations for He i were also been performed +by Becken et al. (1999). +Calculations using variational approaches have been performed +for systems with more electrons such as Li i (Zhao 2018), however +for systems with more than three to four electrons, ff-FCI becomes +numerically intractable due to the factorial scaling in computation +time. +Fortunately, while white dwarfs with heavy elements in their at- +mospheres have been known for more than a century, those with +magnetic fields have hitherto not been observed with field strengths +exceeding ∼ 10 MG, where atoms are safely in the Paschen-Back +regime. White dwarfs with heavier elements fall into two main +classes: the DQs containing spectral features from carbon, and the +DZs containing features from heavier metals (Sion et al. 1983) such +as calcium and iron. +DQ white dwarfs, those with spectral features from carbon in their +atmospheres (detected from C2 Swan bands at low 𝑇eff and C i/ii at +higher 𝑇eff) are generally understood to originate from convective +dredge up of carbon from the core into the surrounding helium en- +velope (Fontaine et al. 1984; Pelletier et al. 1986; MacDonald et al. +1998), though a separate population of massive DQs are thought to +originate as the product of mergers (Dufour et al. 2007; Dunlap & +Clemens 2015; Williams et al. 2016; Kawka et al. 2020; Hollands +et al. 2020). Of these hot suspected merged DQs, a moderate frac- +tion are also magnetic, showing Zeeman split C i/ii lines – some +with field strengths of a few MG (e.g. Dufour et al. 2008). At lower +𝑇eff some peculiar DQs (such as LHS 2229) show highly distorted +and shifted Swan bands which have previously been hypothesised to +arise from strong (100s of MG) magnetic fields. However, Kowal- +ski (2010) demonstrated that the distorted molecular bands primar- +ily result from pressure-effects occurring in high-density, low 𝑇eff, +helium-dominated white dwarf atmospheres. To date, no predictions +for the wavelengths of atomic or molecular carbon transitions in +strong magnetic fields have been performed. +White dwarfs with metals in their atmospheres are denoted with +a Z in their spectral type, e.g. DAZ, DBZ, or DZ, depending which +other lines are visible in their spectra. DZs specifically (the subject +of this work) usually have helium dominated atmospheres, though +are too cool to exhibit He i lines (𝑇eff < 11, 000 K), although for +𝑇eff < 5000 K hydrogen lines are also diminished in strength, and so +in some cases hydrogen atmosphere white dwarfs can also be classed +DZ. Unlike the carbon in DQs, the metals observed in DZs (and +DAZs/DBZs etc.) require an external source, as gravitational settling +should deplete white dwarf atmospheres of metals on timescales that +are always much shorter than white dwarf ages (Paquette et al. 1986) +– specifically in the case of cool DZs, sinking timescales are on the +order of 106–7 yr, whereas their ages range from 109–10 yr (see Wyatt +et al. 2014, Figure 1). +A vast array of evidence now supports accretion of exoplanetesi- +mals from an accompanying planetary system as the source of this +metal pollution. Many metal-rich white dwarfs are observed with +infra-red excesses resulting from circumstellar debris disks (Zucker- +man & Becklin 1987; Jura 2003; Rocchetto et al. 2015; Swan et al. +2019a), with a sub-population of those also exhibiting gaseous emis- +sion from the sublimated part of the disk (Gänsicke et al. 2006, 2007; +Dennihy et al. 2016; Manser et al. 2020, 2021). In a few cases, when +the disk is viewed edge-on, irregular transits are observed demon- +strating the tidal disruption of exoplanetesimals close to the white +dwarf Roche radius (Vanderburg et al. 2015; Vanderbosch et al. 2020, +2021; Guidry et al. 2021; Farihi et al. 2022). In two cases the pres- +ence of planets themselves has been directly inferred, firstly from the +accretion of an evaporating gas giant by WD J0914+1914 (Gänsicke +et al. 2019), and secondly from planetary transits at WD 1856+534 +(Vanderburg et al. 2020). Despite these various sources of evidence +for white dwarf planetary systems, white dwarf spectra containing +metal lines remains the most common observable, and can be used to +infer the composition of the accreted exoplanetesimals (Zuckerman +et al. 2007; Klein et al. 2010; Gänsicke et al. 2012; Dufour et al. 2012; +Farihi et al. 2013; Xu et al. 2014; Wilson et al. 2015; Hollands et al. +2017, 2018b; Blouin et al. 2019; Doyle et al. 2019; Swan et al. 2019b; +Hoskin et al. 2020; Izquierdo et al. 2021; Hollands et al. 2022). A +sub-population of DZs have also been found to exhibit magnetism. +The first discovered magnetic DZ (spectral type DZH) was +LHS 2534 (Reid et al. 2001), which was found to have a 1.9 MG +field strength from Zeeman split lines of Na i, Mg i, and blended +Zeeman components from Ca i/ii. The field strength of LHS 2534 +was recently revised to 2.1 MG by Hollands et al. (2021) along with +the detection of Zeeman splitting of Li i and K i. Since this initial +discovery, additional DZHs were identified by Schmidt et al. (2003) +and Dufour et al. (2006) (WD 0155+003 and G 165−7, respectively). +With the advent of data release 10 (DR10) of the Sloan Digital Sky +Survey (SDSS), Hollands et al. (2015) identified a further seven ob- +jects, bringing the known sample to ten, and finding a high magnetic +incidence of 13±4 percent for DZs. With SDSS DR12, Hollands et al. +(2017) measured the fields of an additional 15 DZs1, with the range +of surface averaged field strengths, 𝐵𝑠, spanning 0.57 ± 0.04 MG to +10.70 ± 0.07 MG. Like LHS 2534, most of these DZs were identi- +fied from Zeeman triplets arising from the Na i resonance doublet +(𝜆 ≃ 5890 Å), and the Mg i triplet (𝜆 ≃ 5180 Å). Several magnetic +DAZ white dwarfs have also been identified, i.e. those with hydrogen +dominated atmospheres, though their field strengths are typically be- +low 1 MG (Kawka & Vennes 2011; Farihi et al. 2011; Zuckerman +et al. 2011; Kawka & Vennes 2014; Kawka et al. 2019). With none of +the objects published so far demonstrating fields exceeding 11 MG, +calculations of metals in ultra-strong magnetic fields have thus far +not been essential for the analysis of DZH spectra. +In this work we investigate SDSS J114333.48+661531.83 (here- +after SDSS J1143+6615), a faint (𝐺=20.1 mag) magnetic DZ white +dwarf with a peculiar spectrum with a sufficiently strong magnetic +field that spectral features are almost entirely unrecognisable. In +Section 2 we present our observations as well as public data on +SDSS J1143+6615. In Section 3 we discuss our finite-field coupled- +cluster calculations for metals in strong magnetic fields. In Section 4, +we make use of our atomic data calculations to identify the spec- +tral lines of SDSS J1143+6615 while simultaneously measuring the +strength of its magnetic field. In Section 5, we attempt to model the +field structure of SDSS J1143+6615, while in Section 6 we discuss +the applicability of our atomic data to higher field strengths and use +in model atmospheres, with our conclusions presented in Section 7. +1 Note that the thesis of Hollands (2017) identified a further seven low-field +magnetic objects in the Hollands et al. (2017) DZ sample, with field strengths +between 250 ± 30 kG to 510 ± 40 kG. +MNRAS 000, 1–14 (2022) + +A DZ white dwarf with a 30 MG magnetic field +3 +4000 +4500 +5000 +5500 +6000 +6500 +Wavelength [˚A] +0 +2 +4 +6 +8 +Flux [×10−17 erg s−1 cm−2 ˚A +−1] +SDSS BOSS +Gemini GMOS +Ca i +Mg i +Na i +Figure 1. SDSS BOSS and Gemini GMOS spectra of SDSS J1143+6615 (𝐺=20.1 mag). The SDSS spectrum is shifted upwards by 4 × 10−17 erg s−1 cm−2 Å−1. +Behind the Gemini spectrum, we show the SDSS spectrum again (light grey), but convolved to a resolving power of 𝑅 = 1100 for direct comparison, +demonstrating the virtually unchanged spectrum over two years. The zero-field air wavelengths of Ca i, Mg i, and Na i are shown by the solid vertical lines. +2 OBSERVATIONS +2.1 SDSS +SDSS J1143+6615 was originally observed in SDSS using the BOSS +spectrograph (Baryon Oscillation Spectroscopic Survey), first pub- +lished in SDSS Data Release 12 (plate-MJD-fiberID 7114-56748- +0973). The SDSS spectrum is shown at the top of Figure 1. This +spectrum was first classified as a candidate DZH white dwarf by +Kepler et al. (2016). This object also appeared in the DZ sample of +Hollands et al. (2017), where it was suggested to have a magnetic +field exceeding 20 MG. +The overall slope of the spectrum appears consistent with a cool +white dwarf with effective temperature (𝑇eff) in the range 5000– +7000 K, but is otherwise highly unusual, exhibiting a myriad of +unidentified features. In particular several bands of broad features +are seen near 4700 Å, 5500 Å, and 6400 Å. However, two sharper +absorption features stand out as resembling atomic lines. One of +these appears at about 5890 Å, and so could be from the Na i-D +resonance doublet (which in the absence of a magnetic field would +appear blended here). The other sharp feature is located at ≃ 5125 Å, +and due to its asymmetry resembles the Mg i-b triplet which is com- +monly observed in cool DZ white dwarfs where the asymmetry arises +from neutral broadening by helium atoms in a dense, helium dom- +inated atmosphere (Allard et al. 2016; Hollands et al. 2017; Blouin +2020). However, while the asymmetry appears qualitatively similar, +the wavelength is bluer by about 50 Å than should be the case for the +Mg triplet. While the SDSS spectrum does extend to 10, 400 Å, we +see no evidence for other absorption features beyond what is shown +in Figure 1. With none of the spectral features firmly identified, we +speculated that SDSS J1143+6615 is a strongly magnetic DZ white +dwarf, where the quadratic Zeeman effect is no longer negligible, +causing additional shifts of Zeeman-split spectral lines, and result- +ing in the appearance of many unidentified features in the spectrum. +The SDSS spectrum itself is composed of four sub-spectra, each +taken with 900 s exposure times. While these individual spectra +are extremely noisy, owing to the faintness of SDSS J1143+6615, +smoothing the data and down-sampling hinted at possible variability +between exposures. Because magnetic white dwarfs are known to +have rotation periods of minutes to days (Brinkworth et al. 2013; +Kilic et al. 2021), we considered the possibility of spectral line +shapes/positions evolving with rotational phase. We therefore sought +to obtain higher quality spectra of SDSS J1143+6615 in order to +confirm this rotation, as well potentially identify spectral lines. +2.2 Gemini +We obtained additional spectra using the GMOS (Gemini Multi Ob- +ject Spectrograph) instrument on the Gemini North telescope on +April 1st 2016 (exactly two years after the SDSS spectrum was +taken). The instrumental setup used the B600_G5307 grating with a +0.75 arcsec slit, giving us a resolving power of about 1100 at 4600 Å. +In total we took 17 exposures lasting 628 s each, separated by 15 s of +MNRAS 000, 1–14 (2022) + +4 +M. A. Hollands et al. +readout time. The GMOS detector uses three CCDs which covered +4100–7000 Å with our instrumental setup. This results in two ≃ 25 Å +gaps between each CCD with no spectral coverage, though these did +not cover any important features identified from the SDSS spectrum +(Figure 1). +We reduced the GMOS spectra using the starlink distribution +of software for bias-subtraction, flat-fielding, and optimal-extraction +(Horne 1986; Marsh 1989) of the spectral trace. Wavelength- +calibration was performed using molly2. For flux-calibration, we +initially used our observed flux standard, EG 131, but found this gave +unsatisfactory results, since it was observed at the end of the night, +whereas our science observations were observed at the start. We in- +stead made use of the SDSS spectrum from Section 2.1, as the SDSS +flux calibration are typically accurate to 1 percent. For each chip we +took the ratio of the spectra (in units of counts) and the already flux- +calibrated SDSS spectrum, re-binned onto the same wavelengths as +the GMOS spectra. We then fitted third-order polynomials to these +ratios to define a calibration function, which we then used to re- +scale the Gemini spectra into flux units. Note that fluxes redwards +of 6700 Å were dominated by telluric absorption and so data beyond +this wavelength were ignored and are not shown in Figure 1. +Our initial goal for these time-resolved spectra was to search for +variability, which may arise from rotation of a magnetic white dwarf, +bringing different parts of the magnetic field structure into view, and +thus causing Zeeman components to change in shape and wavelength. +We show the trailed, normalised Gemini spectra in Figure 2 for chip- +2 of GMOS. This chip has the largest spectral signal-to-noise ratio +(S/N), and contains many of the unassigned spectral features, includ- +ing the proposed Mg i and Na i lines. In the bottom panel, we show a +zoom-in of the suggested Mg i line, which because of the large shift +from the rest-wavelength, should be particularly sensitive to changes +in the magnetic field (if it is indeed Mg). We do not detect variability +in any of the spectral features, suggesting a lack of rotation on time +scales of a few hours. +Given the lack of variability between our 17 spectra, we chose to +co-add these into a single high S/N spectrum. We show this in the +bottom of Figure 1 (dark grey). This is compared with the SDSS +spectrum (light grey) which has been convolved to the same spectral +resolution as our Gemini data. Almost all features appear unchanged, +with perhaps only minor differences in the core strengths of the +5400 Å and 5500 Å features, and a slight change in wavelength of the +feature at 4650 Å. This comparison demonstrates a lack of variability +on a time scale of two years. +With the higher S/N spectrum, the proposed Na i and Mg i lines +are seen to be blue shifted by 5.5 Å and 52 Å respectively. The asym- +metric nature of the latter (discussed in Section 2.1), is also much +clearer. For the proposed Na i line this could be plausibly explained +as a ≃ 300km s−1 blue shift (not including any gravitational redshift +from the white dwarf) if SDSS J1143+6615 is a halo object. That be- +ing said, the much slower 18±2 km s−1 tangential velocity from Gaia +EDR3 (see Section 2.3) argues against this explanation. Furthermore, +such an explanation is effectively ruled out by the proposed Mg i line, +since its much larger wavelength shift would correspond to a veloc- +ity shift of about 3000 km s−1. Therefore magnetism remains a more +likely hypothesis for explaining the spectrum of SDSS J1143+6615. +In addition to the lines observed from the SDSS spectrum, the Gem- +ini spectrum also reveals the possible presence of the Ca i resonance +line (Figure 1, purple), with a small blue shift of 1.6 Å. +2 The software molly can be found at https://cygnus.astro.warwick. +ac.uk/phsaap/software/ +5200 +5400 +5600 +5800 +Wavelength [˚A] +0 +50 +100 +150 +Time [min] +5080 +5100 +5120 +5140 +5160 +Wavelength [˚A] +0 +50 +100 +150 +Time [min] +Figure 2. Trailed continuum-normalised spectra for our Gemini observations +of SDSS J1143+6615. The top panel shows the entirety of chip-2, which +contains both of the sharp features suggested to be from Mg and Na. The +bottom panel shows a Zoom-in of the suggested Mg line, demonstrating an +absence of spectral variability on a 3 hr timescale. +2.3 Gaia +Despite its curious spectrum containing many anomalous features +precluding obvious spectroscopic classification, the measured non- +zero proper-motion by SDSS confirms that SDSS J1143+6615 is a +galactic object. However, without knowing the absolute brightness +of this star, SDSS J1143+6615 could not be claimed to be a white +dwarf with certainty. +In April 2018, the second data release (DR2) from Gaia space +mission made public approximately 1 billion parallaxes (Gaia Col- +laboration et al. 2018). This included SDSS J1143+6615 which had +a measured parallax of 7.79 ± 0.68 mas, confirming this stars loca- +tion along the white dwarf cooling track within the Hertzsprung- +Russel diagram (HR-diagram). In December 2020 a refined par- +allax of 7.24 ± 0.46 mas was made available from Gaia EDR3 +(early data release 3, Gaia Collaboration et al. 2021) correspond- +ing to a distance of 138.8 ± 9.0 pc. The EDR3 HR-diagram is +shown in Figure 3. SDSS J1143+6615 is indicated by the red +point, and is compared against a background of white dwarfs se- +lected from Gentile Fusillo et al. (2021) with PWD > 0.75 and +parallax_over_error > 20. From its location in the HR-diagram, +MNRAS 000, 1–14 (2022) + +A DZ white dwarf with a 30 MG magnetic field +5 +−0.5 +0.0 +0.5 +1.0 +1.5 +GBP − GRP [mag] +8 +10 +12 +14 +16 +Gabs. [mag] +Figure 3. Gaia EDR3 Hertzsprung-Russel diagram showing the location of +SDSS J1143+6615 (red) compared with the white dwarf cooling sequence +(grey histogram). The error bars represent 1𝜎 uncertainties. +it is clear that SDSS J1143+6615 is a cool white dwarf with a +typical mass. Therefore Gentile Fusillo et al. (2021) found 𝑇eff = +5810 ± 460 K and log 𝑔 = 8.17 ± 0.33 fitting the Gaia photometry +with pure hydrogen atmosphere models, and 𝑇eff = 5680 ± 470 K +and log 𝑔 = 8.08 ± 0.33 for a pure helium atmosphere mod- +els. Interestingly, if Figure 3 is recreated using Gaia DR2 data, +SDSS J1143+6615 appears to be offset from the white warf sequence +towards higher masses, with Gentile Fusillo et al. (2019) finding +𝑇eff = 6990 ± 710 K and log 𝑔 = 8.73 ± 0.29 for hydrogen atmo- +sphere models, and 𝑇eff = 6870 ± 750 K and log 𝑔 = 8.67 ± 0.33 for +helium atmosphere models. That said, these parameter shifts amount +to only 1.4𝜎 changes at most and so are in statistical agreement. +3 ATOMIC DATA CALCULATIONS +To test our hypothesis that SDSS J1143+6615 is a highly magnetic +DZ white dwarf, we required accurate wavelengths of (at the very +least) the Na i and Mg i lines as a function of the magnetic field. For +large magnetic field strengths, however, approaches that are based on +a perturbative treatment of the magnetic field are no longer adequate +and hence accurate finite-field quantum-chemical methods need to be +employed. In these methods, the magnetic field is treated explicitly +in the calculation of ground-state energies, excitation energies, and +transition-dipole moments, thereby employing the electronic Hamil- +tonian for an 𝑁-electron system in a static magnetic field in the +𝑧-direction (the gauge-origin is here in the origin of the coordinate +system) +ˆ𝐻 = ˆ𝐻0 + 1 +2 𝐵 ˆ𝐿𝑧 + 𝐵 ˆ𝑆𝑧 + 1 +8 𝐵2 +𝑁 +∑︁ +𝑖 +(𝑥2 +𝑖 + 𝑦2 +𝑖 ), +(1) +where 𝐵 is the magnetic-field strength and ˆ𝐻0 is the field-free atom- +istic (or molecular) Hamiltonian, consisting of the kinetic energy +of the electrons, the nuclear-electronic potential and the electron- +electron repulsion. ˆ𝐿𝑧 = �𝑁 +𝑖 +ˆ𝑙𝑖,𝑧 and ˆ𝑆𝑧 = �𝑁 +𝑖 +ˆ𝑠𝑖,𝑧 are the 𝑧- +components of the angular momentum operator, and spin, respec- +tively. The terms linear in the magnetic field are the orbital-Zeeman +(responsible for the splitting of the orbitals) and spin-Zeeman terms +(responsible for splitting according to spin parallel or antiparallel to +the magnetic field), respectively. The quadratic term is referred to as +diamagnetic contribution which always increases the energy of the +system. As in the field-free case in quantum chemistry, FCI theory is +not applicable for problems like ours due to its high computational +cost. Instead, Coupled-Cluster (CC) theory (Shavitt & Bartlett 2009) +can be used, which has a more economical computational scaling. +CC methods work with an exponential parametrization of the wave +function ΨCC = e ˆ𝑇 Φ0, where ˆ𝑇 = ˆ𝑇1 + ˆ𝑇2 + · · · + ˆ𝑇𝑁 is the so- +called cluster operator generating excitations. ˆ𝑇 contains amplitudes +(weighting coefficients in the wave functions) that are determined by +solving the CC equations +⟨Φ𝐼 | e− ˆ𝑇 ˆ𝐻e ˆ𝑇 | Φ0⟩ = 0. +(2) +The CC energy is then given as +𝐸CC = ⟨Φ0 | e− ˆ𝑇 ˆ𝐻e ˆ𝑇 | Φ0⟩. +(3) +Truncations in ˆ𝑇 as well as limiting the projection space define ap- +proximate CC schemes. For example, CC ‘singles doubles’ (CCSD) +is defined with +ˆ𝑇CCSD = ˆ𝑇1 + ˆ𝑇2 +and projection on singly and doubly excited determinants. Analo- +gously, in CC ‘singles doubles triples’ (CCSDT), ˆ𝑇 is truncated to +ˆ𝑇CCSDT = ˆ𝑇1 + ˆ𝑇2 + ˆ𝑇3 +and projection is additionally also performed on triply excited de- +terminants. While CC is used to describe the ground-state wave +function, Equation-of-Motion-CC (EOM-CC) (Shavitt & Bartlett +2009) can also describe electronically excited states (EE). An op- +erator ˆ𝑅, parametrized similarly as ˆ𝑇 acts on a CC wave function +Ψexc = ˆ𝑅ΨCC. The corresponding amplitudes are determined via the +solution of the eigenvalue problem in matrix form +¯Hr = ΔEexcr +(4) +in which an element of the matrix ¯H is given as +¯𝐻𝐼 𝐽 = ⟨Φ𝐼 | e− ˆ𝑇 ( ˆ𝐻 − 𝐸CC)e ˆ𝑇 | Φ𝐽 ⟩ +(5) +and the vector r contains the amplitudes for the excitations. An +overview of ff-CC and ff-EOM-CC methods can be found in Stop- +kowicz (2017). In this work, we have used various flavors of ff-CC +theory (Stopkowicz et al. 2015; Kitsaras & Stopkowicz 2022a) and +ff-EOM CC theory, implemented within the QCUMBRE program +package (Hampe & Stopkowicz 2017), to determine excited states and +hence transition wavelengths (Hampe & Stopkowicz 2017; Hampe +et al. 2020; Kitsaras & Stopkowicz 2022a). The underlying calcula- +tion of the reference |Φ0⟩ is performed with the CFOUR program +package (Matthews et al. 2020). In the EOM-framework, we have +employed the methods for electronic excitations (EE), spin flip (SF), +adding electrons (EA, electron attachment) and removal of electrons +(IP, ionization potential). Oscillator strengths are also treated at the +expectation value ff-EOM-CC level (Hampe & Stopkowicz 2019) +which enables the prediction of field-dependent intensities. The tran- +sitions for which we have performed ff-calculations are displayed +in Table 1. The data for Na has partly already been available in +MNRAS 000, 1–14 (2022) + +6 +M. A. Hollands et al. +Table 1. +Level information for the transitions we have performed ff- +calculations for. Wavelengths (air) correspond to field-free transitions, which +in the case of multiplets corresponds to the average wavelength given in the +NIST database (weighted by oscillator strength). +Ion +Wavelength [Å] +Lower state +Upper state +Na i +5892 +2𝑆𝑔 ([Ne]3𝑠) +2𝑃𝑢 ([Ne]3𝑝) +Mg i +5178 +3𝑃𝑢 ([Ne]3𝑠3𝑝) +3𝑆𝑔 ([Ne]3𝑠4𝑠) +Ca i +4227 +1𝑆𝑔 ([Ar]4𝑠2) +1𝑃𝑢 ([Ar]4𝑠4𝑝) +Ca i +6142 +3𝑃𝑢 ([Ar]4𝑠4𝑝) +3𝑆𝑔 ([Ar]4𝑠5𝑠) +Ca ii +3945 +2𝑆𝑔 ([Ar]4𝑠) +2𝑃𝑢 ([Ar]4𝑝) +Hampe et al. (2020). The latter work is also the basis for the com- +putational protocol. We will here only mention the most important +points and refer to Hampe et al. (2020) for further details. For all tran- +sitions, the calculations were performed for magnetic fields ranging +between 0.00–0.04 B0, with the atomic unit of the magnetic field, B0 +≃ 2350.518 MG, using a 0.004 B0 step and between 0.04–0.20 B0 +using a 0.02 B0 step. In the protocol, a corrected excitation energy is +computed according to +Δ𝐸corrected +exc += Δ𝐸exc + Δ𝐸basis + Δ𝐸triples, +(6) +where Δ𝐸exc is the excitation energy computed using a large uncon- +tracted augmented one-electron basis set. Δ𝐸basis is a term correcting +the one-electron basis-set error as described in Halkier et al. (1998) +by extrapolating a basis-set limit 𝐸∞ based on uncontracted basis sets +of the type aug-cc-pCVXZ (Kendall et al. 1992; Woon & Dunning, +Jr. 1995), abbreviated as aCXZ, where X is the cardinal number. It +is given as Δ𝐸basis = Δ𝐸∞ − Δ𝐸exc with +Δ𝐸∞ = Δ𝐸aCXZ +exc +𝑋3 − Δ𝐸aCYZ +exc +𝑌3 +𝑋3 − 𝑌3 +. +(7) +The Δ𝐸triples = 𝐸aCXZ +triples − 𝐸aCXZ +exc +correction accounts for the error +which stems from truncating the CC expansion and involves com- +putations at the ff-EOM-CCSDT (Hampe et al. 2020), ff-EOM-CC3 +(Kitsaras & Stopkowicz 2022a) or ff-EOM-CCSD(T)(a)* (Matthews +& Stanton 2016; Kitsaras & Stopkowicz 2022b) levels of theory +for 𝐸aCXZ +triples using a smaller basis set. The accuracy and cost is typ- +ically CCSDT (𝑂(𝑀8))> CC3 (𝑂(𝑀7))> CCSD(T)(a)* (𝑂(𝑀7)) +where 𝑀 is the number of basis functions. In the latter two, triple- +excitations are treated in a perturbative manner. CC3 is iterative while +CCSD(T)(a)* is not. The latter is a very good and relatively cheap +option when the target-states are characterised mostly by single- +excitation character. The dimensionless oscillator strengths 𝑓𝐼 𝐽 were +calculated according to +𝑓𝐼 𝐽 = 2 +3 (Δ𝐸𝐼 𝐽 )|𝜇𝐼 𝐽 |2, +(8) +where Δ𝐸𝐼 𝐽 is the excitation energy from states 𝐼 to 𝐽 and 𝜇𝐼 𝐽 is +the corresponding transition-dipole moment, and where both Δ𝐸𝐼 𝐽 +and 𝜇𝐼 𝐽 are in atomic units. After converting the (field-dependent) +excitation energies to transition wavelengths, the resulting 𝐵 − 𝜆 +curves were shifted to start at the zero-field values taken from the +NIST database (Kramida et al. 2022) thereby correcting for remaining +errors of our predictions at zero field. The spin-orbit contributions +have been averaged out as their contribution is expected to be small for +stronger fields. By the shift made to the NIST data, field-free scalar- +relativistic effects are implicitly accounted for. For the time being, we +are neglecting relativistic effects and in particular their dependence +on the magnetic field in our calculations as the effects are expected to +be small for strong magnetic fields. This approximation is better for +the lighter elements Na and Mg than for the heavier Ca. The specific +details on the calculations are collected in Table 2. +The predicted transition wavelengths and oscillator strengths can +be found in Tables A1–A5. Additionally, the obtained 𝐵 − 𝜆 curves +are shown in Fig. 4. The intensity of the transitions, i.e. oscillator +strengths, are indicated via the opacities of the curves. As all of the +investigated transitions are of 𝑛𝑠 → 𝑛𝑝 or 𝑛𝑝 → (𝑛 + 1)𝑠 type, +where 𝑛 is the main quantum number of the orbital (without field), +there is in all cases a splitting into three components, i.e., the central +𝜋 (transition from/into a 𝑝0 orbital) as well as the two 𝜎 (transi- +tion from/into 𝑝+1 and 𝑝−1) components3. As can be seen here, the +splitting is only linear for fields below about 5–10 MG while for +higher field strengths, the form of the 𝐵 − 𝜆 curves becomes much +more complicated. The distortion from a simple Zeeman behaviour +is transition dependent: For the 𝑛𝑝 → (𝑛 + 1)𝑠 transitions (Mg and +Ca i 6142), the influence of the magnetic field on the central 𝜋 compo- +nent is much more pronounced than for the 𝑛𝑠 → 𝑛𝑝 transitions (Na, +Ca ii, Ca i 4227). The principal reason for this behaviour is that in +the former case the transitions are between orbitals of different main +quantum numbers. The orbitals hence experience a different amount +of polarisation through the magnetic field, i.e. those of higher main +quantum number are polarised more strongly due to their more dif- +fuse nature. Effectively this means that the 𝑠 and 𝑝0 orbitals and the +respective electronic states, don’t evolve in a parallel manner with +increasing magnetic field. Hence, in contrast to the simple pertur- +bative picture, the central 𝜋 component is no longer constant with +increasing magnetic field strength. In addition, the transitions with +decreasing energy difference in the magnetic field, i.e., 𝑛𝑠 → 𝑛𝑝−1 +and 𝑛𝑝+1 → (𝑛 + 1)𝑠 become less relevant for observations, as they +decrease in intensity (see Equation (8)). In addition, small changes in +the magnetic field lead to large changes in the transition wavelength +and hence such transitions will be blurred out in the spectra for strong +fields. A more detailed discussion on the form of the energy levels +and the resulting for of the 𝐵 − 𝜆 curve of the Mg transition can be +found in Kitsaras & Stopkowicz (2022a). As noted in Hampe et al. +(2020), high-accuracy predictions are required as even the prediction +for the transition least affected by the magnetic field, i.e., the central +𝜋 component of Na can vary by up to 100 Å depending on the level +of theory and basis set used.4 +4 LINE IDENTIFICATION +With the wavelengths and oscillator strengths calculated in Sec- +tion 3, we were able to compare these with the spectrum of +SDSS J1143+6615. With no immediate indication of which spec- +tral features could correspond to the 𝜎-components of the calculated +transitions, we began by restricting ourselves to the 𝜋-components +only. In Section 2 we identified possible 𝜋-components of Na i, Mg i, +and Ca i in the SDSS and GMOS spectra, based on the sharpness +of the lines, rough proximity in wavelengths to the field-free values, +and characteristic asymmetry in the case of Mg. +We compare these lines to our calculated wavelengths as a function +of field strength in the top panels of Figure 5. From the bottom-right +panel, it is clear that the Na line shift could be explained by either a +relatively small field of ≃ 30 MG or much larger field of ≃ 410 MG, +3 Note that in the magnetic field, the SO(3) symmetry is lowered to 𝐶∞ℎ but +we will, for simplicity, still refer to field-free state and orbital classifications. +4 Note that the uncertainty of the predicted transition wavelengths is not +only dependent on the accuracy of the method but also on the position of the +absorption peak. +MNRAS 000, 1–14 (2022) + +A DZ white dwarf with a 30 MG magnetic field +7 +Table 2. Detailed information on ff-EOM calculations for the respective transitions. If not specified otherwise, Δ𝐸𝐼 𝐽, see Eq. (8), has been calculated at the +same level as 𝜇𝐼 𝐽 +Transition +Basis functions +Δ𝐸exc +Δ𝐸basis +Δ𝐸triples +𝜇𝐼 𝐽 +Na i +Cartesian +EE-CCSD/aCQZ +EE-CCSD/aCXZ, X=T, Q +CCSDT/aCTZ +EE-CCSD/aCQZ +Mg i +Spherical +EE-CCSD/aC5Z +EE-CCSD/aCXZ, X=Q, 5 +CC3/aCQZ +EE-CCSD/aC5Z +Ca i 4227 +Spherical +EE-CCSD/aC5Z +EE-CCSD/aCXZ, X=Q, 5 +EE-CC3/aCQZ +EE-CCSD/aCQZ(𝑎) +Ca i 6142 +Spherical +SF-CCSD(T)(a)*/aC5Z +SF-CCSD(T)(a)*/aCXZ, X=Q, 5 +No further triples correction +SF-CCSD/aC5Z(𝑏) +Ca ii +Spherical +EA-CCSD/aC5Z +EA-CCSD/aCXZ, X=Q, 5 +EE-CCSD(T)(a)*/aCQZ +EA-CCSD/aC5Z(𝑐) +Notes: (𝑎) 𝐸𝐼 𝐽 calculated using EE-CC3, (𝑏) Reference for SF calculations: 1𝑆𝑔 ([Ar] 4𝑠2), (𝑐) Reference for EA calculations: 1𝑆𝑔([Ar]) +2000 +3000 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +Wavelength [˚A] +0 +100 +200 +300 +400 +B [MG] +Na i +Mg i +Ca i 4227 +Ca i 6142 +Ca ii +Figure 4. Calculated transition wavelengths as a function of field strength. For each Zeeman triplet, the line opacities are scaled to the oscillator strengths. +owing to a turnaround in wavelength at ≃ 240 MG. This degeneracy +is entirely resolved by the large shift of the Mg line which has only one +wavelength solution and is also consistent with a field of ≃ 30 MG. +Thus, to our surprise, the peculiar spectrum of SDSS J1143+6615 +(Figure 1) can not result from a field in the regime of 100s of MG, +but is best explained by a field strength an order of magnitude lower, +though notably still a factor three higher than all previously identified +DZH white dwarfs (Hollands et al. 2015, 2017; Dufour et al. 2015). +For Ca i the match in wavelength is quite poor, though thus far we +have neglected wavelength shifts that may arise from radial motion +and gravitational redshift, the latter of which could be on the order +of 100 km s−1 if SDSS J1143+6615 is particularly massive, which is +typically the case for magnetic white dwarfs (Liebert 1988; Kawka +et al. 2020; Ferrario et al. 2020). Additionally, the absent treatment of +relativistic effects may here play a role in the quality of the prediction. +It is also clear that at 30 MG, the predicted wavelength for Mg is +a similar amount bluer than the line centre (though with greater +relative accuracy). To account for this we fitted the field strength and +radial velocity simultaneously. We measured the line centres for all +three 𝜋-components by simply fitting parabolas to the central few +pixels (five for Ca and seven for Mg and Na), constraining them with +uncertainties of 0.1–0.3 Å. Performing a least squares fit to the three +line centres, we found a magnetic field strength of 29.92 ± 0.05 MG +and a redshift of 74 ± 8 km s−1. With these best fitting values the +residuals are −0.7 Å, 0.0 Å, and 1.8 Å for the Ca, Mg, and Na lines, +respectively. This was a clear improvement for Ca i and Mg i, though +provides a somewhat worse result for the Na i line. +With the field strength established from the 𝜋-components, we +could then determine the expected wavelengths of the 𝜎-components. +We make this comparison in Figure 6. We first investigated the com- +ponents of Na and Mg, with their 𝜎-components identified with +relative ease. In particular the large broad feature at ≃ 6350 Å is es- +tablished as the 𝜎+ component of Na, which does not appear blended +with any of the other nearby features. Near 5500 Å both the Na 𝜎− +and Mg 𝜎+ components are observed, though notably the order of +their wavelengths has swapped due to the components crossing at +a field strength of ≃ 25 MG. The Mg 𝜎− component is inferred to +be the broad, asymmetric feature at ≃ 4800 Å. The asymmetry ap- +MNRAS 000, 1–14 (2022) + +8 +M. A. Hollands et al. +0.0 +0.5 +1.0 +Normalised flux +Ca i +4220 +4240 +Wavelength [˚A] +101 +102 +B [MG] +Mg i +5050 +5100 +5150 +5200 +Wavelength [˚A] +Na i +5800 +5850 +5900 +5950 +Wavelength [˚A] +Figure 5. Top row: Spectral regions covering the suspected 𝜋-components of Ca i, Mg i, and Na i. Bottom row: Predicted wavelengths for the corresponding +𝜋-components as a function of field strength. In all panels the black dashed lines indicate the field-free vacuum wavelengths for each line, whereas the dotted +lines indicate the wavelengths expected for a 30 MG field. +pears more extreme than for the 𝜋-component, which itself is more +asymmetric than the 𝜎+ component. This may imply that the degree +of neutral broadening affects each component differently, which per- +haps is not surprising given that both the perturbations from neutral +helium atoms and the magnetic field both alter the energy levels of +Mg. +Having identified all components from Na i and Mg i, we pro- +ceeded with classifying transitions from Ca. For the Ca i resonance +line, we had already identified the 𝜋-component (rest wavelength at +4227 Å; see Figure 5, left). As our Gemini GMOS spectrum does +not go bluer than about 4090 Å, the 𝜎−-component is not covered, +and so we were only able to search for the 𝜎+ component which, +at 30 MG, has an expected wavelength of 4475 Å. Indeed, a spectral +feature was found at this wavelength which we attribute to the 𝜎+ +component (Figure 6). +The final Ca transitions are less certain, though we still make +some attempt at their classification. For the Ca ii Zeeman triplet +(H+K resonance doublet in the absence of an external magnetic +field), only the 𝜎+ component is expected to be covered by our +GMOS spectrum at a field strength of 30 MG. While we detect a +feature at the expected wavelength of 4160 Å (Figure 6), the signal- +to-noise ratio is somewhat poor at this end of the spectrum, making +this assignment less secure. However, it is worth noting that for a +𝑇eff between 5000 K and 7000 K, both Ca i and Ca ii resonance lines +are typically observed together in non-magnetic DZs (Hollands et al. +2017). +Finally we consider the Ca i 4𝑝 → 5𝑠 transition, which in the +absence of an external magnetic field appears as a triplet (due to +the spin-orbit interaction) centred on 6142 Å. In the presence of a +strong magnetic field, this transition appears as a Zeeman triplet ex- +hibiting the strongest quadratic shift of all the transitions calculated in +Section 3. Nevertheless, weak transitions are observed at all of the ex- +pected wavelengths. Whether this assignment is correct is debatable: +the identified central component at around 6060 Å shows some asym- +metry, as is observed in the field-free case (see SDSS J0916+2540 in +Figure 10 of Hollands et al. 2018a). On the other hand, the 6142 Å +triplet is typically much weaker than the Ca i 4227 Å resonance line, +and is only usually visible for extremely large calcium abundances. +Yet, in the case of SDSS J1143+6615, the established components of +the 4227 Å Ca i Zeeman triplet are not particularly strong, suggesting +that the 6142 Å components would likely be too weak to be visible. +Given the sheer number of unknown features in the spectrum of +SDSS J1143+6615, it is probable that our assignments to the 6142 Å +triplet in Figure 6 might also originate from some other source. +Many anomalous features in the spectrum of SDSS J1143+6615 +remain unclassified. In particular two strong and broad features are +observed at wavelengths of ≃ 4570 Å and ≃ 4660 Å, between the 𝜎+- +component of the Ca i resonance line, and the 𝜎−-component of Mg i. +The strength of these features suggest they originate from another +element commonly observed in DZ spectra. With the strongest Na, +Mg and Ca lines already accounted for, the most likely candidate is +therefore Fe. In the field-free case, a large number of Fe i lines can be +found between 4000–4500 Å (see Hollands et al. 2018a, Figure 7). +Among the strongest transitions in this range are the 3𝐹 → 5𝐺 and +3𝐹 → 3𝐺 multiplets, which share the same lower level. We therefore +suggest that the unidentified features at ≃ 4570 Å and ≃ 4660 Å arise +from these iron transitions. Additional unidentified features include +broad absorption around 4300 Å (between the 𝜋- and 𝜎+-components +MNRAS 000, 1–14 (2022) + +A DZ white dwarf with a 30 MG magnetic field +9 +4000 +4500 +5000 +5500 +6000 +6500 +Wavelength [˚A] +0 +10 +20 +30 +40 +50 +60 +B [MG] +Na i +Mg i +Ca i 4227 +Ca i 6142 +Ca ii +Figure 6. Line identification diagram for SDSS J1143+6615. The Zeeman triplets from our finite-field coupled-cluster calculations are shown by the solid +curves, with the naïve wavelengths from the linear Zeeman effect indicated by the dotted lines. These are plotted over the spectrum of SDSS J1143+6615 (grey), +where black dashed lines match Zeeman components to features in the spectrum for a field strength of approximately 30 MG (light grey horizontal band). +of Ca i), sharp features at ≃ 5200/5330/5580 Å, and several other +features at ≃ 6030/6450/6530/6620 Å (some of which we were +unable to conclusively assign to the Ca i 6142 Å multiplet). We note +that the feature near 5200 Å is close to the field-free wavelength +of the Cr i 4𝑠 → 4𝑝 triplet (5208 Å, vacuum), and so that feature +could plausibly correspond to the 𝜋-component of the Cr i transition. +Firmly establishing the origin of these remaining features necessarily +will require additional finite-field coupled-cluster calculations in the +future, with the above Fe and Cr transitions as the highest priority. +For these systems, treatment of field-dependent relativistic effects +and a robust treatment of multi-reference character in the electronic +structure will be important. +5 MAGNETIC FIELD MODELLING +With several of the spectral features of SDSS J1143+6615 identified, +we finally sought to model the magnetic field distribution across +its surface. For a purely dipolar magnetic field, the field strength +spans a factor of two between the equator and poles. This results +in broadened spectral lines, particularly the 𝜎-components due to +their stronger wavelength dependence of the field strength. It is clear +from the width of the Na i 𝜎+ component that the range of magnetic +field strengths on the visible hemisphere of SDSS J1143+6615 spans +a much narrower field range, with Figure 6 suggesting about 24– +31 MG. Thus it is necessary to invoke a field structure more complex +than a centred dipole. +5.1 The offset dipole model +We chose to use the offset-dipole model, which has been com- +monly used in the analysis of magnetic white dwarf field structures +(Achilleos & Wickramasinghe 1989). This model is similar to a +centred-dipole, but allows for the origin of the field to be shifted +within the white dwarf. In principle this shift can be applied in three +dimensions, but typically it is only applied along the magnetic field +axis by a fractional amount of the white dwarf radius, 𝑎𝑧. The offset- +dipole model has been successfully applied to many different white +dwarfs (Achilleos et al. 1992; Putney & Jordan 1995; Külebi et al. +2009; Hollands et al. 2015) leading to much improved fits with only +a single additional free-parameter, which is advantageous compared +to a more general multi-pole expansion. +For a centred-dipole with the magnetic field aligned with the 𝑧- +axis, the value of the magnetic field at any point on (and external to) +the stellar surface in Cartesian coordinates (𝑥/𝑦/𝑧) is given by, +������ +𝐵𝑥 +𝐵𝑦 +𝐵𝑧 +������ += 𝐵𝑑 +2𝑟5 +������ +3𝑥𝑧 +3𝑦𝑧 +3𝑧2 − 𝑟2 +������ +, +(9) +where 𝐵𝑥/𝑦/𝑧 are the Cartesian components of the magnetic field, +𝐵𝑑 is the polar field strength, and 𝑟2 = 𝑥2 + 𝑦2 + 𝑧2. The offset- +dipole model simply requires making the translation 𝑧 ↦→ 𝑧 − 𝑎𝑧, in +Equation (9) and in the definition of 𝑟2. To complete the offset-dipole +model we also allow rotation between the magnetic field axis and the +observer. We implement this by considering coordinate systems for +both the magnetic field and the viewing direction of the observer, +with a rotation matrix used to convert between them. +Using the above model of the white dwarf magnetic field struc- +MNRAS 000, 1–14 (2022) + +10 +M. A. Hollands et al. +ture, we construct a toy model spectrum by randomly sampling +10,000 points uniformly across the stellar disc (i.e. sampled uni- +formly within the unit circle). For each point on the stellar disc, +𝑖, we used Equation (9) to calculate the magnetic field vector (ac- +counting for the chosen inclination). Then for each transition, 𝑗, we +compute a Zeeman-triplet of three Lorentzian profiles, using our +atomic data from Section 3 to determine their wavelengths and oscil- +lator strengths. Furthermore, the 𝜋-component is scaled by a factor +sin2 𝜓/2, and the 𝜎-components by a factor (1 + cos2 𝜓)/4, which +accounts for linear and circular polarisation effects respectively (Put- +ney & Jordan 1995), and where 𝜓 is the angle between the field line +and the observer’s line of sight5. These three Lorentzian components +are then summed to form an opacity function +𝜅𝑖 𝑗 (𝜆; 𝐵𝑖, 𝜓𝑖) = ++1 +∑︁ +Δ𝑚𝑙=−1 +𝐿 𝑗 (𝜆; 𝐵𝑖, 𝜓𝑖, Δ𝑚𝑙), +(10) +where 𝐿 𝑗 are the Lorentzian profiles per transition. Finally, the nor- +malised flux for all transitions at point 𝑖 is given by +𝐹𝑖(𝜆; 𝐵𝑖, 𝜓𝑖) = exp +��� +��� +− +∑︁ +𝑗 +𝐴 𝑗𝜅𝑖 𝑗 (𝜆; 𝐵𝑖, 𝜓𝑖) +��� +��� +, +(11) +where 𝐴 𝑗 is a pseudo-abundance which we use to arbitrarily scale the +strength of each Zeeman-triplet. Finally, we compute the integrated +flux over the stellar disc as a weighted sum based on the centre-to- +limb intensity of the stellar disc +𝐹(𝜆) = +� +𝑖 𝐹𝑖(𝜆; 𝐵𝑖, 𝜓𝑖)𝐼(𝜇𝑖) +� +𝑖 𝐼(𝜇𝑖) +, +(12) +where 𝐼(𝜇𝑖) is the intensity across the stellar disc, and where 𝜇𝑖 is +equivalent to the 𝑧 coordinate of the 𝑖-th point on the stellar disc in the +observers frame of reference. We use the logarithmic limb-darkening +law for a 6000 K, log 𝑔 = 8 white dwarf from Gianninas et al. (2013). +5.2 Application to SDSS J1143+6615 +We applied the offset dipole model to SDSS J1143+6615 initially fo- +cusing on the Na triplet. From analysing the 𝜋-components of Mg and +Na in Section 4, we established a surface averaged field of ≃ 30 MG, +and hence located the features corresponding to the 𝜎-components. +Due to the asymmetry of the Mg components we decided to begin +our focus on the Na triplet. However, the 𝜎− component of Na and +the 𝜎+ component of Mg are somewhat overlapping (≃ 5500 Å), and +so we chose to restrict ourselves to the 𝜋 and 𝜎+ components of Na +(≃ 6400 Å). Overall we therefore had five parameters to adjust: the +polar field strength 𝐵𝑑, the dipole-inclination, and the dipole-offset +𝑎𝑧, which controlled the field distribution; plus the Lorentzian line +strength (𝐴 𝑗 in Section 5.1) and width which are most easily inferred +by the relatively static 𝜋-component. +As described at the start of Section 5, the width of the 𝜎+ compo- +nent of Na implies a field strength distribution narrower than the fac- +tor of two for a centred dipole. In the offset-dipole model a narrower +distribution can be achieved for negative values of 𝑎𝑧, combined with +a low inclination (i.e. viewed close to pole-on). This implies a wider +5 These oscillator strength scaling factors mean that when the observer looks +down a field line, the 𝜋-component vanishes and the 𝜎-components are at +maximum intensity, and when the observer looks perpendicular to a field line +the 𝜋-component is at maximum intensity with the 𝜎-components at half +intensity. In the absence of a magnetic field where all components overlap, +all three scaling factors sum to one for all angles of 𝜓. +distribution of field strengths on the opposite hemisphere of the star. +Because 𝐵𝑑 in Equation (9) no longer corresponds to the field at the +poles, for finite 𝑎𝑧, both parameters must be adjusted simultaneously +to maintain a polar field strength of 30 MG on the visible hemisphere. +Manipulating Equation 9, and making the substitution 𝑧 ↦→ 𝑧 − 𝑎𝑧, +it can be shown that +𝐵𝑑 = (1 − 𝑎𝑧)3𝐵𝑧=1, +(13) +where 𝐵𝑧=1 is the near-side pole strength of 30 MG. Adjusting these +parameters by hand6, we found good agreement with the shape of +the Na 𝜎+-component could be achieved with 𝑎𝑧 = −0.15 (imply- +ing 𝐵𝑑 = 45.6 MG from Equation (13)) and a dipole-inclination of +15 degrees (Figure 7). This also yields a reasonable agreement with +the 𝜎−-component (at wavelengths where it is not blended with the +𝜎+-component from Mg). We then included all other transitions from +Section 3 into the model adjusting only the strengths and widths of +the Lorentzian profiles. A further refinement is required for the Mg i +and Ca i 6142 Å triplets as these are 𝑛𝑝 → (𝑛 + 1)𝑠 transitions (the +others are all 𝑛𝑠 → 𝑛𝑝), and so we scale the component strengths +by Boltzmann factors reflecting the different occupation levels of the +lower states. +Unsurprisingly, the Lorentzian profiles used provide a poor fit +for the asymmetric 𝜋- and 𝜎−-components of Mg i, though reason- +able agreement is found for the 𝜎+-component. As discussed pre- +viously, this may indicate that the degree of neutral broadening is +field-dependent, and affects the bluer components more strongly. For +the Ca i 4227 Å resonance line, when the width and strength param- +eters are adjusted to match the 𝜋-component, the strength and shape +of the 𝜎+-component (≃ 4090 Å) also agree well with the observa- +tions. This demonstrates that the values of 𝐵𝑑, 𝑎𝑧, and the inclination +found from the Na lines are also appropriate for this transition. For +the Ca ii triplet, the width of the 𝜎+ component is also seen to be in +agreement with the data, though the signal-to-noise ratio in this part +of the spectrum is too poor to compare the shape of the line with the +data. Finally for the Ca i 6142 Å Zeeman-triplet, only the shape of +the 𝜎+-component in is reasonable agreement with the data, further- +ing the argument from Section 4 that these transitions may originate +from another source. +6 DISCUSSION +6.1 DZs with much stronger fields +In Section 5, we constructed a toy-model for generating simplified +magnetic DZ spectra, including atomic data from Section 3. While it +turned out that SDSS J1143+6615 has only a 30 MG field, in principle +our model allows us to generate synthetic spectra for much larger +fields, with 470 MG covering all the transitions we calculated in +Section 3. Since ongoing/upcoming spectroscopic surveys such as +WEAVE, DESI, SDSS V, and 4MOST, are expected to yield hundreds +of thousands of white dwarf spectra in the next decade, we investigate +which transitions ought to be focused on for identifying even higher +field DZ stars in the future. +In Figure 8, we show models with average surface fields span- +ning 25–400 MG against the same curves from Figure 4. For all five +models, we used the same inclination and dipole offset as found +for SDSS J1143+6615, i.e. 15 degrees and −0.15 respectively. Note +6 While we did attempt a more rigorous least-squares fit to the data, the lack +of a well-defined continuum led to worse results than manual adjustment of +the model parameters. +MNRAS 000, 1–14 (2022) + +A DZ white dwarf with a 30 MG magnetic field +11 +−1.5 +−1.0 +−0.5 +0.0 +0.5 +1.0 +1.5 +x +−1.5 +−1.0 +−0.5 +0.0 +0.5 +1.0 +1.5 +y +i = 15◦ +az = −0.15 +4500 +5000 +5500 +6000 +6500 +Wavelength [˚A] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Normalised flux +Bd = 45.6 MG +visible: 22.8 − 30.0 MG +Figure 7. Left: Visualisation of the field structure of SDSS J1143+6615 modeled with an offset-dipole. Right: The simulated absorption spectrum of +SDSS J1143+6615 (red) using data from our finite-field coupled cluster calculations. +that the 𝐵𝑠 values are the surface averaged field strengths whereas +the dipole field strength, 𝐵𝑑, is approximately 52 percent larger (see +Equation 13). +The bottom model has a field 𝐵𝑠 = 25 MG, similar in strength to +that found for SDSS J1143+6615, and thus shows a similar spectrum. +Despite the relatively uniform field for an inclination of 15 degrees +and 𝑎𝑧 = −0.15, as the field increases, the 𝜎-components still be- +come washed out, and for most of the transitions are almost invisible +at fields of around 100 MG and above. For Mg i the 𝜎+ component +still remains visible above 100 MG due to its increase in oscillator +strength. +On the other hand, most of the 𝜋-components remain relatively +steady in wavelength. For the Na 𝜋-component, as already noted +in Hampe et al. (2020), the wavelength changes very little below +100 MG, leaving this line similarly sharp as for a 25 MG field. The +Na line reaches a maximum in blue-shift at 240 MG (100 Å bluer +than the field free wavelength), before rapidly turning around and +moving to redder wavelengths. Therefore for 𝐵𝑠 = 400 MG, the line +becomes much broader, but remains clearly visible. Therefore this +transition ought to be used as a primary marker for identifying cool +magnetic DZ white dwarfs with > 100 MG fields. +Similarly the Ca i 𝜋-component remains relatively stationary up +to 100 MG, but becomes more washed out for larger fields due to +the quadratic Zeeman effect, and becoming broadened to a width of +100 Å at 400 MG. Therefore this line is likely to be less reliable than +the Na 𝜋-component for identifying the highest field DZs, but will +still remain reliable up to 200 MG. +The Ca ii 𝜋-component is also near stationary, and should still be +recognisable even at 400 MG, making this a more obvious choice +for identifying warmer high field DZs where the Na i and Ca i lines +may be too weak to identify. Note that at 300 MG, the Ca i and Ca ii +𝜋-components overlap producing a blended spectral feature. +Finally, the Mg i 𝜋-component experiences a much larger quadratic +shift than the other transitions considered here. Therefore at 400 MG, +the line appears broad and asymmetric though is notably still visible, +in part due to the increased oscillator strength for this component, +which is close to four times larger than in the field-free case, thereby +also showcasing the importance of considering field-dependent in- +tensities. Note that this toy-model does not consider the intrinsic +asymmetry caused by neutral broadening, which itself could be a +function of field strength. +A final consideration is that we have not yet identified all the fea- +tures in the spectrum of SDSS J1143+6615. Therefore at very high +field strengths of 100s of MG, these unclassified features will also ap- +pear shifted into other parts of the spectrum further complicating the +identification of the transitions discussed above. Furthermore, other +strong lines outside the optical such as the Mg i and Mg ii resonance +lines (field free wavelengths at 2853 Å and 2799 Å, respectively), +may find some of their Zeeman split components shifted into the +optical providing other atomic features requiring identification. +6.2 Use in model atmospheres +Ideally the atomic data we have presented in Section 3 can be utilised +in white dwarf model atmospheres for more detailed analyses of +magnetic DZ stars. As we have shown in this work, however, this +is not necessary for a basic assessment. For simply determining the +surface-average field strength, 𝐵𝑠, and which ions are present in +the atmosphere, it is sufficient to simply compare our atomic data +with the spectrum in question, as was demonstrated in Section 4 +for SDSS J1143+6615. Furthermore, determining the field structure +of a white dwarf can be achieved with a simple model such as the +toy-model we demonstrated in Section 5. Importantly our toy-model +is computationally efficient, taking only a few seconds to produce +Figure 7. +Of course, much can still be learned from incorporating our atomic +data into model atmospheres. In our toy-model from Section 5, the +strength and widths of the Lorentzian profiles we used have no phys- +ical basis, and are simply adjusted to give acceptable agreement with +the data. In a model atmosphere, the strengths and widths of the fea- +tures seen in the spectrum of SDSS J1143+6615 can be investigated +by adjusting the abundances and 𝑇eff (and to some extent the surface +gravity) of the model, allowing these atmospheric properties to be +measured in a physically meaningful way. +The main challenge of such an approach is the computation time +required. In the field-free case, the final model spectrum is inte- +grated over the stellar disc from spectra calculated at different angles +between the surface-normal and the observer. For finite-fields, how- +ever, the synthetic spectra must also be calculated over a grid of field +strengths and angles between the field and observer. In particular, +the field strength axis of the grid must be computed at sufficiently +fine steps so that artefacts from undersampling are not present when +MNRAS 000, 1–14 (2022) + +12 +M. A. Hollands et al. +4000 +5000 +6000 +7000 +8000 +Wavelength [˚A] +0 +1 +2 +3 +4 +5 +Normalised flux +Bs = 25 MG +Bs = 50 MG +Bs = 100 MG +Bs = 200 MG +Bs = 400 MG +1.2 +1.4 +1.6 +1.8 +2.0 +2.2 +2.4 +2.6 +log(B/MG) +Figure 8. Simulated magnetic DZ spectra for five different surface averaged field strengths (𝐵𝑠), with each spectrum offset from one another by 1 in normalised +flux. The inclination and dipole offset parameters are fixed to the values found for SDSS J1143+6615 (i.e. 15 degrees and −0.15, respectively). The background +Zeeman triplets have the same meaning as in Figure 4, with the field strength scale given on the right-hand axis. +integrating over the stellar disc. Therefore, depending on the range +of field strengths required, computation may take hundreds to thou- +sands of times longer than in the field-free case. If the 𝑇eff, log 𝑔, or +abundances require refinement when comparing against a particular +spectrum, the grid must then be recomputed with updated atmo- +spheric parameters, leading to an even larger amount of computation +time. +For that reason we have refrained from including our atomic data +within model atmospheres at the present time, and also because it +exceeds the scope of our primary goals of classifying the spectral +features of SDSS J1143+6615 and measuring its field strength. How- +ever, future work should perform a detailed atmospheric analysis of +SDSS J1143+6615 utilising the atomic data presented here to mea- +sure its 𝑇eff, log 𝑔, and abundances. +7 CONCLUSIONS +We have identified SDSS J1143+6615 as DZ white dwarf with strong +magnetic field resulting in its unique spectrum. Using finite-field, +coupled-cluster calculations we were able to identify lines from Na i, +Mg i, and Ca i–ii that were split and shifted by the linear and quadratic +Zeeman effects. This also allowed us to establish a field strength of +≃ 30 MG, demonstrating that DZ spectra are challenging to interpret +at only a few 10 MG, due to multiple overlapping transitions from a +variety of chemical elements, which is not the case for magnetic DAs +or DBs at the same field strength. Using the offset-dipole model, we +were able to obtain an adequate fit to the spectral features of Na with +an almost pole-on observation angle, and the dipole offset away from +the observer. +Despite our success in elucidating some of the peculiar features in +the spectrum of SDSS J1143+6615, many transitions still lack classi- +fication at the present time. Giving consideration to the elements and +lines most commonly encountered in non-magnetic cool DZ stars, +future atomic data calculations should concentrate on Fe and Cr lines, +as well as additional transitions of Ca. Because SDSS J1143+6615 +is currently the only available test for these calculations, and only +samples the relatively low-field end, searching for additional high- +field DZs within ongoing and future spectroscopic surveys (such as +SDSS V, WEAVE, and DESI) is imperative to test the accuracy of +our atomic data further at field strengths of many 100 MG. +MNRAS 000, 1–14 (2022) + +A DZ white dwarf with a 30 MG magnetic field +13 +ACKNOWLEDGEMENTS +MAH was supported by grant ST/V000853/1 from the Science and +Technology Facilities Council (STFC). S.S. acknowledges support +by the Deutsche Forschungsgemeinschaft (DFG) grant number STO +1239/1-1 S.S. and within project B5 of the TRR 146 (Project No. 233 +630 050). Based on observations obtained at the international Gemini +Observatory, a program of NSF’s NOIRLab, which is managed by the +Association of Universities for Research in Astronomy (AURA) un- +der a cooperative agreement with the National Science Foundation on +behalf of the Gemini Observatory partnership: the National Science +Foundation (United States), National Research Council (Canada), +Agencia Nacional de Investigación y Desarrollo (Chile), Ministe- +rio de Ciencia, Tecnología e Innovación (Argentina), Ministério da +Ciência, Tecnologia, Inovações e Comunicações (Brazil), and Korea +Astronomy and Space Science Institute (Republic of Korea). +DATA AVAILABILITY +All observations of SDSS J1143+6615 are either public (SDSS, Gaia) +or no longer proprietary (Gemini). The Gemini data can be be ob- +tained from the Gemini Observatory Archive, using program ID +GN-2015B-FT-29. 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Oscillator strengths are calculated according to Equation (8). +Wavelength [Å] +Oscillator strength +𝐵 [B0] +𝐵 [MG] +𝜎− +𝜋 +𝜎+ +𝜎− +𝜋 +𝜎+ +0.000 +0.0 +5894.571 +5894.571 +5894.571 +0.324 +0.324 +0.324 +0.004 +9.4 +5742.745 +5894.121 +6048.521 +0.332 +0.324 +0.316 +0.008 +18.8 +5593.316 +5892.750 +6204.298 +0.341 +0.325 +0.307 +0.012 +28.2 +5446.622 +5890.503 +6361.706 +0.349 +0.325 +0.299 +0.016 +37.6 +5302.977 +5887.427 +6520.591 +0.358 +0.325 +0.291 +0.020 +47.0 +5162.697 +5883.594 +6680.899 +0.367 +0.325 +0.284 +0.024 +56.4 +5026.005 +5879.071 +6842.550 +0.375 +0.325 +0.275 +0.028 +65.8 +4893.157 +5873.968 +7005.660 +0.383 +0.325 +0.268 +0.032 +75.2 +4764.303 +5868.374 +7170.305 +0.391 +0.326 +0.260 +0.036 +84.6 +4639.560 +5862.401 +7336.642 +0.400 +0.326 +0.253 +0.040 +94.0 +4519.034 +5856.224 +7504.982 +0.408 +0.326 +0.246 +0.060 +141.0 +3978.205 +5824.551 +8382.905 +0.444 +0.327 +0.211 +0.080 +188.0 +3532.705 +5800.229 +9351.385 +0.475 +0.328 +0.180 +0.100 +235.1 +3166.158 +5790.578 +10456.342 +0.501 +0.328 +0.152 +0.120 +282.1 +2862.247 +5797.779 +11753.431 +0.523 +0.327 +0.127 +0.140 +329.1 +2607.352 +5820.542 +13310.699 +0.541 +0.325 +0.106 +0.160 +376.1 +2390.966 +5855.747 +15218.789 +0.557 +0.324 +0.087 +0.180 +423.1 +2205.210 +5899.663 +17609.247 +0.570 +0.322 +0.071 +0.200 +470.1 +2044.197 +5948.392 +20689.341 +0.581 +0.320 +0.057 +0.220 +517.1 +1903.490 +5998.674 +24813.862 +0.590 +0.319 +0.045 +0.240 +564.1 +1779.628 +6048.357 +30631.655 +0.597 +0.318 +0.035 +0.260 +611.1 +1669.853 +6095.463 +39457.890 +0.603 +0.318 +0.026 +0.280 +658.1 +1571.985 +6139.070 +54464.836 +0.608 +0.319 +0.018 +0.300 +705.2 +1484.230 +6179.092 +85612.964 +0.611 +0.319 +0.011 +0.320 +752.2 +1405.100 +6215.124 +188444.928 +0.613 +0.320 +0.005 +0.340 +799.2 +1333.383 +6247.316 +1320875.762 +0.614 +0.322 +0.001 +0.360 +846.2 +1268.067 +6275.954 +153882.824 +0.615 +0.324 +0.005 +0.380 +893.2 +1208.306 +6301.160 +84066.792 +0.614 +0.325 +0.009 +0.400 +940.2 +1153.390 +6322.883 +59099.755 +0.613 +0.327 +0.012 +0.420 +987.2 +1102.746 +6341.015 +46374.747 +0.611 +0.329 +0.015 +0.440 +1034.2 +1055.874 +6355.731 +38758.544 +0.609 +0.331 +0.017 +0.460 +1081.2 +1010.555 +6366.008 +35866.670 +0.608 +0.333 +0.017 +0.480 +1128.2 +971.855 +6371.932 +30295.527 +0.604 +0.335 +0.019 +0.500 +1175.3 +934.065 +6372.374 +27799.155 +0.601 +0.337 +0.020 +Table A2. Atomic data for the Mg i Zeeman triplet under an applied magnetic field. Columns have the same meaning as in Table A1. +Wavelength [Å] +Oscillator strength +𝐵 [B0] +𝐵 [MG] +𝜎− +𝜋 +𝜎+ +𝜎− +𝜋 +𝜎+ +0.000 +0.0 +5179.597 +5179.597 +5179.597 +0.138 +0.135 +0.137 +0.004 +9.4 +5061.068 +5174.347 +5294.864 +0.157 +0.136 +0.120 +0.008 +18.8 +4940.130 +5158.717 +5406.135 +0.177 +0.137 +0.105 +0.012 +28.2 +4817.641 +5133.057 +5512.746 +0.199 +0.139 +0.092 +0.016 +37.6 +4694.469 +5097.935 +5614.167 +0.224 +0.142 +0.080 +0.020 +47.0 +4571.460 +5054.112 +5710.030 +0.250 +0.146 +0.070 +0.024 +56.4 +4449.419 +5002.504 +5800.152 +0.279 +0.150 +0.060 +0.028 +65.8 +4329.099 +4944.155 +5884.567 +0.309 +0.156 +0.052 +0.032 +75.2 +4211.181 +4880.187 +5963.528 +0.341 +0.163 +0.045 +0.036 +84.6 +4096.271 +4811.771 +6037.528 +0.373 +0.171 +0.038 +0.040 +94.0 +3984.896 +4740.081 +6107.286 +0.407 +0.181 +0.033 +0.060 +141.0 +3493.822 +4370.457 +6432.470 +0.556 +0.249 +0.013 +0.080 +188.0 +3125.114 +4054.666 +6862.945 +0.576 +0.345 +0.004 +0.100 +235.1 +2863.579 +3828.322 +7609.333 +0.451 +0.430 +0.001 +0.120 +282.1 +2672.345 +3666.824 +8859.285 +0.307 +0.479 +0.000 +0.140 +329.1 +2521.311 +3537.491 +10899.854 +0.201 +0.504 +0.000 +0.160 +376.1 +2394.409 +3422.271 +14450.113 +0.127 +0.518 +0.000 +0.180 +423.1 +2283.908 +3313.842 +21816.811 +0.073 +0.529 +0.000 +0.200 +470.1 +2185.904 +3209.926 +45703.577 +0.034 +0.539 +0.000 +MNRAS 000, 1–14 (2022) + +16 +M. A. Hollands et al. +Table A3. Atomic data for the Ca ii Zeeman triplet under an applied magnetic field. Columns have the same meaning as in Table A1. +Wavelength [Å] +Oscillator strength +𝐵 [B0] +𝐵 [MG] +𝜎− +𝜋 +𝜎+ +𝜎− +𝜋 +𝜎+ +0.000 +0.0 +3946.314 +3946.314 +3946.314 +0.320 +0.320 +0.320 +0.004 +9.4 +3876.686 +3946.392 +4017.343 +0.326 +0.320 +0.314 +0.008 +18.8 +3808.443 +3946.626 +4089.790 +0.331 +0.320 +0.309 +0.012 +28.2 +3741.567 +3947.016 +4163.676 +0.337 +0.320 +0.303 +0.016 +37.6 +3676.043 +3947.561 +4239.019 +0.342 +0.320 +0.297 +0.020 +47.0 +3611.854 +3948.260 +4315.843 +0.348 +0.320 +0.292 +0.024 +56.4 +3548.987 +3949.111 +4394.170 +0.353 +0.320 +0.286 +0.028 +65.8 +3487.425 +3950.113 +4474.025 +0.358 +0.321 +0.280 +0.032 +75.2 +3427.153 +3951.265 +4555.436 +0.363 +0.321 +0.275 +0.036 +84.6 +3368.155 +3952.564 +4638.432 +0.369 +0.321 +0.269 +0.040 +94.0 +3310.414 +3954.008 +4723.043 +0.374 +0.321 +0.263 +0.060 +141.0 +3039.968 +3963.325 +5171.589 +0.398 +0.324 +0.236 +0.080 +188.0 +2798.221 +3975.903 +5666.558 +0.419 +0.326 +0.209 +0.100 +235.1 +2582.582 +3991.409 +6214.523 +0.436 +0.331 +0.184 +0.120 +282.1 +2390.297 +4009.463 +6823.401 +0.450 +0.336 +0.160 +0.140 +329.1 +2218.604 +4029.513 +7501.885 +0.459 +0.343 +0.138 +0.160 +376.1 +2064.851 +4050.637 +8258.054 +0.462 +0.351 +0.118 +0.180 +423.1 +1926.562 +4071.303 +9096.442 +0.458 +0.360 +0.100 +0.200 +470.1 +1801.483 +4089.169 +10012.688 +0.445 +0.370 +0.083 +Table A4. Atomic data for the Ca i 4227 Å Zeeman triplet under an applied magnetic field. Columns have the same meaning as in Table A1. +Wavelength [Å] +Oscillator strength +𝐵 [B0] +𝐵 [MG] +𝜎− +𝜋 +𝜎+ +𝜎− +𝜋 +𝜎+ +0.000 +0.0 +4227.920 +4227.920 +4227.920 +0.612 +0.612 +0.612 +0.004 +9.4 +4148.624 +4227.625 +4307.822 +0.624 +0.612 +0.601 +0.008 +18.8 +4070.040 +4226.742 +4388.241 +0.635 +0.612 +0.589 +0.012 +28.2 +3992.294 +4225.272 +4469.117 +0.647 +0.613 +0.578 +0.016 +37.6 +3915.516 +4223.223 +4550.413 +0.659 +0.613 +0.567 +0.020 +47.0 +3839.838 +4220.600 +4632.124 +0.670 +0.613 +0.557 +0.024 +56.4 +3765.388 +4217.412 +4714.272 +0.682 +0.614 +0.546 +0.028 +65.8 +3692.284 +4213.667 +4796.905 +0.694 +0.615 +0.535 +0.032 +75.2 +3620.629 +4209.378 +4880.090 +0.705 +0.615 +0.524 +0.036 +84.6 +3550.512 +4204.555 +4963.919 +0.716 +0.616 +0.514 +0.040 +94.0 +3482.000 +4199.209 +5048.494 +0.727 +0.617 +0.503 +0.060 +141.0 +3164.874 +4165.022 +5487.059 +0.778 +0.621 +0.450 +0.080 +188.0 +2890.269 +4119.406 +5966.238 +0.819 +0.626 +0.399 +0.100 +235.1 +2654.834 +4063.819 +6511.058 +0.852 +0.631 +0.349 +0.120 +282.1 +2453.542 +4000.033 +7156.578 +0.876 +0.637 +0.301 +0.140 +329.1 +2281.503 +3930.006 +7957.549 +0.891 +0.642 +0.255 +0.160 +376.1 +2134.652 +3855.464 +9009.840 +0.898 +0.648 +0.211 +0.180 +423.1 +2009.768 +3777.517 +10498.296 +0.895 +0.655 +0.168 +0.200 +470.1 +1904.166 +3696.491 +12819.115 +0.882 +0.662 +0.127 +MNRAS 000, 1–14 (2022) + +A DZ white dwarf with a 30 MG magnetic field +17 +Table A5. Atomic data for the Ca i 6142 Å Zeeman triplet under an applied magnetic field. Columns have the same meaning as in Table A1. +Wavelength [Å] +Oscillator strength +𝐵 [B0] +𝐵 [MG] +𝜎− +𝜋 +𝜎+ +𝜎− +𝜋 +𝜎+ +0.000 +0.0 +6143.862 +6143.862 +6143.862 +0.149 +0.149 +0.149 +0.004 +9.4 +5976.357 +6134.649 +6306.090 +0.170 +0.150 +0.130 +0.008 +18.8 +5805.312 +6107.587 +6461.705 +0.192 +0.152 +0.114 +0.012 +28.2 +5632.852 +6063.618 +6610.116 +0.216 +0.156 +0.098 +0.016 +37.6 +5460.710 +6004.725 +6750.751 +0.241 +0.161 +0.085 +0.020 +47.0 +5290.838 +5933.170 +6884.019 +0.268 +0.168 +0.072 +0.024 +56.4 +5124.852 +5851.294 +7010.699 +0.295 +0.175 +0.062 +0.028 +65.8 +4964.118 +5761.377 +7132.077 +0.321 +0.184 +0.052 +0.032 +75.2 +4809.669 +5666.004 +7249.747 +0.347 +0.194 +0.043 +0.036 +84.6 +4662.011 +5567.124 +7364.999 +0.372 +0.205 +0.036 +0.040 +94.0 +4521.526 +5466.431 +7479.478 +0.394 +0.216 +0.030 +0.060 +141.0 +3928.656 +4976.764 +8105.437 +0.457 +0.273 +0.010 +0.080 +188.0 +3494.930 +4562.450 +8986.627 +0.436 +0.325 +0.003 +0.100 +235.1 +3177.291 +4234.160 +10395.103 +0.367 +0.364 +0.001 +0.120 +282.1 +2935.563 +3968.106 +12750.000 +0.290 +0.391 +0.000 +0.140 +329.1 +2740.263 +3737.364 +16942.725 +0.223 +0.410 +0.000 +0.160 +376.1 +2574.017 +3525.150 +25704.466 +0.169 +0.425 +0.000 +0.180 +423.1 +2428.204 +3325.838 +142223.979 +0.125 +0.440 +0.000 +0.200 +470.1 +2299.061 +3139.785 +498426.017 +0.087 +0.457 +0.000 +MNRAS 000, 1–14 (2022) + diff --git a/_tE3T4oBgHgl3EQfsAry/content/tmp_files/load_file.txt b/_tE3T4oBgHgl3EQfsAry/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fd73b6d2a5200eca9e0148ebf6c3a12b536fe2cc --- /dev/null +++ b/_tE3T4oBgHgl3EQfsAry/content/tmp_files/load_file.txt @@ -0,0 +1,2307 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf,len=2306 +page_content='MNRAS 000, 1–14 (2022) Preprint 13 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 A DZ white dwarf with a 30 MG magnetic field M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hollands,1★ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Stopkowicz,2,3,4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Kitsaras,3 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hampe,3 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Blaschke,3 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hermes5 1 Department of Physics and Astronomy, University of Sheffield, Sheffield, S3 7RH, UK 2 Fachrichtung Chemie, Universität des Saarlandes, D-66123 Saarbrücken, Germany 3 Department Chemie, Johannes Gutenberg-Universität Mainz, Duesbergweg 10-14, D-55128 Mainz, Germany 4 Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Box 1033 Blindern, N-0315 Oslo, Norway 5 Department of Astronomy & Institute for Astrophysical Research, Boston University, 725 Commonwealth Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=', Boston, MA 02215, USA Accepted 2023 January 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Received 2023 January 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' in original form 2022 November 28 ABSTRACT Magnetic white dwarfs with field strengths below 10 MG are easy to recognise since the Zeeman splitting of spectral lines appears proportional to the magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For fields ≳ 100 MG, however, transition wavelengths become chaotic, requiring quantum-chemical predictions of wavelengths and oscillator strengths with a non-perturbative treatment of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' While highly accurate calculations have previously been performed for hydrogen and helium, the variational techniques employed become computationally intractable for systems with more than three to four electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Modern computational techniques, such as finite-field coupled-cluster theory, allow the calculation of many-electron systems in arbitrarily strong magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Because around 25 percent of white dwarfs have metal lines in their spectra, and some of those are also magnetic, the possibility arises for some metals to be observed in very strong magnetic fields, resulting in unrecognisable spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We have identified SDSS J114333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='48+661531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='83 as a magnetic DZ white dwarf, with a spectrum exhibiting many unusually shaped lines at unknown wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Using atomic data calculated from computational finite-field coupled-cluster methods, we have identified some of these lines arising from Na, Mg, and Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Surprisingly, we find a relatively low field strength of 30 MG, where the large number of overlapping lines from different elements make the spectrum challenging to interpret at a much lower field strength than for DAs and DBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Finally we model the field structure of SDSS J1143+6615 finding the data are consistent with an offset dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Key words: white dwarfs – stars: magnetic field – atomic data 1 INTRODUCTION The first magnetic white dwarf was discovered by Kemp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (1970), through the detection of circularly polarised light from GJ 742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Since then, many hundreds of magnetic white dwarfs have been discovered (Kawka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Kepler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2013), with observed fields strengths spanning a few 10 kG up to about 1000 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For fields ranging be- tween a few 100 kG to a few 10 MG, magnetic DA white dwarfs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' those with spectra dominated by hydrogen absorption lines) are easy to identify in intensity spectra and their field strengths are simple to measure, as many hydrogen lines split into three components, where the degree of splitting is proportional to field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For smaller fields, where such splitting is unresolved, spectropolarimetry can be used instead (Bagnulo & Landstreet 2018, 2019, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Landstreet & Bagnulo 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' However, due to reduced throughput, spectropo- larimetry is limited to only the brightest white dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For higher fields, particularly those beyond 100 MG, identification is often still straightforward, though measuring the field strength is no-longer trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The diamagnetic term in the Hamiltonian of the hydrogen atom (Wickramasinghe & Ferrario 2000) (resulting in the quadratic Zeeman effect due to its 𝐵2 dependence), quickly exceeds the interaction strength of the paramagnetic term (linear Zeeman ★ E-mail: m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='hollands@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='uk effect), and eventually even the electrostatic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This results in large shifts in wavelength, which ostensibly appear chaotic in their field strength dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Due to the 𝑛4 dependence on the quadratic Zeeman effect (where 𝑛 is the principle quantum number, Wickramasinghe & Ferrario 2000), the shifts are first observed in the higher order Balmer lines, but beyond a few 10 MG also causes the wavelengths of the H𝛼 components to become chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Because the size of the diamagnetic term in the Hamiltonian becomes comparable to the other terms, and overall the magnetic field is no longer a small perturbation to the system, the energies (and hence transition wavelengths), cannot be determined using perturbation theory, and instead must be determined numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For hydrogen, the first detailed atomistic calculations were per- formed in the 1980s (Roesner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Forster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Henry & O’Connell 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Wunner 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The results of these calcula- tions quickly found application to assignment of lines in strongly magnetic white dwarf spectra (Greenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Angel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' More recent calculations have refined the atomic data for hydrogen in strong fields (Schimeczek & Wunner 2014b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Schimeczek & Wunner 2014a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Even at these early stages, however, the magnetic white dwarf GD 229 was found to defy assignment of hydrogen spectral lines, leading to speculation that it may instead have a helium dominated atmosphere (Green & Liebert 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1990, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='04665v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='SR] 11 Jan 2023 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' hypothesis was proved correct when the first calculations of He i by Jordan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (1998) were matched to lines in the spectrum of GD 229, implying a surface field varying between 300–700 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The calcu- lations themselves relied on finite-field full configuration interaction (ff-FCI) theory, a variational technique providing near-exact solu- tions to the time-independent electronic Schrödinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Such a description is needed due to electron-electron repulsion term in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Similar calculations for He i were also been performed by Becken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Calculations using variational approaches have been performed for systems with more electrons such as Li i (Zhao 2018), however for systems with more than three to four electrons, ff-FCI becomes numerically intractable due to the factorial scaling in computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Fortunately, while white dwarfs with heavy elements in their at- mospheres have been known for more than a century, those with magnetic fields have hitherto not been observed with field strengths exceeding ∼ 10 MG, where atoms are safely in the Paschen-Back regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' White dwarfs with heavier elements fall into two main classes: the DQs containing spectral features from carbon, and the DZs containing features from heavier metals (Sion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1983) such as calcium and iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' DQ white dwarfs, those with spectral features from carbon in their atmospheres (detected from C2 Swan bands at low 𝑇eff and C i/ii at higher 𝑇eff) are generally understood to originate from convective dredge up of carbon from the core into the surrounding helium en- velope (Fontaine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Pelletier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' MacDonald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1998), though a separate population of massive DQs are thought to originate as the product of mergers (Dufour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Dunlap & Clemens 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Kawka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Of these hot suspected merged DQs, a moderate frac- tion are also magnetic, showing Zeeman split C i/ii lines – some with field strengths of a few MG (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Dufour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' At lower 𝑇eff some peculiar DQs (such as LHS 2229) show highly distorted and shifted Swan bands which have previously been hypothesised to arise from strong (100s of MG) magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' However, Kowal- ski (2010) demonstrated that the distorted molecular bands primar- ily result from pressure-effects occurring in high-density, low 𝑇eff, helium-dominated white dwarf atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' To date, no predictions for the wavelengths of atomic or molecular carbon transitions in strong magnetic fields have been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' White dwarfs with metals in their atmospheres are denoted with a Z in their spectral type, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' DAZ, DBZ, or DZ, depending which other lines are visible in their spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' DZs specifically (the subject of this work) usually have helium dominated atmospheres, though are too cool to exhibit He i lines (𝑇eff < 11, 000 K), although for 𝑇eff < 5000 K hydrogen lines are also diminished in strength, and so in some cases hydrogen atmosphere white dwarfs can also be classed DZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Unlike the carbon in DQs, the metals observed in DZs (and DAZs/DBZs etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=') require an external source, as gravitational settling should deplete white dwarf atmospheres of metals on timescales that are always much shorter than white dwarf ages (Paquette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1986) – specifically in the case of cool DZs, sinking timescales are on the order of 106–7 yr, whereas their ages range from 109–10 yr (see Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2014, Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' A vast array of evidence now supports accretion of exoplanetesi- mals from an accompanying planetary system as the source of this metal pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Many metal-rich white dwarfs are observed with infra-red excesses resulting from circumstellar debris disks (Zucker- man & Becklin 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Jura 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Rocchetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Swan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2019a), with a sub-population of those also exhibiting gaseous emis- sion from the sublimated part of the disk (Gänsicke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2006, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Dennihy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Manser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In a few cases, when the disk is viewed edge-on, irregular transits are observed demon- strating the tidal disruption of exoplanetesimals close to the white dwarf Roche radius (Vanderburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Vanderbosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Guidry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Farihi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In two cases the pres- ence of planets themselves has been directly inferred, firstly from the accretion of an evaporating gas giant by WD J0914+1914 (Gänsicke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2019), and secondly from planetary transits at WD 1856+534 (Vanderburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Despite these various sources of evidence for white dwarf planetary systems, white dwarf spectra containing metal lines remains the most common observable, and can be used to infer the composition of the accreted exoplanetesimals (Zuckerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Gänsicke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Dufour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Farihi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2017, 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Blouin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Doyle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Swan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hoskin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Izquierdo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' A sub-population of DZs have also been found to exhibit magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The first discovered magnetic DZ (spectral type DZH) was LHS 2534 (Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2001), which was found to have a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='9 MG field strength from Zeeman split lines of Na i, Mg i, and blended Zeeman components from Ca i/ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The field strength of LHS 2534 was recently revised to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='1 MG by Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2021) along with the detection of Zeeman splitting of Li i and K i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Since this initial discovery, additional DZHs were identified by Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2003) and Dufour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2006) (WD 0155+003 and G 165−7, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' With the advent of data release 10 (DR10) of the Sloan Digital Sky Survey (SDSS), Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2015) identified a further seven ob- jects, bringing the known sample to ten, and finding a high magnetic incidence of 13±4 percent for DZs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' With SDSS DR12, Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2017) measured the fields of an additional 15 DZs1, with the range of surface averaged field strengths, 𝐵𝑠, spanning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='04 MG to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='07 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Like LHS 2534, most of these DZs were identi- fied from Zeeman triplets arising from the Na i resonance doublet (𝜆 ≃ 5890 Å), and the Mg i triplet (𝜆 ≃ 5180 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Several magnetic DAZ white dwarfs have also been identified, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' those with hydrogen dominated atmospheres, though their field strengths are typically be- low 1 MG (Kawka & Vennes 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Farihi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Zuckerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Kawka & Vennes 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Kawka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' With none of the objects published so far demonstrating fields exceeding 11 MG, calculations of metals in ultra-strong magnetic fields have thus far not been essential for the analysis of DZH spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In this work we investigate SDSS J114333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='48+661531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='83 (here- after SDSS J1143+6615), a faint (𝐺=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='1 mag) magnetic DZ white dwarf with a peculiar spectrum with a sufficiently strong magnetic field that spectral features are almost entirely unrecognisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In Section 2 we present our observations as well as public data on SDSS J1143+6615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In Section 3 we discuss our finite-field coupled- cluster calculations for metals in strong magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In Section 4, we make use of our atomic data calculations to identify the spec- tral lines of SDSS J1143+6615 while simultaneously measuring the strength of its magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In Section 5, we attempt to model the field structure of SDSS J1143+6615, while in Section 6 we discuss the applicability of our atomic data to higher field strengths and use in model atmospheres, with our conclusions presented in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1 Note that the thesis of Hollands (2017) identified a further seven low-field magnetic objects in the Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2017) DZ sample, with field strengths between 250 ± 30 kG to 510 ± 40 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) A DZ white dwarf with a 30 MG magnetic field 3 4000 4500 5000 5500 6000 6500 Wavelength [˚A] 0 2 4 6 8 Flux [×10−17 erg s−1 cm−2 ˚A −1] SDSS BOSS Gemini GMOS Ca i Mg i Na i Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' SDSS BOSS and Gemini GMOS spectra of SDSS J1143+6615 (𝐺=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='1 mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The SDSS spectrum is shifted upwards by 4 × 10−17 erg s−1 cm−2 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Behind the Gemini spectrum, we show the SDSS spectrum again (light grey), but convolved to a resolving power of 𝑅 = 1100 for direct comparison, demonstrating the virtually unchanged spectrum over two years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The zero-field air wavelengths of Ca i, Mg i, and Na i are shown by the solid vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2 OBSERVATIONS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='1 SDSS SDSS J1143+6615 was originally observed in SDSS using the BOSS spectrograph (Baryon Oscillation Spectroscopic Survey), first pub- lished in SDSS Data Release 12 (plate-MJD-fiberID 7114-56748- 0973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The SDSS spectrum is shown at the top of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This spectrum was first classified as a candidate DZH white dwarf by Kepler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This object also appeared in the DZ sample of Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2017), where it was suggested to have a magnetic field exceeding 20 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The overall slope of the spectrum appears consistent with a cool white dwarf with effective temperature (𝑇eff) in the range 5000– 7000 K, but is otherwise highly unusual, exhibiting a myriad of unidentified features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In particular several bands of broad features are seen near 4700 Å, 5500 Å, and 6400 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' However, two sharper absorption features stand out as resembling atomic lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' One of these appears at about 5890 Å, and so could be from the Na i-D resonance doublet (which in the absence of a magnetic field would appear blended here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The other sharp feature is located at ≃ 5125 Å, and due to its asymmetry resembles the Mg i-b triplet which is com- monly observed in cool DZ white dwarfs where the asymmetry arises from neutral broadening by helium atoms in a dense, helium dom- inated atmosphere (Allard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Blouin 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' However, while the asymmetry appears qualitatively similar, the wavelength is bluer by about 50 Å than should be the case for the Mg triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' While the SDSS spectrum does extend to 10, 400 Å, we see no evidence for other absorption features beyond what is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' With none of the spectral features firmly identified, we speculated that SDSS J1143+6615 is a strongly magnetic DZ white dwarf, where the quadratic Zeeman effect is no longer negligible, causing additional shifts of Zeeman-split spectral lines, and result- ing in the appearance of many unidentified features in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The SDSS spectrum itself is composed of four sub-spectra, each taken with 900 s exposure times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' While these individual spectra are extremely noisy, owing to the faintness of SDSS J1143+6615, smoothing the data and down-sampling hinted at possible variability between exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Because magnetic white dwarfs are known to have rotation periods of minutes to days (Brinkworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Kilic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2021), we considered the possibility of spectral line shapes/positions evolving with rotational phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We therefore sought to obtain higher quality spectra of SDSS J1143+6615 in order to confirm this rotation, as well potentially identify spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='2 Gemini We obtained additional spectra using the GMOS (Gemini Multi Ob- ject Spectrograph) instrument on the Gemini North telescope on April 1st 2016 (exactly two years after the SDSS spectrum was taken).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The instrumental setup used the B600_G5307 grating with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='75 arcsec slit, giving us a resolving power of about 1100 at 4600 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In total we took 17 exposures lasting 628 s each, separated by 15 s of MNRAS 000, 1–14 (2022) 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' readout time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The GMOS detector uses three CCDs which covered 4100–7000 Å with our instrumental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This results in two ≃ 25 Å gaps between each CCD with no spectral coverage, though these did not cover any important features identified from the SDSS spectrum (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We reduced the GMOS spectra using the starlink distribution of software for bias-subtraction, flat-fielding, and optimal-extraction (Horne 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Marsh 1989) of the spectral trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Wavelength- calibration was performed using molly2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For flux-calibration, we initially used our observed flux standard, EG 131, but found this gave unsatisfactory results, since it was observed at the end of the night, whereas our science observations were observed at the start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We in- stead made use of the SDSS spectrum from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='1, as the SDSS flux calibration are typically accurate to 1 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For each chip we took the ratio of the spectra (in units of counts) and the already flux- calibrated SDSS spectrum, re-binned onto the same wavelengths as the GMOS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We then fitted third-order polynomials to these ratios to define a calibration function, which we then used to re- scale the Gemini spectra into flux units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Note that fluxes redwards of 6700 Å were dominated by telluric absorption and so data beyond this wavelength were ignored and are not shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Our initial goal for these time-resolved spectra was to search for variability, which may arise from rotation of a magnetic white dwarf, bringing different parts of the magnetic field structure into view, and thus causing Zeeman components to change in shape and wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We show the trailed, normalised Gemini spectra in Figure 2 for chip- 2 of GMOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This chip has the largest spectral signal-to-noise ratio (S/N), and contains many of the unassigned spectral features, includ- ing the proposed Mg i and Na i lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In the bottom panel, we show a zoom-in of the suggested Mg i line, which because of the large shift from the rest-wavelength, should be particularly sensitive to changes in the magnetic field (if it is indeed Mg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We do not detect variability in any of the spectral features, suggesting a lack of rotation on time scales of a few hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Given the lack of variability between our 17 spectra, we chose to co-add these into a single high S/N spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We show this in the bottom of Figure 1 (dark grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This is compared with the SDSS spectrum (light grey) which has been convolved to the same spectral resolution as our Gemini data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Almost all features appear unchanged, with perhaps only minor differences in the core strengths of the 5400 Å and 5500 Å features, and a slight change in wavelength of the feature at 4650 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This comparison demonstrates a lack of variability on a time scale of two years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' With the higher S/N spectrum, the proposed Na i and Mg i lines are seen to be blue shifted by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='5 Å and 52 Å respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The asym- metric nature of the latter (discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='1), is also much clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For the proposed Na i line this could be plausibly explained as a ≃ 300km s−1 blue shift (not including any gravitational redshift from the white dwarf) if SDSS J1143+6615 is a halo object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' That be- ing said, the much slower 18±2 km s−1 tangential velocity from Gaia EDR3 (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='3) argues against this explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Furthermore, such an explanation is effectively ruled out by the proposed Mg i line, since its much larger wavelength shift would correspond to a veloc- ity shift of about 3000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Therefore magnetism remains a more likely hypothesis for explaining the spectrum of SDSS J1143+6615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In addition to the lines observed from the SDSS spectrum, the Gem- ini spectrum also reveals the possible presence of the Ca i resonance line (Figure 1, purple), with a small blue shift of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='6 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2 The software molly can be found at https://cygnus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='uk/phsaap/software/ 5200 5400 5600 5800 Wavelength [˚A] 0 50 100 150 Time [min] 5080 5100 5120 5140 5160 Wavelength [˚A] 0 50 100 150 Time [min] Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Trailed continuum-normalised spectra for our Gemini observations of SDSS J1143+6615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The top panel shows the entirety of chip-2, which contains both of the sharp features suggested to be from Mg and Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The bottom panel shows a Zoom-in of the suggested Mg line, demonstrating an absence of spectral variability on a 3 hr timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='3 Gaia Despite its curious spectrum containing many anomalous features precluding obvious spectroscopic classification, the measured non- zero proper-motion by SDSS confirms that SDSS J1143+6615 is a galactic object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' However, without knowing the absolute brightness of this star, SDSS J1143+6615 could not be claimed to be a white dwarf with certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In April 2018, the second data release (DR2) from Gaia space mission made public approximately 1 billion parallaxes (Gaia Col- laboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This included SDSS J1143+6615 which had a measured parallax of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='68 mas, confirming this stars loca- tion along the white dwarf cooling track within the Hertzsprung- Russel diagram (HR-diagram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In December 2020 a refined par- allax of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='46 mas was made available from Gaia EDR3 (early data release 3, Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2021) correspond- ing to a distance of 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='8 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The EDR3 HR-diagram is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' SDSS J1143+6615 is indicated by the red point, and is compared against a background of white dwarfs se- lected from Gentile Fusillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2021) with PWD > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='75 and parallax_over_error > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' From its location in the HR-diagram, MNRAS 000, 1–14 (2022) A DZ white dwarf with a 30 MG magnetic field 5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='5 GBP − GRP [mag] 8 10 12 14 16 Gabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' [mag] Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Gaia EDR3 Hertzsprung-Russel diagram showing the location of SDSS J1143+6615 (red) compared with the white dwarf cooling sequence (grey histogram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The error bars represent 1𝜎 uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' it is clear that SDSS J1143+6615 is a cool white dwarf with a typical mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Therefore Gentile Fusillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2021) found 𝑇eff = 5810 ± 460 K and log 𝑔 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='33 fitting the Gaia photometry with pure hydrogen atmosphere models, and 𝑇eff = 5680 ± 470 K and log 𝑔 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='33 for a pure helium atmosphere mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Interestingly, if Figure 3 is recreated using Gaia DR2 data, SDSS J1143+6615 appears to be offset from the white warf sequence towards higher masses, with Gentile Fusillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2019) finding 𝑇eff = 6990 ± 710 K and log 𝑔 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='29 for hydrogen atmo- sphere models, and 𝑇eff = 6870 ± 750 K and log 𝑔 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='33 for helium atmosphere models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' That said, these parameter shifts amount to only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='4𝜎 changes at most and so are in statistical agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 3 ATOMIC DATA CALCULATIONS To test our hypothesis that SDSS J1143+6615 is a highly magnetic DZ white dwarf, we required accurate wavelengths of (at the very least) the Na i and Mg i lines as a function of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For large magnetic field strengths, however, approaches that are based on a perturbative treatment of the magnetic field are no longer adequate and hence accurate finite-field quantum-chemical methods need to be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In these methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' the magnetic field is treated explicitly in the calculation of ground-state energies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' excitation energies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' and transition-dipole moments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' thereby employing the electronic Hamil- tonian for an 𝑁-electron system in a static magnetic field in the 𝑧-direction (the gauge-origin is here in the origin of the coordinate system) ˆ𝐻 = ˆ𝐻0 + 1 2 𝐵 ˆ𝐿𝑧 + 𝐵 ˆ𝑆𝑧 + 1 8 𝐵2 𝑁 ∑︁ 𝑖 (𝑥2 𝑖 + 𝑦2 𝑖 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (1) where 𝐵 is the magnetic-field strength and ˆ𝐻0 is the field-free atom- istic (or molecular) Hamiltonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' consisting of the kinetic energy of the electrons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' the nuclear-electronic potential and the electron- electron repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' ˆ𝐿𝑧 = �𝑁 𝑖 ˆ𝑙𝑖,𝑧 and ˆ𝑆𝑧 = �𝑁 𝑖 ˆ𝑠𝑖,𝑧 are the 𝑧- components of the angular momentum operator, and spin, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The terms linear in the magnetic field are the orbital-Zeeman (responsible for the splitting of the orbitals) and spin-Zeeman terms (responsible for splitting according to spin parallel or antiparallel to the magnetic field), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The quadratic term is referred to as diamagnetic contribution which always increases the energy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' As in the field-free case in quantum chemistry, FCI theory is not applicable for problems like ours due to its high computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Instead, Coupled-Cluster (CC) theory (Shavitt & Bartlett 2009) can be used, which has a more economical computational scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' CC methods work with an exponential parametrization of the wave function ΨCC = e ˆ𝑇 Φ0, where ˆ𝑇 = ˆ𝑇1 + ˆ𝑇2 + · · · + ˆ𝑇𝑁 is the so- called cluster operator generating excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' ˆ𝑇 contains amplitudes (weighting coefficients in the wave functions) that are determined by solving the CC equations ⟨Φ𝐼 | e− ˆ𝑇 ˆ𝐻e ˆ𝑇 | Φ0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2) The CC energy is then given as 𝐸CC = ⟨Φ0 | e− ˆ𝑇 ˆ𝐻e ˆ𝑇 | Φ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (3) Truncations in ˆ𝑇 as well as limiting the projection space define ap- proximate CC schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For example, CC ‘singles doubles’ (CCSD) is defined with ˆ𝑇CCSD = ˆ𝑇1 + ˆ𝑇2 and projection on singly and doubly excited determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Analo- gously, in CC ‘singles doubles triples’ (CCSDT), ˆ𝑇 is truncated to ˆ𝑇CCSDT = ˆ𝑇1 + ˆ𝑇2 + ˆ𝑇3 and projection is additionally also performed on triply excited de- terminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' While CC is used to describe the ground-state wave function, Equation-of-Motion-CC (EOM-CC) (Shavitt & Bartlett 2009) can also describe electronically excited states (EE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' An op- erator ˆ𝑅, parametrized similarly as ˆ𝑇 acts on a CC wave function Ψexc = ˆ𝑅ΨCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The corresponding amplitudes are determined via the solution of the eigenvalue problem in matrix form ¯Hr = ΔEexcr (4) in which an element of the matrix ¯H is given as ¯𝐻𝐼 𝐽 = ⟨Φ𝐼 | e− ˆ𝑇 ( ˆ𝐻 − 𝐸CC)e ˆ𝑇 | Φ𝐽 ⟩ (5) and the vector r contains the amplitudes for the excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' An overview of ff-CC and ff-EOM-CC methods can be found in Stop- kowicz (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In this work, we have used various flavors of ff-CC theory (Stopkowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Kitsaras & Stopkowicz 2022a) and ff-EOM CC theory, implemented within the QCUMBRE program package (Hampe & Stopkowicz 2017), to determine excited states and hence transition wavelengths (Hampe & Stopkowicz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hampe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Kitsaras & Stopkowicz 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The underlying calcula- tion of the reference |Φ0⟩ is performed with the CFOUR program package (Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In the EOM-framework, we have employed the methods for electronic excitations (EE), spin flip (SF), adding electrons (EA, electron attachment) and removal of electrons (IP, ionization potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Oscillator strengths are also treated at the expectation value ff-EOM-CC level (Hampe & Stopkowicz 2019) which enables the prediction of field-dependent intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The tran- sitions for which we have performed ff-calculations are displayed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The data for Na has partly already been available in MNRAS 000, 1–14 (2022) 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Level information for the transitions we have performed ff- calculations for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Wavelengths (air) correspond to field-free transitions, which in the case of multiplets corresponds to the average wavelength given in the NIST database (weighted by oscillator strength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Ion Wavelength [Å] Lower state Upper state Na i 5892 2𝑆𝑔 ([Ne]3𝑠) 2𝑃𝑢 ([Ne]3𝑝) Mg i 5178 3𝑃𝑢 ([Ne]3𝑠3𝑝) 3𝑆𝑔 ([Ne]3𝑠4𝑠) Ca i 4227 1𝑆𝑔 ([Ar]4𝑠2) 1𝑃𝑢 ([Ar]4𝑠4𝑝) Ca i 6142 3𝑃𝑢 ([Ar]4𝑠4𝑝) 3𝑆𝑔 ([Ar]4𝑠5𝑠) Ca ii 3945 2𝑆𝑔 ([Ar]4𝑠) 2𝑃𝑢 ([Ar]4𝑝) Hampe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The latter work is also the basis for the com- putational protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We will here only mention the most important points and refer to Hampe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2020) for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For all tran- sitions, the calculations were performed for magnetic fields ranging between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='00–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='04 B0, with the atomic unit of the magnetic field, B0 ≃ 2350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='518 MG, using a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='004 B0 step and between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='04–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='20 B0 using a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='02 B0 step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In the protocol, a corrected excitation energy is computed according to Δ𝐸corrected exc = Δ𝐸exc + Δ𝐸basis + Δ𝐸triples, (6) where Δ𝐸exc is the excitation energy computed using a large uncon- tracted augmented one-electron basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Δ𝐸basis is a term correcting the one-electron basis-set error as described in Halkier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (1998) by extrapolating a basis-set limit 𝐸∞ based on uncontracted basis sets of the type aug-cc-pCVXZ (Kendall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Woon & Dunning, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1995), abbreviated as aCXZ, where X is the cardinal number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' It is given as Δ𝐸basis = Δ𝐸∞ − Δ𝐸exc with Δ𝐸∞ = Δ𝐸aCXZ exc 𝑋3 − Δ𝐸aCYZ exc 𝑌3 𝑋3 − 𝑌3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (7) The Δ𝐸triples = 𝐸aCXZ triples − 𝐸aCXZ exc correction accounts for the error which stems from truncating the CC expansion and involves com- putations at the ff-EOM-CCSDT (Hampe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2020), ff-EOM-CC3 (Kitsaras & Stopkowicz 2022a) or ff-EOM-CCSD(T)(a)* (Matthews & Stanton 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Kitsaras & Stopkowicz 2022b) levels of theory for 𝐸aCXZ triples using a smaller basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The accuracy and cost is typ- ically CCSDT (𝑂(𝑀8))> CC3 (𝑂(𝑀7))> CCSD(T)(a)* (𝑂(𝑀7)) where 𝑀 is the number of basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In the latter two, triple- excitations are treated in a perturbative manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' CC3 is iterative while CCSD(T)(a)* is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The latter is a very good and relatively cheap option when the target-states are characterised mostly by single- excitation character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The dimensionless oscillator strengths 𝑓𝐼 𝐽 were calculated according to 𝑓𝐼 𝐽 = 2 3 (Δ𝐸𝐼 𝐽 )|𝜇𝐼 𝐽 |2, (8) where Δ𝐸𝐼 𝐽 is the excitation energy from states 𝐼 to 𝐽 and 𝜇𝐼 𝐽 is the corresponding transition-dipole moment, and where both Δ𝐸𝐼 𝐽 and 𝜇𝐼 𝐽 are in atomic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' After converting the (field-dependent) excitation energies to transition wavelengths, the resulting 𝐵 − 𝜆 curves were shifted to start at the zero-field values taken from the NIST database (Kramida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2022) thereby correcting for remaining errors of our predictions at zero field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The spin-orbit contributions have been averaged out as their contribution is expected to be small for stronger fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' By the shift made to the NIST data, field-free scalar- relativistic effects are implicitly accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For the time being, we are neglecting relativistic effects and in particular their dependence on the magnetic field in our calculations as the effects are expected to be small for strong magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This approximation is better for the lighter elements Na and Mg than for the heavier Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The specific details on the calculations are collected in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The predicted transition wavelengths and oscillator strengths can be found in Tables A1–A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Additionally, the obtained 𝐵 − 𝜆 curves are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The intensity of the transitions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' oscillator strengths, are indicated via the opacities of the curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' As all of the investigated transitions are of 𝑛𝑠 → 𝑛𝑝 or 𝑛𝑝 → (𝑛 + 1)𝑠 type, where 𝑛 is the main quantum number of the orbital (without field), there is in all cases a splitting into three components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=', the central 𝜋 (transition from/into a 𝑝0 orbital) as well as the two 𝜎 (transi- tion from/into 𝑝+1 and 𝑝−1) components3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' As can be seen here, the splitting is only linear for fields below about 5–10 MG while for higher field strengths, the form of the 𝐵 − 𝜆 curves becomes much more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The distortion from a simple Zeeman behaviour is transition dependent: For the 𝑛𝑝 → (𝑛 + 1)𝑠 transitions (Mg and Ca i 6142), the influence of the magnetic field on the central 𝜋 compo- nent is much more pronounced than for the 𝑛𝑠 → 𝑛𝑝 transitions (Na, Ca ii, Ca i 4227).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The principal reason for this behaviour is that in the former case the transitions are between orbitals of different main quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The orbitals hence experience a different amount of polarisation through the magnetic field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' those of higher main quantum number are polarised more strongly due to their more dif- fuse nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Effectively this means that the 𝑠 and 𝑝0 orbitals and the respective electronic states, don’t evolve in a parallel manner with increasing magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hence, in contrast to the simple pertur- bative picture, the central 𝜋 component is no longer constant with increasing magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In addition, the transitions with decreasing energy difference in the magnetic field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=', 𝑛𝑠 → 𝑛𝑝−1 and 𝑛𝑝+1 → (𝑛 + 1)𝑠 become less relevant for observations, as they decrease in intensity (see Equation (8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In addition, small changes in the magnetic field lead to large changes in the transition wavelength and hence such transitions will be blurred out in the spectra for strong fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' A more detailed discussion on the form of the energy levels and the resulting for of the 𝐵 − 𝜆 curve of the Mg transition can be found in Kitsaras & Stopkowicz (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' As noted in Hampe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2020), high-accuracy predictions are required as even the prediction for the transition least affected by the magnetic field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=', the central 𝜋 component of Na can vary by up to 100 Å depending on the level of theory and basis set used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='4 4 LINE IDENTIFICATION With the wavelengths and oscillator strengths calculated in Sec- tion 3, we were able to compare these with the spectrum of SDSS J1143+6615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' With no immediate indication of which spec- tral features could correspond to the 𝜎-components of the calculated transitions, we began by restricting ourselves to the 𝜋-components only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In Section 2 we identified possible 𝜋-components of Na i, Mg i, and Ca i in the SDSS and GMOS spectra, based on the sharpness of the lines, rough proximity in wavelengths to the field-free values, and characteristic asymmetry in the case of Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We compare these lines to our calculated wavelengths as a function of field strength in the top panels of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' From the bottom-right panel, it is clear that the Na line shift could be explained by either a relatively small field of ≃ 30 MG or much larger field of ≃ 410 MG, 3 Note that in the magnetic field, the SO(3) symmetry is lowered to 𝐶∞ℎ but we will, for simplicity, still refer to field-free state and orbital classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 4 Note that the uncertainty of the predicted transition wavelengths is not only dependent on the accuracy of the method but also on the position of the absorption peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) A DZ white dwarf with a 30 MG magnetic field 7 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Detailed information on ff-EOM calculations for the respective transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' If not specified otherwise, Δ𝐸𝐼 𝐽, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' has been calculated at the same level as 𝜇𝐼 𝐽 Transition Basis functions Δ𝐸exc Δ𝐸basis Δ𝐸triples 𝜇𝐼 𝐽 Na i Cartesian EE-CCSD/aCQZ EE-CCSD/aCXZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' X=T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Q CCSDT/aCTZ EE-CCSD/aCQZ Mg i Spherical EE-CCSD/aC5Z EE-CCSD/aCXZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' X=Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 5 CC3/aCQZ EE-CCSD/aC5Z Ca i 4227 Spherical EE-CCSD/aC5Z EE-CCSD/aCXZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' X=Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 5 EE-CC3/aCQZ EE-CCSD/aCQZ(𝑎) Ca i 6142 Spherical SF-CCSD(T)(a)*/aC5Z SF-CCSD(T)(a)*/aCXZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' X=Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 5 No further triples correction SF-CCSD/aC5Z(𝑏) Ca ii Spherical EA-CCSD/aC5Z EA-CCSD/aCXZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' X=Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 5 EE-CCSD(T)(a)*/aCQZ EA-CCSD/aC5Z(𝑐) Notes: (𝑎) 𝐸𝐼 𝐽 calculated using EE-CC3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (𝑏) Reference for SF calculations: 1𝑆𝑔 ([Ar] 4𝑠2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (𝑐) Reference for EA calculations: 1𝑆𝑔([Ar]) 2000 3000 4000 5000 6000 7000 8000 9000 10000 Wavelength [˚A] 0 100 200 300 400 B [MG] Na i Mg i Ca i 4227 Ca i 6142 Ca ii Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Calculated transition wavelengths as a function of field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For each Zeeman triplet, the line opacities are scaled to the oscillator strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' owing to a turnaround in wavelength at ≃ 240 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This degeneracy is entirely resolved by the large shift of the Mg line which has only one wavelength solution and is also consistent with a field of ≃ 30 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Thus, to our surprise, the peculiar spectrum of SDSS J1143+6615 (Figure 1) can not result from a field in the regime of 100s of MG, but is best explained by a field strength an order of magnitude lower, though notably still a factor three higher than all previously identified DZH white dwarfs (Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2015, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Dufour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For Ca i the match in wavelength is quite poor, though thus far we have neglected wavelength shifts that may arise from radial motion and gravitational redshift, the latter of which could be on the order of 100 km s−1 if SDSS J1143+6615 is particularly massive, which is typically the case for magnetic white dwarfs (Liebert 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Kawka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Ferrario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Additionally, the absent treatment of relativistic effects may here play a role in the quality of the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' It is also clear that at 30 MG, the predicted wavelength for Mg is a similar amount bluer than the line centre (though with greater relative accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' To account for this we fitted the field strength and radial velocity simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We measured the line centres for all three 𝜋-components by simply fitting parabolas to the central few pixels (five for Ca and seven for Mg and Na), constraining them with uncertainties of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='1–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='3 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Performing a least squares fit to the three line centres, we found a magnetic field strength of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='05 MG and a redshift of 74 ± 8 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' With these best fitting values the residuals are −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='7 Å, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 Å, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='8 Å for the Ca, Mg, and Na lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This was a clear improvement for Ca i and Mg i, though provides a somewhat worse result for the Na i line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' With the field strength established from the 𝜋-components, we could then determine the expected wavelengths of the 𝜎-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We make this comparison in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We first investigated the com- ponents of Na and Mg, with their 𝜎-components identified with relative ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In particular the large broad feature at ≃ 6350 Å is es- tablished as the 𝜎+ component of Na, which does not appear blended with any of the other nearby features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Near 5500 Å both the Na 𝜎− and Mg 𝜎+ components are observed, though notably the order of their wavelengths has swapped due to the components crossing at a field strength of ≃ 25 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The Mg 𝜎− component is inferred to be the broad, asymmetric feature at ≃ 4800 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The asymmetry ap- MNRAS 000, 1–14 (2022) 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 Normalised flux Ca i 4220 4240 Wavelength [˚A] 101 102 B [MG] Mg i 5050 5100 5150 5200 Wavelength [˚A] Na i 5800 5850 5900 5950 Wavelength [˚A] Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Top row: Spectral regions covering the suspected 𝜋-components of Ca i, Mg i, and Na i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Bottom row: Predicted wavelengths for the corresponding 𝜋-components as a function of field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In all panels the black dashed lines indicate the field-free vacuum wavelengths for each line, whereas the dotted lines indicate the wavelengths expected for a 30 MG field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' pears more extreme than for the 𝜋-component, which itself is more asymmetric than the 𝜎+ component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This may imply that the degree of neutral broadening affects each component differently, which per- haps is not surprising given that both the perturbations from neutral helium atoms and the magnetic field both alter the energy levels of Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Having identified all components from Na i and Mg i, we pro- ceeded with classifying transitions from Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For the Ca i resonance line, we had already identified the 𝜋-component (rest wavelength at 4227 Å;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' see Figure 5, left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' As our Gemini GMOS spectrum does not go bluer than about 4090 Å, the 𝜎−-component is not covered, and so we were only able to search for the 𝜎+ component which, at 30 MG, has an expected wavelength of 4475 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Indeed, a spectral feature was found at this wavelength which we attribute to the 𝜎+ component (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The final Ca transitions are less certain, though we still make some attempt at their classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For the Ca ii Zeeman triplet (H+K resonance doublet in the absence of an external magnetic field), only the 𝜎+ component is expected to be covered by our GMOS spectrum at a field strength of 30 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' While we detect a feature at the expected wavelength of 4160 Å (Figure 6), the signal- to-noise ratio is somewhat poor at this end of the spectrum, making this assignment less secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' However, it is worth noting that for a 𝑇eff between 5000 K and 7000 K, both Ca i and Ca ii resonance lines are typically observed together in non-magnetic DZs (Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Finally we consider the Ca i 4𝑝 → 5𝑠 transition, which in the absence of an external magnetic field appears as a triplet (due to the spin-orbit interaction) centred on 6142 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In the presence of a strong magnetic field, this transition appears as a Zeeman triplet ex- hibiting the strongest quadratic shift of all the transitions calculated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Nevertheless, weak transitions are observed at all of the ex- pected wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Whether this assignment is correct is debatable: the identified central component at around 6060 Å shows some asym- metry, as is observed in the field-free case (see SDSS J0916+2540 in Figure 10 of Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' On the other hand, the 6142 Å triplet is typically much weaker than the Ca i 4227 Å resonance line, and is only usually visible for extremely large calcium abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Yet, in the case of SDSS J1143+6615, the established components of the 4227 Å Ca i Zeeman triplet are not particularly strong, suggesting that the 6142 Å components would likely be too weak to be visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Given the sheer number of unknown features in the spectrum of SDSS J1143+6615, it is probable that our assignments to the 6142 Å triplet in Figure 6 might also originate from some other source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Many anomalous features in the spectrum of SDSS J1143+6615 remain unclassified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In particular two strong and broad features are observed at wavelengths of ≃ 4570 Å and ≃ 4660 Å, between the 𝜎+- component of the Ca i resonance line, and the 𝜎−-component of Mg i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The strength of these features suggest they originate from another element commonly observed in DZ spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' With the strongest Na, Mg and Ca lines already accounted for, the most likely candidate is therefore Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In the field-free case, a large number of Fe i lines can be found between 4000–4500 Å (see Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2018a, Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Among the strongest transitions in this range are the 3𝐹 → 5𝐺 and 3𝐹 → 3𝐺 multiplets, which share the same lower level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We therefore suggest that the unidentified features at ≃ 4570 Å and ≃ 4660 Å arise from these iron transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Additional unidentified features include broad absorption around 4300 Å (between the 𝜋- and 𝜎+-components MNRAS 000, 1–14 (2022) A DZ white dwarf with a 30 MG magnetic field 9 4000 4500 5000 5500 6000 6500 Wavelength [˚A] 0 10 20 30 40 50 60 B [MG] Na i Mg i Ca i 4227 Ca i 6142 Ca ii Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Line identification diagram for SDSS J1143+6615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The Zeeman triplets from our finite-field coupled-cluster calculations are shown by the solid curves, with the naïve wavelengths from the linear Zeeman effect indicated by the dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' These are plotted over the spectrum of SDSS J1143+6615 (grey), where black dashed lines match Zeeman components to features in the spectrum for a field strength of approximately 30 MG (light grey horizontal band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' of Ca i), sharp features at ≃ 5200/5330/5580 Å, and several other features at ≃ 6030/6450/6530/6620 Å (some of which we were unable to conclusively assign to the Ca i 6142 Å multiplet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We note that the feature near 5200 Å is close to the field-free wavelength of the Cr i 4𝑠 → 4𝑝 triplet (5208 Å, vacuum), and so that feature could plausibly correspond to the 𝜋-component of the Cr i transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Firmly establishing the origin of these remaining features necessarily will require additional finite-field coupled-cluster calculations in the future, with the above Fe and Cr transitions as the highest priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For these systems, treatment of field-dependent relativistic effects and a robust treatment of multi-reference character in the electronic structure will be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 5 MAGNETIC FIELD MODELLING With several of the spectral features of SDSS J1143+6615 identified, we finally sought to model the magnetic field distribution across its surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For a purely dipolar magnetic field, the field strength spans a factor of two between the equator and poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This results in broadened spectral lines, particularly the 𝜎-components due to their stronger wavelength dependence of the field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' It is clear from the width of the Na i 𝜎+ component that the range of magnetic field strengths on the visible hemisphere of SDSS J1143+6615 spans a much narrower field range, with Figure 6 suggesting about 24– 31 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Thus it is necessary to invoke a field structure more complex than a centred dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='1 The offset dipole model We chose to use the offset-dipole model, which has been com- monly used in the analysis of magnetic white dwarf field structures (Achilleos & Wickramasinghe 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This model is similar to a centred-dipole, but allows for the origin of the field to be shifted within the white dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In principle this shift can be applied in three dimensions, but typically it is only applied along the magnetic field axis by a fractional amount of the white dwarf radius, 𝑎𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The offset- dipole model has been successfully applied to many different white dwarfs (Achilleos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Putney & Jordan 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Külebi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 2015) leading to much improved fits with only a single additional free-parameter, which is advantageous compared to a more general multi-pole expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For a centred-dipole with the magnetic field aligned with the 𝑧- axis, the value of the magnetic field at any point on (and external to) the stellar surface in Cartesian coordinates (𝑥/𝑦/𝑧) is given by, ������ 𝐵𝑥 𝐵𝑦 𝐵𝑧 ������ = 𝐵𝑑 2𝑟5 ������ 3𝑥𝑧 3𝑦𝑧 3𝑧2 − 𝑟2 ������ , (9) where 𝐵𝑥/𝑦/𝑧 are the Cartesian components of the magnetic field, 𝐵𝑑 is the polar field strength, and 𝑟2 = 𝑥2 + 𝑦2 + 𝑧2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The offset- dipole model simply requires making the translation 𝑧 ↦→ 𝑧 − 𝑎𝑧, in Equation (9) and in the definition of 𝑟2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' To complete the offset-dipole model we also allow rotation between the magnetic field axis and the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We implement this by considering coordinate systems for both the magnetic field and the viewing direction of the observer, with a rotation matrix used to convert between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Using the above model of the white dwarf magnetic field struc- MNRAS 000, 1–14 (2022) 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' ture, we construct a toy model spectrum by randomly sampling 10,000 points uniformly across the stellar disc (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' sampled uni- formly within the unit circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For each point on the stellar disc, 𝑖, we used Equation (9) to calculate the magnetic field vector (ac- counting for the chosen inclination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Then for each transition, 𝑗, we compute a Zeeman-triplet of three Lorentzian profiles, using our atomic data from Section 3 to determine their wavelengths and oscil- lator strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Furthermore, the 𝜋-component is scaled by a factor sin2 𝜓/2, and the 𝜎-components by a factor (1 + cos2 𝜓)/4, which accounts for linear and circular polarisation effects respectively (Put- ney & Jordan 1995), and where 𝜓 is the angle between the field line and the observer’s line of sight5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' These three Lorentzian components are then summed to form an opacity function 𝜅𝑖 𝑗 (𝜆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 𝐵𝑖, 𝜓𝑖) = +1 ∑︁ Δ𝑚𝑙=−1 𝐿 𝑗 (𝜆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 𝐵𝑖, 𝜓𝑖, Δ𝑚𝑙), (10) where 𝐿 𝑗 are the Lorentzian profiles per transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Finally, the nor- malised flux for all transitions at point 𝑖 is given by 𝐹𝑖(𝜆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 𝐵𝑖, 𝜓𝑖) = exp ��� ��� − ∑︁ 𝑗 𝐴 𝑗𝜅𝑖 𝑗 (𝜆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 𝐵𝑖, 𝜓𝑖) ��� ��� , (11) where 𝐴 𝑗 is a pseudo-abundance which we use to arbitrarily scale the strength of each Zeeman-triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Finally, we compute the integrated flux over the stellar disc as a weighted sum based on the centre-to- limb intensity of the stellar disc 𝐹(𝜆) = � 𝑖 𝐹𝑖(𝜆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 𝐵𝑖, 𝜓𝑖)𝐼(𝜇𝑖) � 𝑖 𝐼(𝜇𝑖) , (12) where 𝐼(𝜇𝑖) is the intensity across the stellar disc, and where 𝜇𝑖 is equivalent to the 𝑧 coordinate of the 𝑖-th point on the stellar disc in the observers frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We use the logarithmic limb-darkening law for a 6000 K, log 𝑔 = 8 white dwarf from Gianninas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='2 Application to SDSS J1143+6615 We applied the offset dipole model to SDSS J1143+6615 initially fo- cusing on the Na triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' From analysing the 𝜋-components of Mg and Na in Section 4, we established a surface averaged field of ≃ 30 MG, and hence located the features corresponding to the 𝜎-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Due to the asymmetry of the Mg components we decided to begin our focus on the Na triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' However, the 𝜎− component of Na and the 𝜎+ component of Mg are somewhat overlapping (≃ 5500 Å), and so we chose to restrict ourselves to the 𝜋 and 𝜎+ components of Na (≃ 6400 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Overall we therefore had five parameters to adjust: the polar field strength 𝐵𝑑, the dipole-inclination, and the dipole-offset 𝑎𝑧, which controlled the field distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' plus the Lorentzian line strength (𝐴 𝑗 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='1) and width which are most easily inferred by the relatively static 𝜋-component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' As described at the start of Section 5, the width of the 𝜎+ compo- nent of Na implies a field strength distribution narrower than the fac- tor of two for a centred dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In the offset-dipole model a narrower distribution can be achieved for negative values of 𝑎𝑧, combined with a low inclination (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' viewed close to pole-on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This implies a wider 5 These oscillator strength scaling factors mean that when the observer looks down a field line, the 𝜋-component vanishes and the 𝜎-components are at maximum intensity, and when the observer looks perpendicular to a field line the 𝜋-component is at maximum intensity with the 𝜎-components at half intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In the absence of a magnetic field where all components overlap, all three scaling factors sum to one for all angles of 𝜓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' distribution of field strengths on the opposite hemisphere of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Because 𝐵𝑑 in Equation (9) no longer corresponds to the field at the poles, for finite 𝑎𝑧, both parameters must be adjusted simultaneously to maintain a polar field strength of 30 MG on the visible hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Manipulating Equation 9, and making the substitution 𝑧 ↦→ 𝑧 − 𝑎𝑧, it can be shown that 𝐵𝑑 = (1 − 𝑎𝑧)3𝐵𝑧=1, (13) where 𝐵𝑧=1 is the near-side pole strength of 30 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Adjusting these parameters by hand6, we found good agreement with the shape of the Na 𝜎+-component could be achieved with 𝑎𝑧 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='15 (imply- ing 𝐵𝑑 = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='6 MG from Equation (13)) and a dipole-inclination of 15 degrees (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This also yields a reasonable agreement with the 𝜎−-component (at wavelengths where it is not blended with the 𝜎+-component from Mg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' We then included all other transitions from Section 3 into the model adjusting only the strengths and widths of the Lorentzian profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' A further refinement is required for the Mg i and Ca i 6142 Å triplets as these are 𝑛𝑝 → (𝑛 + 1)𝑠 transitions (the others are all 𝑛𝑠 → 𝑛𝑝), and so we scale the component strengths by Boltzmann factors reflecting the different occupation levels of the lower states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Unsurprisingly, the Lorentzian profiles used provide a poor fit for the asymmetric 𝜋- and 𝜎−-components of Mg i, though reason- able agreement is found for the 𝜎+-component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' As discussed pre- viously, this may indicate that the degree of neutral broadening is field-dependent, and affects the bluer components more strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For the Ca i 4227 Å resonance line, when the width and strength param- eters are adjusted to match the 𝜋-component, the strength and shape of the 𝜎+-component (≃ 4090 Å) also agree well with the observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This demonstrates that the values of 𝐵𝑑, 𝑎𝑧, and the inclination found from the Na lines are also appropriate for this transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For the Ca ii triplet, the width of the 𝜎+ component is also seen to be in agreement with the data, though the signal-to-noise ratio in this part of the spectrum is too poor to compare the shape of the line with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Finally for the Ca i 6142 Å Zeeman-triplet, only the shape of the 𝜎+-component in is reasonable agreement with the data, further- ing the argument from Section 4 that these transitions may originate from another source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 6 DISCUSSION 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='1 DZs with much stronger fields In Section 5, we constructed a toy-model for generating simplified magnetic DZ spectra, including atomic data from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' While it turned out that SDSS J1143+6615 has only a 30 MG field, in principle our model allows us to generate synthetic spectra for much larger fields, with 470 MG covering all the transitions we calculated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Since ongoing/upcoming spectroscopic surveys such as WEAVE, DESI, SDSS V, and 4MOST, are expected to yield hundreds of thousands of white dwarf spectra in the next decade, we investigate which transitions ought to be focused on for identifying even higher field DZ stars in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In Figure 8, we show models with average surface fields span- ning 25–400 MG against the same curves from Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For all five models, we used the same inclination and dipole offset as found for SDSS J1143+6615, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 15 degrees and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='15 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Note 6 While we did attempt a more rigorous least-squares fit to the data, the lack of a well-defined continuum led to worse results than manual adjustment of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) A DZ white dwarf with a 30 MG magnetic field 11 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='5 x −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='5 y i = 15◦ az = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='15 4500 5000 5500 6000 6500 Wavelength [˚A] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='2 Normalised flux Bd = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='6 MG visible: 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='8 − 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 MG Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Left: Visualisation of the field structure of SDSS J1143+6615 modeled with an offset-dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Right: The simulated absorption spectrum of SDSS J1143+6615 (red) using data from our finite-field coupled cluster calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' that the 𝐵𝑠 values are the surface averaged field strengths whereas the dipole field strength, 𝐵𝑑, is approximately 52 percent larger (see Equation 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The bottom model has a field 𝐵𝑠 = 25 MG, similar in strength to that found for SDSS J1143+6615, and thus shows a similar spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Despite the relatively uniform field for an inclination of 15 degrees and 𝑎𝑧 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='15, as the field increases, the 𝜎-components still be- come washed out, and for most of the transitions are almost invisible at fields of around 100 MG and above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For Mg i the 𝜎+ component still remains visible above 100 MG due to its increase in oscillator strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' On the other hand, most of the 𝜋-components remain relatively steady in wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For the Na 𝜋-component, as already noted in Hampe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' (2020), the wavelength changes very little below 100 MG, leaving this line similarly sharp as for a 25 MG field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The Na line reaches a maximum in blue-shift at 240 MG (100 Å bluer than the field free wavelength), before rapidly turning around and moving to redder wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Therefore for 𝐵𝑠 = 400 MG, the line becomes much broader, but remains clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Therefore this transition ought to be used as a primary marker for identifying cool magnetic DZ white dwarfs with > 100 MG fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Similarly the Ca i 𝜋-component remains relatively stationary up to 100 MG, but becomes more washed out for larger fields due to the quadratic Zeeman effect, and becoming broadened to a width of 100 Å at 400 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Therefore this line is likely to be less reliable than the Na 𝜋-component for identifying the highest field DZs, but will still remain reliable up to 200 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The Ca ii 𝜋-component is also near stationary, and should still be recognisable even at 400 MG, making this a more obvious choice for identifying warmer high field DZs where the Na i and Ca i lines may be too weak to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Note that at 300 MG, the Ca i and Ca ii 𝜋-components overlap producing a blended spectral feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Finally, the Mg i 𝜋-component experiences a much larger quadratic shift than the other transitions considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Therefore at 400 MG, the line appears broad and asymmetric though is notably still visible, in part due to the increased oscillator strength for this component, which is close to four times larger than in the field-free case, thereby also showcasing the importance of considering field-dependent in- tensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Note that this toy-model does not consider the intrinsic asymmetry caused by neutral broadening, which itself could be a function of field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' A final consideration is that we have not yet identified all the fea- tures in the spectrum of SDSS J1143+6615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Therefore at very high field strengths of 100s of MG, these unclassified features will also ap- pear shifted into other parts of the spectrum further complicating the identification of the transitions discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Furthermore, other strong lines outside the optical such as the Mg i and Mg ii resonance lines (field free wavelengths at 2853 Å and 2799 Å, respectively), may find some of their Zeeman split components shifted into the optical providing other atomic features requiring identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='2 Use in model atmospheres Ideally the atomic data we have presented in Section 3 can be utilised in white dwarf model atmospheres for more detailed analyses of magnetic DZ stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' As we have shown in this work, however, this is not necessary for a basic assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For simply determining the surface-average field strength, 𝐵𝑠, and which ions are present in the atmosphere, it is sufficient to simply compare our atomic data with the spectrum in question, as was demonstrated in Section 4 for SDSS J1143+6615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Furthermore, determining the field structure of a white dwarf can be achieved with a simple model such as the toy-model we demonstrated in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Importantly our toy-model is computationally efficient, taking only a few seconds to produce Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Of course, much can still be learned from incorporating our atomic data into model atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In our toy-model from Section 5, the strength and widths of the Lorentzian profiles we used have no phys- ical basis, and are simply adjusted to give acceptable agreement with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In a model atmosphere, the strengths and widths of the fea- tures seen in the spectrum of SDSS J1143+6615 can be investigated by adjusting the abundances and 𝑇eff (and to some extent the surface gravity) of the model, allowing these atmospheric properties to be measured in a physically meaningful way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The main challenge of such an approach is the computation time required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In the field-free case, the final model spectrum is inte- grated over the stellar disc from spectra calculated at different angles between the surface-normal and the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For finite-fields, how- ever, the synthetic spectra must also be calculated over a grid of field strengths and angles between the field and observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' In particular, the field strength axis of the grid must be computed at sufficiently fine steps so that artefacts from undersampling are not present when MNRAS 000, 1–14 (2022) 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Hollands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 4000 5000 6000 7000 8000 Wavelength [˚A] 0 1 2 3 4 5 Normalised flux Bs = 25 MG Bs = 50 MG Bs = 100 MG Bs = 200 MG Bs = 400 MG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='6 log(B/MG) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Simulated magnetic DZ spectra for five different surface averaged field strengths (𝐵𝑠), with each spectrum offset from one another by 1 in normalised flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The inclination and dipole offset parameters are fixed to the values found for SDSS J1143+6615 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 15 degrees and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='15, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The background Zeeman triplets have the same meaning as in Figure 4, with the field strength scale given on the right-hand axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' integrating over the stellar disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Therefore, depending on the range of field strengths required, computation may take hundreds to thou- sands of times longer than in the field-free case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' If the 𝑇eff, log 𝑔, or abundances require refinement when comparing against a particular spectrum, the grid must then be recomputed with updated atmo- spheric parameters, leading to an even larger amount of computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' For that reason we have refrained from including our atomic data within model atmospheres at the present time, and also because it exceeds the scope of our primary goals of classifying the spectral features of SDSS J1143+6615 and measuring its field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' How- ever, future work should perform a detailed atmospheric analysis of SDSS J1143+6615 utilising the atomic data presented here to mea- sure its 𝑇eff, log 𝑔, and abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 7 CONCLUSIONS We have identified SDSS J1143+6615 as DZ white dwarf with strong magnetic field resulting in its unique spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Using finite-field, coupled-cluster calculations we were able to identify lines from Na i, Mg i, and Ca i–ii that were split and shifted by the linear and quadratic Zeeman effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' This also allowed us to establish a field strength of ≃ 30 MG, demonstrating that DZ spectra are challenging to interpret at only a few 10 MG, due to multiple overlapping transitions from a variety of chemical elements, which is not the case for magnetic DAs or DBs at the same field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Using the offset-dipole model, we were able to obtain an adequate fit to the spectral features of Na with an almost pole-on observation angle, and the dipole offset away from the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Despite our success in elucidating some of the peculiar features in the spectrum of SDSS J1143+6615, many transitions still lack classi- fication at the present time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Giving consideration to the elements and lines most commonly encountered in non-magnetic cool DZ stars, future atomic data calculations should concentrate on Fe and Cr lines, as well as additional transitions of Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Because SDSS J1143+6615 is currently the only available test for these calculations, and only samples the relatively low-field end, searching for additional high- field DZs within ongoing and future spectroscopic surveys (such as SDSS V, WEAVE, and DESI) is imperative to test the accuracy of our atomic data further at field strengths of many 100 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) A DZ white dwarf with a 30 MG magnetic field 13 ACKNOWLEDGEMENTS MAH was supported by grant ST/V000853/1 from the Science and Technology Facilities Council (STFC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) grant number STO 1239/1-1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' and within project B5 of the TRR 146 (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' 233 630 050).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Based on observations obtained at the international Gemini Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' a program of NSF’s NOIRLab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' which is managed by the Association of Universities for Research in Astronomy (AURA) un- der a cooperative agreement with the National Science Foundation on behalf of the Gemini Observatory partnership: the National Science Foundation (United States),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' National Research Council (Canada),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Agencia Nacional de Investigación y Desarrollo (Chile),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Ministe- rio de Ciencia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Tecnología e Innovación (Argentina),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Ministério da Ciência,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Tecnologia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Inovações e Comunicações (Brazil),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' and Korea Astronomy and Space Science Institute (Republic of Korea).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' DATA AVAILABILITY All observations of SDSS J1143+6615 are either public (SDSS, Gaia) or no longer proprietary (Gemini).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The Gemini data can be be ob- tained from the Gemini Observatory Archive, using program ID GN-2015B-FT-29.' 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Koester D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=', Dufour P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=', Melis C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=', Klein B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=', Jura M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=', 2011, ApJ, 739, 101 APPENDIX A: ATOMIC DATA TABLES This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) A DZ white dwarf with a 30 MG magnetic field 15 Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Atomic data for the Na i Zeeman triplet under an applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' The magnetic field strength 𝐵, is given in both atomic units and in MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Wavelengths are given in vacuum form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Oscillator strengths are calculated according to Equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE3T4oBgHgl3EQfsAry/content/2301.04665v1.pdf'} +page_content=' Wavelength [Å] 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@@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:acb83f2e880f171963957ae7e6ff862956d9de1af81b54b6c33784a2cc04fe85 +size 103146 diff --git a/fNE4T4oBgHgl3EQfRAzN/content/tmp_files/2301.04987v1.pdf.txt b/fNE4T4oBgHgl3EQfRAzN/content/tmp_files/2301.04987v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..576091f30908f477f2db93bcd7801400b2d84457 --- /dev/null +++ b/fNE4T4oBgHgl3EQfRAzN/content/tmp_files/2301.04987v1.pdf.txt @@ -0,0 +1,564 @@ +arXiv:2301.04987v1 [math.CO] 12 Jan 2023 +On perfect coverings of two-dimensional grids +Elias Heikkil¨a, Pyry Herva and Jarkko Kari +Abstract +We study perfect multiple coverings in translation invariant graphs with vertex set +Z2 using an algebraic approach. In this approach we consider any such covering as a +two-dimensional binary configuration which we then express as a two-variate formal +power series. +Using known results, we conclude that any perfect multiple covering +has a non-trivial periodizer, that is, there exists a non-zero polynomial whose formal +product with the power series presenting the covering is a two-periodic configuration. +If a non-trivial periodizer has line polynomial factors in at most one direction, then +the configuration is known to be periodic. Using this result we find many setups where +perfect multiple coverings of infinite grids are necessarily periodic. We also consider +some algorithmic questions on finding perfect multiple coverings. +1 +Introduction and preliminaries +A perfect multiple covering in a graph is a set of vertices, a code, such that the number of +codewords in the neighborhood of an arbitrary vertex depends only on whether the vertex is +in the code or not. In this paper we study these codes on translation invariant graphs with +the vertex set Z2. We present codes as two-dimensional binary configurations and observe +that the perfect covering condition provides an algebraic condition that can be treated with +the algebraic tools developed in [8]. We focus on periodic codes and, in particular, study +setups where all codes are necessarily periodic. The approach we take was initially mentioned +in an example in the survey [6] by the third author, and considered in the Master’s thesis +[5] by the first author. +We start by giving the basic definitions, presenting the aforementioned algebraic approach +and stating some past results relevant to us. In Section 2 we describe an algorithm to find the +line polynomial factors of any given (Laurent) polynomial. In Section 3 we formally define +the perfect multiple coverings in graphs and prove some periodicity results concerning them. +We give new algebraic proofs of some known results concerning perfect multiple coverings +on the infinite square grid and on the triangular grid +[1, 12], and provide a new result +on the forced periodicity of such coverings on the king grid. Furthermore, we generalize +the definition of perfect coverings for two-dimensional binary configurations with respect to +different neighborhoods and covering constants. In Section 4 we consider some algorithmic +questions concerning perfect coverings. Using a standard argument by H. Wang we show +1 + +that under certain constraints it is algorithmically decidable to determine whether there +exist any perfect coverings with given neighborhood and given covering constants. +Configurations, periodicity, finite patterns and subshifts +A d-dimensional configuration is a coloring of the infinite grid Zd using finitely many colors, +that is, an element of AZd which we call the d-dimensional configuration space where A +is some finite alphabet. For a configuration c we let cu = c(u) to be the symbol or color +that c has in cell u. The translation τ t by a vector t ∈ Zd shifts a configuration c such +that τ t(c)u = cu−t for all u ∈ Zd. A configuration c is t-periodic if τ t(c) = c and c is +periodic if c is t-periodic for some non-zero t ∈ Zd. We also say that a configuration c is +periodic in direction v ∈ Zd \ {0} if c is kv-periodic for some k ∈ Q. A d-dimensional +configuration c is strongly periodic if it has d linearly independent vectors of periodicity. +Strongly periodic configurations are then periodic in all directions. Two-dimensional strongly +periodic configurations are called two-periodic. +A finite pattern is an assignment of symbols on some finite shape D ⊆ Zd, that is, an +element of AD where A is some fixed alphabet. In particular, the finite patterns in AD +are called D-patterns. Let us denote by A∗ the set of all finite patterns over alphabet A +where the dimension d is known from the context. A finite pattern p ∈ AD appears in a +configuration c ∈ AZd if τ t(c)|D = p for some t ∈ Zd. A configuration c contains the pattern +p if it appears in c. For a fixed shape D, the set of all D-patterns that appear in c is the set +LD(c) = {τ t(c)|D | t ∈ Zd} and the set of all finite patterns in c is denoted by L(c) which +we call the language of c. For a set S ⊆ AZd of configurations we define LD(S) and L(S) as +the unions of LD(c) and L(c) over all c ∈ S, respectively. +Let us review some basic concepts of symbolic dynamics we need. For a reference see +e.g. [3, 10, 11]. The configuration space AZd can be made a compact topological space by +endowing A with the discrete topology and considering the product topology it induces on +AZd – the prodiscrete topology. This topology is induced by a metric where two configurations +are close if they agree on a large area around the origin. Thus AZd is a compact metric space. +A subset S ⊆ AZd of the configuration space is a subshift if it is topologically closed and +translation-invariant meaning that if c ∈ S then for any t ∈ Zd also τ t(c) ∈ S. Equivalently +we can define subshifts using forbidden patterns: Given a set F ⊆ A∗ of forbidden finite +patterns, the set +XF = {c ∈ AZd | L(c) ∩ F = ∅} +of configurations that avoid all forbidden patterns is a subshift, and every subshift is obtained +by forbidding some set of finite patterns. If F ⊆ A∗ is finite then we say that XF is a subshift +of finite type (SFT). +The orbit of a configuration c is the set O(c) = {τ t(c) | t ∈ Zd} of its every translate. +The orbit closure O(c) is the topological closure of its orbit under the prodiscrete topology. +The orbit closure of a configuration c is the smallest subshift that contains c. It consists of +all configurations c′ such that L(c′) ⊆ L(c). +2 + +The algebraic approach +To present a configuration c ∈ AZd algebraically we make the assumption that A ⊆ Z. Then +we identify the configuration c with the formal power series +c(X) = +� +u∈Zd +cuXu +over d variables x1, . . . , xd where we have denoted X = (x1, . . . , xd) and Xu = xu1 +1 · · · xud +d for +any u = (u1, . . . , ud) ∈ Zd. For d = 2 we usually denote X = (x, y). More generally we study +the set of all formal power series over d variables x1, . . . , xd with complex coefficients which +we denote by C[[X±1]] = C[[x±1 +1 , . . . , x±1 +d ]]. A power series is finitary if it has only finitely +many different coefficients and integral if its coefficients are all integers. Thus we identify +configurations with finitary and integral power series. +We also use Laurent polynomials which we call from now on simply polynomials. We +use the term “proper” when we talk about proper (i.e., non-Laurent) polynomials. Let us +denote by C[X±1] = C[x±1 +1 , . . . , x±1 +d ] the set of all (Laurent) polynomials over d variables +x1, . . . , xd with complex coefficients, which is the Laurent polynomial ring. We say that two +polynomials have no common factors if all of their common factors are units and that they +have a common factor if they have a non–unit common factor. +A product of a polynomial and a power series is well defined. We say that a polynomial +f = f(X) annihilates (or is an annihilator of) a power series c = c(X) if fc = 0, that is, +if their product is the zero power series. We say that a formal power series c = c(X) is +periodic if it is annihilated by a difference polynomial Xt − 1 where t is non-zero. Note that +this definition is consistent with the definition of periodicity of configurations defined above. +Indeed if c = c(X) is a configuration then multiplying it by a monomial Xt produces the +translated configuration τ t(c) and hence c is t-periodic if and only if c = τ t(c) = Xtc, which +is equivalent to (Xt − 1)c = 0. So it is natural to study the annihilator ideal +Ann(c) = {f ∈ C[X±1] | fc = 0} +of a power series c ∈ C[[X±1]], which indeed is an ideal of the Laurent polynomial ring. Hence +the question whether a configuration (or any formal power series) is periodic is equivalent to +asking whether its annihilator ideal contains a difference polynomial. Another useful ideal +that we study is the periodizer ideal +Per(c) = {f ∈ C[X±1] | fc is strongly periodic}. +Note that clearly Ann(c) is a subset of Per(c). Note also that a configuration c has a non- +trivial (= non-zero) annihilator if and only if it has a non-trivial periodizer. The following +theorem states that if a configuration has a non-trivial periodizer then it has in fact an +annihilator of a particular simple form – a product of difference polynomials. +Theorem 1 ([8]). Let c be a configuration in any dimension that has a non-trivial periodizer. +Then there exist pairwise linearly independent vectors t1, . . . , tm with m ≥ 1 such that +(Xt1 − 1) · · · (Xtm − 1) ∈ Ann(c). +3 + +Line polynomials +The support of a power series c = � +u∈Zd cuXu is the set supp(c) = {u ∈ Zd | cu ̸= 0}. +Thus a polynomial is a power series with a finite support. A line polynomial is a polynomial +whose support contains at least two points and the points of the support lie on a unique line. +In other words, a polynomial f is a line polynomial if it is not a monomial and there exist +vectors u, v ∈ Zd such that supp(f) ⊆ u+Qv. In this case we say that f is a line polynomial +in direction v. We say that non-zero vectors v, v′ ∈ Zd are parallel if v′ ∈ Qv, and clearly +then a line polynomial in direction v is also a line polynomial in any parallel direction. A +vector v ∈ Zd is primitive if its components are pairwise relatively prime. If v is primitive +then Qv∩Zd = Zv. For any non-zero v ∈ Zd there exists a parallel primitive vector v′ ∈ Zd. +It follows that we may assume the vector v in the definition of a line polynomial f to be +primitive so that supp(f) ⊆ u+Zv. In the following our preferred presentations of directions +are in terms of primitive vectors. +Any line polynomial φ in a (primitive) direction v can be written uniquely in the form +φ = Xu(a0 + a1Xv + . . . + anXnv) = Xu(a0 + a1t + . . . + antn) +where u ∈ Zd, n ≥ 1, a0 ̸= 0, an ̸= 0 and t = Xv. Let us call the single variable proper +polynomial a0 + a1t + . . . + antn ∈ C[t] the normal form of φ. Moreover, for a monomial +aXu we define its normal form to be a. Thus two line polynomials in the direction v have +the same normal form if and only if they are the same polynomial up to multiplication by +Xu, for some u ∈ Zd. +Difference polynomials are line polynomials and hence the annihilator provided by The- +orem 1 is a product of line polynomials. Annihilation by a difference polynomial means +periodicity. +More generally, annihilation of a configuration c by a line polynomial in a +primitive direction v can be understood as the annihilation of the one-dimensional v-fibers +� +k∈Z cu+kvXu+kv of c in direction v, and since annihilation in the one-dimensional setting +implies periodicity we conclude that a configuration is periodic if and only if it is annihilated +by a line polynomial. It is known that if c has a periodizer with line polynomial factors in +at most one direction then c is periodic: +Theorem 2 ([9]). Let c be a two-dimensional configuration and f ∈ Per(c). +Then the +following conditions hold. +• If f does not have any line polynomial factors then c is two-periodic. +• If all line polynomial factors of f are in the same direction then c is periodic in this +direction. +Proof sketch. The periodizer ideal Per(c) is a principal ideal generated by a polynomial +g = φ1 · · · φm where φ1, . . . , φm are line polynomials in pairwise non-parallel directions [9]. +Because f ∈ Per(c) we know that g divides f. If f does not have any line polynomial factors +then g = 1 and thus c = gc is two-periodic. If f has line polynomial factors and they are +in the same primitive direction v then g is a line polynomial in this direction. Since gc is +4 + +two-periodic it is annihilated by (Xkv − 1) for some k ∈ Z. Then the configuration c is +annihilated by the line polynomial (Xkv − 1)g in direction v. We conclude that c is periodic +in direction v. +(See the Appendix for an alternative proof that mimics the usage of resultants in [7], instead +of relying on the structure of the ideal Per(c).) +2 +Line polynomial factors +The open and closed discrete half planes determined by a non-zero vector v ∈ Z2 are the +sets Hv = {u ∈ Z2 | ⟨u, v⊥⟩ > 0} and Hv = {u ∈ Z2 | ⟨u, v⊥⟩ ≥ 0}, respectively, where +v⊥ = (v2, −v1) is orthogonal to v = (v1, v2). Let us also denote by lv = Hv \ Hv the discrete +line parallel to v that goes through the origin. In other words, the half plane determined by +v is the half plane “to the right” of the line lv when moving along the line in the direction of +v. We say that a finite set D ⊆ Z2 has an outer edge in direction v if there exists a vector +t ∈ Z2 such that D ⊆ Hv + t and |D ∩ (lv + t)| ≥ 2. We then call D ∩ (lv + t) an outer edge +of D in direction v. An outer edge corresponding to v means that the convex hull of D has +an edge in direction v in the clockwise orientation around D. +If a finite non-empty set D does not have an outer edge in direction v then there exists +a vector t ∈ Z2 such that D ⊆ Hv + t and |D ∩ (lv + t)| = 1 and then we say that D has a +vertex in direction v and we call D ∩ (lv + t) a vertex of D in direction v. We say that a +polynomial f has an outer edge or a vertex in direction v if its support has an outer edge or +a vertex in direction v, respectively. Note that every finite shape D has either an edge or a +vertex in any non-zero direction. Note also that in this context directions v and −v are not +the same: a shape may have an outer edge in direction v but no outer edge in direction −v. +The following lemma shows that a polynomial can have line polynomial factors only in the +directions of its outer edges. +Lemma 3 ([7]). Let f be a non-zero polynomial with a line polynomial factor in direction +v. Then f has outer edges in directions v and −v. +Let v ∈ Z2 \ {0} be any non-zero primitive vector and let f = � fuXu be a polynomial. +Recall that a v-fiber of f is a polynomial of the form � +k∈Z fu+kvXu+kv for some u ∈ Z2. +Thus a non-zero v-fiber of a polynomial is either a line polynomial or a monomial. Let us +denote by Fv(f) the set of different normal forms of all non-zero v-fibers of a polynomial f, +which is thus a finite set. The following simple example illustrates the concept of fibers and +their normal forms. +Example 4. Let us determine the set Fv(f) for f = f(X) = f(x, y) = 3x + y + xy2 + xy + +x3y3 + x4y4 and v = (1, 1). By grouping the terms we can write +f = 3x + y(1 + xy) + xy(1 + x2y2 + x3y3) = X(1,0) · 3 + X(0,1)(1 + t) + X(1,1)(1 + t2 + t3) +where t = X(1,1) = xy. Hence Fv(f) = {3, 1 + t, 1 + t2 + t3}. See Figure 1 for a pictorial +illustration. +5 + +3x +y +xy2 +xy +x3y3 +x4y4 +Figure 1: The support of f = 3x + y + xy2 + xy + x3y3 + x4y4 and its different (1, 1)-fibers. +As noticed in the example above, polynomials are linear combinations of their fibers: for any +polynomial f and any non-zero primitive vector v we can write +f = Xu1ψ1 + . . . + Xunψn +for some u1, . . . , un ∈ Z2 where ψ1, . . . , ψn ∈ Fv(f). We use this in the proof of the next +theorem. +Theorem 5. A polynomial f has a line polynomial factor in direction v if and only if the +polynomials in Fv(f) have a common factor. +Proof. For any line polynomial φ in direction v, and for any polynomial g, the v-fibers of the +product φg have a common factor φ. In other words, if a polynomial f has a line polynomial +factor φ in direction v then the polynomials in Fv(f) have the normal form of φ as a common +factor. +For the converse direction, assume that the polynomials in Fv(f) have a common factor +φ which is thus a line polynomial in direction v. Then there exist vectors u1, . . . , un ∈ Z2 +and polynomials φψ1, . . . , φψn ∈ Fv(f) such that +f = Xu1φψ1 + . . . + Xunφψn. +Hence φ is a line polynomial factor of f in direction v. +Note that Lemma 3 actually follows immediately from Theorem 5: A vertex instead of +an outer edge in direction v or −v provides a non-zero monomial v-fiber, which implies that +the polynomials in Fv(f) have no common factors. +Thus to find out the line polynomial factors of f we first need to find out the possible +directions of the line polynomials, that is, the directions of the (finitely many) outer edges +of f, and then we need to check for which of these possible directions v the polynomials in +Fv(f) have a common factor. There are clearly algorithms to find the outer edges of a given +polynomial and to determine whether finitely many line polynomials have a common factor. +If such a factor exists then f has a line polynomial factor in this direction by Theorem 5. +Thus we have proved the following theorem. +Theorem 6. There is an algorithm to find the line polynomial factors of a given (Laurent) +polynomial. +6 + +(a) The square grid +(b) The king grid +(c) The triangular grid +Figure 2: The 1-neighborhoods of the black vertex in (a) the square grid, (b) the king grid, +and (c) the triangular grid. +3 +Perfect coverings +In this paper a graph is a tuple G = (V, E) where V is the (possibly infinite) vertex set of +G and E ⊆ {{u, v} | u, v ∈ V, u ̸= v} is the edge set of G. Thus the graphs we consider are +simple and undirected. We also assume that all vertices have only finitely many neighbors in +the graph. For a graph G = (V, E) we call any subset S ⊆ V of the vertex set a code in G. +The distance d(u, v) of two vertices u, v ∈ V is the length of a shortest path between them. +The (closed) r-neighborhood of a vertex u ∈ V is the set Nr(u) = {v ∈ V | d(v, u) ≤ r}, that +is, the ball of radius r centered at u. Let us now give the definition of the family of codes +we consider. +Definition 7. Let G = (V, E) be a graph. A code S ⊆ V is an (r, b, a)-covering in G for +non-negative integers b and a if the r-neighborhood of every vertex in S contains exactly b +elements of S and the r-neighborhood of every vertex not in S contains exactly a elements +of S, that is, if for every u ∈ V +|Nr(u) ∩ S| = +� +b if u ∈ S +a if u ̸∈ S +. +By a perfect (multiple) covering we mean any (r, b, a)-covering. +3.1 +Infinite grids +An infinite grid is a translation invariant graph with the vertex set Z2. In other words, +in infinite grids we have Nr(u) = u + Nr(0) for all u ∈ Z2. The square grid is the graph +(Z2, ES) with ES = {{u, v} | u−v ∈ {(±1, 0), (0, ±1)}}, the king grid is the graph (Z2, EK) +with EK = {{u, v} | u − v ∈ {(±1, 0), (0, ±1), (±1, ±1)}} and the triangular grid is the +graph (Z2, ET ) with ET = {{u, v} | u − v ∈ {(±1, 0), (0, ±1), (1, 1), (−1, −1)}}. See Figure +2 for the 1-neighborhoods of a vertex in these graphs. A code S ⊆ Z2 is periodic if S = S +t +for some non-zero t ∈ Z2. It is two-periodic if S = S + t1 and S = S + t2 where t1 and t2 +are linearly independent. The following result is by Axenovich. +Theorem 8 ([1]). If b − a ̸= 1 then any (1, b, a)-covering in the square grid is two-periodic. +A code S ⊆ Z2 in any infinite grid can be presented as a configuration c ∈ {0, 1}Z2 which is +defined such that cu = 1 if u ∈ S and cu = 0 if u ̸∈ S. The positioning of the codewords in +the r-neighborhood of any vertex u ∈ Z2 is then presented as a finite pattern c|u+Nr(0). +7 + +Definition 9. A configuration c ∈ {0, 1}Z2 is a (D, b, a)-covering for a finite shape D ⊆ Z2 +(the neighborhood) and non-negative integers b and a (the covering constants) if for all +u ∈ Z2 the pattern c|u+D contains exactly b symbols 1 if cu = 1 and exactly a symbols 1 if +cu = 0. +We call also any (D, b, a)-covering perfect and hence a perfect covering is either a code in a +graph or a two-dimensional binary configuration. +Definitions 7 and 9 are consistent in infinite grids: a code S in an infinite grid G is +an (r, b, a)-covering if and only if the configuration c ∈ {0, 1}Z2 presenting S is a (D, b, a)- +covering where D is the r-neighborhood of 0 in G. For a set D ⊆ Z2 we define its char- +acteristic polynomial to be fD(X) = � +u∈D X−u. +Let us denote by +1(X) the constant +power series � +u∈Z2 Xu. +If c is a (D, b, a)-covering then from the definition we get that +fD(X)c(X) = (b − a)c(X) + a1(X) which is equivalent to (fD(X) − (b − a)) c(X) = a1(X). +Thus if c is a (D, b, a)-covering then fD(X) − (b − a) ∈ Per(c). Using our formulation we get +a simple proof for Theorem 8: +Reformulation of Theorem 8. Let D be the 1-neighborhood of 0 in the square grid and +assume that b − a ̸= 1. Then every (D, b, a)-covering is two-periodic. +Proof. Let c be an arbitrary (D, b, a)-covering. We show that g = fD − (b − a) = x−1 + +y−1 + 1 − (b − a) + x + y ∈ Per(c) has no line polynomial factors. Then c is two-periodic by +Theorem 2. The outer edges of g are in directions (1, 1), (−1, −1), (1, −1) and (−1, 1) and +hence by Lemma 3 any line polynomial factor of g is either in direction (1, 1) or (1, −1). For +v ∈ {(1, 1), (1, −1)} we have Fv(g) = {1 + t, 1 − (b − a)}. See Figure 3 for an illustration. +Since 1 − (b − a) is a non-trivial monomial, by Theorem 5 the periodizer g ∈ Per(c) has no +line polynomial factors. +The following result was already proved in a more general form in [12]. We give a short proof +using our algebraic approach. +Theorem 10 ([12]). Let r ≥ 2 and let D be the r-neighborhood of 0 in the square grid. Then +every (D, b, a)-covering is two-periodic. In other words, all (r, b, a)-coverings in the square +grid are two-periodic for all r ≥ 2. +Proof. Let c be an arbitrary (D, b, a)-covering. Again, by Theorem 2, it is enough to show +that g = fD − (b − a) ∈ Per(c) has no line polynomial factors. +By Lemma 3 any line +polynomial factor of g has direction (1, 1) or (1, −1). So assume that v ∈ {(1, 1), (1, −1)}. +We have φ1 = 1+t+. . .+tr ∈ Fv(g) and φ2 = 1+t+. . .+tr−1 ∈ Fv(g). See Figure 3 for an +illustration in the case r = 2. Since φ1−φ2 = tr, the polynomials φ1 and φ2 have no common +factors, and hence by Theorem 5 the periodizer g has no line polynomial factors. +If a ̸= b then for all r ≥ 1 any (r, b, a)-covering in the king grid is two-periodic: +Theorem 11. Let r ≥ 1 be arbitrary and let D be the r-neighborhood of 0 in the king grid +and assume that a ̸= b. Then any (D, b, a)-covering is two-periodic. In other words, all +(r, b, a)-coverings in the king grid are two-periodic whenever a ̸= b. +8 + +1 + t +1 − (b − a) +1 + t + t2 +1 + t +1 + t + t2 + t3 + t4 +1 + t + (1 − (b − a))t2 + t3 + t4 +Figure 3: The constellation on the left illustrates the proof of Theorem 8, the constellation +on the center illustrates the proof of Theorem 10 with r = 2 and the constellation on the +right illustrates the proof of Theorem 11 with r = 2. +Proof. Let c be an arbitrary (D, b, a)-covering. By Theorem 2 it is sufficient to show that +g = fD − (b − a) has no line polynomial factors. The outer edges of g are in directions +(1, 0), (−1, 0), (0, 1) and (0, −1). Hence by Lemma 3 any line polynomial factor of g has +direction (1, 0) or (0, 1). Let v ∈ {(1, 0), (0, 1)}. We have φ1 = 1 + t + . . . + tr−1 + (1 − +(b − a))tr + tr+1 + . . . + t2r ∈ Fv(g) and φ2 = 1 + t + . . . + t2r ∈ Fv(g). See Figure 3 for an +illustration with r = 2. Since φ2 − φ1 = (b − a)tr is a non-trivial monomial, φ1 and φ2 have +no common factors. Thus g has no line polynomial factors by Theorem 5. +Similarly as in the square grid we can give simple proofs for known results from [12] con- +cerning forced periodicity in the triangular grid: +Theorem 12 ([12]). Let D be the 1-neighborhood of 0 in the triangular grid and assume that +b − a ̸= −1. Then every (D, b, a)-covering in the triangular grid is two-periodic. In other +words, all (1, b, a)-coverings in the triangular grid are two-periodic whenever b − a ̸= −1. +Theorem 13 ([12]). Let r ≥ 2 and let D be the r-neighborhood of 0 in the triangular grid. +Then every (D, b, a)-covering is two-periodic. In other words, all (r, b, a)-coverings in the +triangular grid are two-periodic for r ≥ 2. +3.2 +General convex neighborhoods +A shape D ⊆ Z2 is convex if it is the intersection D = conv(D) ∩ Z2 where conv(D) ⊆ R2 is +the real convex hull of D. +Let D ⊆ Z2 be a finite convex shape. +Any (D, b, a)-covering has a periodizer g = +fD − (b − a). As earlier, we study whether g has any line polynomial factors. +For any +v ̸= 0 the set Fv(fD) contains only polynomials φn = 1 + . . . + tn−1 for different n ≥ 1 since +D is convex: if D contains two points then D contains every point between them. Thus +Fv(g) contains only polynomials φn for different n ≥ 1 and, if b − a ̸= 0, also a polynomial +φn0 − (b − a)tm0 for some n0 ≥ 1 such that φn0 ∈ Fv(fD) and for some m0 ≥ 0. If b − a = 0 +then g = fD and thus Fv(g) = Fv(fD). +Two polynomials φm and φn have a common factor if and only if gcd(m, n) > 1. More gen- +erally, the polynomials φn1, . . . , φnr have a common factor if and only if d = gcd(n1, . . . , nr) > +9 + +1 and, in fact, their greatest common factor is the dth cyclotomic polynomial +� +1≤k≤d +gcd(k,d)=1 +(t − ei· 2πk +d ). +Let us introduce the following notation. For any polynomial f, we denote by F ′ +v(f) the +set of normal forms of the non-zero fibers � +k∈Z fu+kvXu+kv for all u ̸∈ Zv. In other words, +we exclude the fiber through the origin. Let us also denote fibv(f) for the normal form of +the fiber � +k∈Z fkvXkv through the origin. We have Fv(f) = F ′ +v(f)∪{fibv(f)} if fibv(f) ̸= 0 +and Fv(f) = F ′ +v(f) if fibv(f) = 0. +Applying Theorems 2 and 5 we have the following theorem that gives sufficient conditions +for every (D, b, a)-covering to be periodic for a finite and convex D. The first part of the +theorem was also mentioned in [4] in a more general form. +Theorem 14. Let D be a finite convex shape, g = fD − (b − a) and let E be the set of the +outer edge directions of g. +• Assume that b − a = 0. For any v ∈ E denote dv = gcd(n1, . . . , nr) where Fv(g) = +{φn1, . . . , φnr}. If dv = 1 holds for all v ∈ E then every (D, b, a)-covering is two- +periodic. If dv = 1 holds for all but some parallel v ∈ E then every (D, b, a)-covering +is periodic. +• Assume that b − a ̸= 0. For any v ∈ E denote dv = gcd(n1, . . . , nr) where F ′ +v(g) = +{φn1, . . . , φnr}. If the dv’th cyclotomic polynomial and fibv(g) have no common factors +for any v ∈ E then every (D, b, a)-covering is two-periodic. If the condition holds for +all but some parallel v ∈ E then every (D, b, a)-covering is periodic. (Note that the +condition is satisfied, in particular, if dv = 1.) +4 +Algorithmic aspects +All coverings are periodic, in particular, if there are no coverings at all! It is useful to be +able to detect such trivial cases. +The set +S(D, b, a) = {c ∈ {0, 1}Z2 | (fD − (b − a))c = a1(X)} +of all (D, b, a)-coverings is an SFT for any given finite shape D and non-negative integers b +and a. Hence the question whether there exist any (D, b, a)-coverings for given neighborhood +D and covering constants b and a is equivalent to the question whether the SFT S = +S(D, b, a) is non-empty. The question of emptiness of a given SFT is in general undecidable, +but if the SFT is known to be not aperiodic then the problem becomes decidable. +In +particular, if g = fD − (b − a) has line polynomial factors in at most one direction then this +question is decidable: +10 + +Theorem 15. Let finite D ⊆ Z2 and non-negative integers b and a be given such that the +polynomial g = fD − (b − a) has line polynomial factors in at most one parallel direction. +Then there exists an algorithm to determine whether there exist any (D, b, a)-coverings. +Proof. Let S = S(D, b, a) be the SFT of all (D, b, a)-coverings. Since g has line polyno- +mial factors in at most one direction, by Theorem 2 every element of S is periodic. Any +two-dimensional SFT that contains periodic configurations contains also two-periodic con- +figurations, so S is either empty or contains a two-periodic configuration. By a standard +argumentation by H. Wang [13] there exist semi-algorithms to determine whether a given +SFT is empty and whether a given SFT contains a two-periodic configuration. Running +these two semi-algorithms in parallel gives us an algorithm to test whether S ̸= ∅. +One may also want to design a perfect (D, b, a)-covering for given D, b and a. This can be +effectively done under the assumptions of Theorem 15: As we have seen, if S = S(D, b, a) is +non-empty it contains a two-periodic configuration. For any two-periodic configuration c it is +easy to check if c contains a forbidden pattern. By enumerating two-periodic configurations +one-by-one one is guaranteed to find eventually one that is in S. +If the polynomial g has no line polynomial factors then the following stronger result holds: +Theorem 16. If the polynomial g = fD − (b − a) has no line polynomial factors for given +finite shape D ⊆ Z2 and non-negative integers b and a then the SFT S = S(D, b, a) is finite. +One can then effectively construct all the finitely many elements of S. +The proof of the first part of above theorem relies on the fact that a two-dimensional subshift +is finite if and only if it contains only two-periodic cofigurations [2]. If g has no line polynomial +factors then every configuration it periodizes (including every configuration in S) is two- +periodic by Theorem 2, and hence S is finite. The “moreover” part of the theorem, i.e., the +fact that one can effectively produce all the finitely many elements of S holds generally for +finite SFTs. (The proof is provided in the Appendix for the sake of completeness.) +References +[1] M. A. Axenovich. On multiple coverings of the infinite rectangular grid with balls of +constant radius. Discrete Mathematics, 268(1):31 – 48, 2003. +[2] A. Ballier, B. Durand, and E. Jeandal. Structural aspects of tilings. In Susanne Albers +and Pascal Weil, editors, 25th International Symposium on Theoretical Aspects of Com- +puter Science, volume 1 of Leibniz International Proceedings in Informatics (LIPIcs), +pages 61–72, Dagstuhl, Germany, 2008. Schloss Dagstuhl–Leibniz-Zentrum fuer Infor- +matik. +[3] T. Ceccherini-Silberstein and M. Coornaert. Cellular Automata and Groups. Springer +Monographs in Mathematics. Springer Berlin Heidelberg, 2010. +11 + +[4] N. Geravker and S. A. Puzynina. Abelian Nivat’s conjecture for non-rectangular pat- +terns. arXiv:2111.04690, December 2021. +[5] E. Heikkil¨a. Algebrallinen n¨ak¨okulma peittokoodeihin. Master’s thesis, University of +Turku, May 2020. +[6] J. Kari. Low-complexity tilings of the plane. In Descriptional Complexity of Formal +Systems - 21st IFIP WG 1.02 International Conference, DCFS 2019, volume 11612 of +Lecture Notes in Computer Science, pages 35–45. Springer, 2019. +[7] J. Kari and E. Moutot. Nivat’s conjecture and pattern complexity in algebraic subshifts. +Theoretical Computer Science, 777:379 – 386, 2019. +[8] J. Kari and M. Szabados. An algebraic geometric approach to Nivat’s conjecture. In +Proceedings of ICALP 2015, part II, volume 9135 of Lecture Notes in Computer Science, +pages 273–285, 2015. +[9] J. Kari and M. Szabados. An algebraic geometric approach to Nivat’s conjecture. In- +formation and Computation, 271, 2020. +[10] P. Kurka. Topological and Symbolic Dynamics. Collection SMF. Soci´et´e math´ematique +de France, 2003. +[11] D. Lind and B. Marcus. An Introduction to Symbolic Dynamics and Coding. Cambridge +University Press, 1995. +[12] S. A. Puzynina. On periodicity of generalized two-dimensional infinite words. Informa- +tion and Computation, 207(11):1315–1328, 2009. +[13] H. Wang. Proving theorems by pattern recognition – II. The Bell System Technical +Journal, 40(1):1–41, 1961. +12 + +1 + t +1 + (1 − (b − a))t + t2 +1 + t + t2 +1 + t + t2 + t3 +Figure 4: The constellation on the left illustrates the proof of Theorem 12 and the constel- +lation on the right illustrates the proof of Theorem 13 with r = 2. +Appendix +Proofs of Theorems 12 and 13 +Theorem 12.. Let D be the 1-neighborhood of 0 in the triangular grid and assume that +b − a ̸= −1. Then every (D, b, a)-covering in the triangular grid is two-periodic. In other +words, all (1, b, a)-coverings in the triangular grid are two-periodic whenever b − a ̸= −1. +Proof. Let c be an arbitrary (D, b, a)-covering. Once again, we show that g = fD −(b−a) = +x−1y−1 + x−1 + y−1 + 1 − (b − a) + x + y + xy has no line polynomial factors, so that +by Theorem 2 the configuration c is two-periodic. +The outer edges of g have directions +(1, 1), (−1, −1), (1, 0), (−1, 0), (0, 1) and (0, −1) and hence by Lemma 3 any line polynomial +factor of g has direction (1, 1), (1, 0) or (0, 1). So, let v ∈ {(1, 1), (1, 0), (0, 1)}. We have +Fv(g) = {1+t, 1+(1−(b−a))t+t2}. See Figure 4 for an illustration. Polynomials φ1 = 1+t +and φ2 = 1 + (1 − (b − a))t + t2 satisfy φ2 +1 − φ2 = (1 + b − a)t so that they do not have any +common factors if b − a ̸= −1. Thus g has no line polynomial factors by Theorem 5. +Theorem +13. Let r ≥ 2 and let D be the r-neighborhood of 0 in the triangular grid. +Then every (D, b, a)-covering is two-periodic. In other words, all (r, b, a)-coverings in the +triangular grid are two-periodic for r ≥ 2. +Proof. Let c be an arbitrary (D, b, a)-covering. We show that g = fD − (b − a) ∈ Per(c) +has no line polynomial factors, which by Theorem 2 implies that the configuration c is two- +periodic. The outer edges of g have directions (1, 1), (−1, −1), (1, 0), (−1, 0), (0, 1) and +(0, −1), and hence by Lemma 3 any line polynomial factor of g has direction (1, 1), (1, 0) or +(0, 1). So, let v ∈ {(1, 1), (1, 0), (0, 1)}. There exists n ≥ 1 such that 1 + t + . . . + tn ∈ Fv(g) +and 1 + t + . . . + tn+1 ∈ Fv(g). See Figure 4 for an illustration with r = 2. Since these two +polynomials have no common factors g has no line polynomial factors by Theorem 5. +An alternative proof of Theorem 2 +Theorem 2.. Let c be a two-dimensional configuration and f ∈ Per(c). Then the following +conditions hold. +13 + +• If f does not have any line polynomial factors then c is two-periodic. +• If all line polynomial factors of f are in the same direction then c is periodic in this +direction. +Second proof sketch. The existence of a non-trivial periodizer f implies by Theorem 1 that +c has a special annihilator g = φ1 · · · φm that is a product of (difference) line polynomials +φ1, . . . , φm in pairwise different directions. All irreducible factors of g are line polynomials. If +f does not have any line polynomial factors then the periodizers f and g do not have common +factors. We can assume that both are proper polynomials as they can be multiplied by a +monomial if needed. The x-resultant of f, g ∈ C[x, y] is a polynomial Resx(f, g) = αf + βg +for some α, β ∈ C[x, y] such that the variable x is eliminated, i.e., Resx(f, g) is a polynomial +in variable y only. Moreover, since f and g do not have common factors, Resx(f, g) is not +identically zero. Because f, g ∈ Per(c) also Resx(f, g) ∈ Per(c), implying that c has a non- +trivial annihilator containing only variable y. This means that c is periodic in the vertical +direction. Analogously, the y-resultant Resy(f, g) shows that c is horizontally periodic, and +hence two-periodic. +The proof for the case that f has line polynomial factors only in one direction v goes +analogously by considering φc instead of c, where φ is the greatest common line polynomial +factor of f and g in the direction v. We get that φc is two-periodic, implying that c is +periodic in the direction v. +An algorithm to find all elements of a given finite SFT +Theorem 17. Given a finite F ⊆ A∗ such that XF is finite, one can effectively construct +the elements of XF. +Proof. Given a finite F ⊆ A∗ and a pattern p ∈ AD, assuming that strongly periodic +configurations are dense in XF, one can effectively check whether p ∈ L(XF). Indeed, we +have a semi-algorithm for the positive instances that guesses a strongly periodic configuration +c and verifies that c ∈ XF and p ∈ L(c). A semi-algorithm for the negative instances exists +for any SFT XF and is a standard compactness argument: guess a finite E ⊆ Zd such that +D ⊆ E and verify that every q ∈ AE such that q|D = p contains a forbidden subpattern. +Consequently, given finite F, G ⊆ A∗, assuming that strongly periodic configurations are +dense in XF and XG, one can effectively determine whether XF = XG. Indeed, XF ⊆ XG +if and only if no p ∈ G is in L(XF), a condition that we have shown above to be decidable. +Analogously we can test XG ⊆ XF. +Finally, let a finite F ⊆ A∗ be given such that XF is known to be finite. All elements +of XF are strongly periodic so that strongly periodic configurations are certainly dense in +XF. One can effectively enumerate all finite sets P of strongly periodic configurations. For +each P that is translation invariant (and hence a finite SFT) one can construct a finite set +G ⊆ A∗ of forbidden patterns such that XG = P. As shown above, there is an algorithm +to test whether XF = XG = P. Since XF is finite, a set P is eventually found such that +XF = P. +14 + diff --git a/fNE4T4oBgHgl3EQfRAzN/content/tmp_files/load_file.txt b/fNE4T4oBgHgl3EQfRAzN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aac10fd8a44b73046e8747f45cd0f1b811580a36 --- /dev/null +++ b/fNE4T4oBgHgl3EQfRAzN/content/tmp_files/load_file.txt @@ -0,0 +1,516 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf,len=515 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content='04987v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content='CO] 12 Jan 2023 On perfect coverings of two-dimensional grids Elias Heikkil¨a, Pyry Herva and Jarkko Kari Abstract We study perfect multiple coverings in translation invariant graphs with vertex set Z2 using an algebraic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In this approach we consider any such covering as a two-dimensional binary configuration which we then express as a two-variate formal power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Using known results, we conclude that any perfect multiple covering has a non-trivial periodizer, that is, there exists a non-zero polynomial whose formal product with the power series presenting the covering is a two-periodic configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If a non-trivial periodizer has line polynomial factors in at most one direction, then the configuration is known to be periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Using this result we find many setups where perfect multiple coverings of infinite grids are necessarily periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We also consider some algorithmic questions on finding perfect multiple coverings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' 1 Introduction and preliminaries A perfect multiple covering in a graph is a set of vertices, a code, such that the number of codewords in the neighborhood of an arbitrary vertex depends only on whether the vertex is in the code or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In this paper we study these codes on translation invariant graphs with the vertex set Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We present codes as two-dimensional binary configurations and observe that the perfect covering condition provides an algebraic condition that can be treated with the algebraic tools developed in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We focus on periodic codes and, in particular, study setups where all codes are necessarily periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The approach we take was initially mentioned in an example in the survey [6] by the third author, and considered in the Master’s thesis [5] by the first author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We start by giving the basic definitions, presenting the aforementioned algebraic approach and stating some past results relevant to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In Section 2 we describe an algorithm to find the line polynomial factors of any given (Laurent) polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In Section 3 we formally define the perfect multiple coverings in graphs and prove some periodicity results concerning them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We give new algebraic proofs of some known results concerning perfect multiple coverings on the infinite square grid and on the triangular grid [1, 12], and provide a new result on the forced periodicity of such coverings on the king grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Furthermore, we generalize the definition of perfect coverings for two-dimensional binary configurations with respect to different neighborhoods and covering constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In Section 4 we consider some algorithmic questions concerning perfect coverings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Using a standard argument by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Wang we show 1 that under certain constraints it is algorithmically decidable to determine whether there exist any perfect coverings with given neighborhood and given covering constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Configurations, periodicity, finite patterns and subshifts A d-dimensional configuration is a coloring of the infinite grid Zd using finitely many colors, that is, an element of AZd which we call the d-dimensional configuration space where A is some finite alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For a configuration c we let cu = c(u) to be the symbol or color that c has in cell u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The translation τ t by a vector t ∈ Zd shifts a configuration c such that τ t(c)u = cu−t for all u ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A configuration c is t-periodic if τ t(c) = c and c is periodic if c is t-periodic for some non-zero t ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We also say that a configuration c is periodic in direction v ∈ Zd \\ {0} if c is kv-periodic for some k ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A d-dimensional configuration c is strongly periodic if it has d linearly independent vectors of periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Strongly periodic configurations are then periodic in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Two-dimensional strongly periodic configurations are called two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A finite pattern is an assignment of symbols on some finite shape D ⊆ Zd, that is, an element of AD where A is some fixed alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In particular, the finite patterns in AD are called D-patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let us denote by A∗ the set of all finite patterns over alphabet A where the dimension d is known from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A finite pattern p ∈ AD appears in a configuration c ∈ AZd if τ t(c)|D = p for some t ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A configuration c contains the pattern p if it appears in c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For a fixed shape D, the set of all D-patterns that appear in c is the set LD(c) = {τ t(c)|D | t ∈ Zd} and the set of all finite patterns in c is denoted by L(c) which we call the language of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For a set S ⊆ AZd of configurations we define LD(S) and L(S) as the unions of LD(c) and L(c) over all c ∈ S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let us review some basic concepts of symbolic dynamics we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For a reference see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' [3, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The configuration space AZd can be made a compact topological space by endowing A with the discrete topology and considering the product topology it induces on AZd – the prodiscrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' This topology is induced by a metric where two configurations are close if they agree on a large area around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Thus AZd is a compact metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A subset S ⊆ AZd of the configuration space is a subshift if it is topologically closed and translation-invariant meaning that if c ∈ S then for any t ∈ Zd also τ t(c) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Equivalently we can define subshifts using forbidden patterns: Given a set F ⊆ A∗ of forbidden finite patterns, the set XF = {c ∈ AZd | L(c) ∩ F = ∅} of configurations that avoid all forbidden patterns is a subshift, and every subshift is obtained by forbidding some set of finite patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If F ⊆ A∗ is finite then we say that XF is a subshift of finite type (SFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The orbit of a configuration c is the set O(c) = {τ t(c) | t ∈ Zd} of its every translate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The orbit closure O(c) is the topological closure of its orbit under the prodiscrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The orbit closure of a configuration c is the smallest subshift that contains c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' It consists of all configurations c′ such that L(c′) ⊆ L(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' 2 The algebraic approach To present a configuration c ∈ AZd algebraically we make the assumption that A ⊆ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then we identify the configuration c with the formal power series c(X) = � u∈Zd cuXu over d variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , xd where we have denoted X = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , xd) and Xu = xu1 1 · · · xud d for any u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , ud) ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For d = 2 we usually denote X = (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' More generally we study the set of all formal power series over d variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , xd with complex coefficients which we denote by C[[X±1]] = C[[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , x±1 d ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A power series is finitary if it has only finitely many different coefficients and integral if its coefficients are all integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Thus we identify configurations with finitary and integral power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We also use Laurent polynomials which we call from now on simply polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We use the term “proper” when we talk about proper (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=', non-Laurent) polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let us denote by C[X±1] = C[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , x±1 d ] the set of all (Laurent) polynomials over d variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , xd with complex coefficients, which is the Laurent polynomial ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We say that two polynomials have no common factors if all of their common factors are units and that they have a common factor if they have a non–unit common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A product of a polynomial and a power series is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We say that a polynomial f = f(X) annihilates (or is an annihilator of) a power series c = c(X) if fc = 0, that is, if their product is the zero power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We say that a formal power series c = c(X) is periodic if it is annihilated by a difference polynomial Xt − 1 where t is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Note that this definition is consistent with the definition of periodicity of configurations defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Indeed if c = c(X) is a configuration then multiplying it by a monomial Xt produces the translated configuration τ t(c) and hence c is t-periodic if and only if c = τ t(c) = Xtc, which is equivalent to (Xt − 1)c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' So it is natural to study the annihilator ideal Ann(c) = {f ∈ C[X±1] | fc = 0} of a power series c ∈ C[[X±1]], which indeed is an ideal of the Laurent polynomial ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Hence the question whether a configuration (or any formal power series) is periodic is equivalent to asking whether its annihilator ideal contains a difference polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Another useful ideal that we study is the periodizer ideal Per(c) = {f ∈ C[X±1] | fc is strongly periodic}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Note that clearly Ann(c) is a subset of Per(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Note also that a configuration c has a non- trivial (= non-zero) annihilator if and only if it has a non-trivial periodizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The following theorem states that if a configuration has a non-trivial periodizer then it has in fact an annihilator of a particular simple form – a product of difference polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Theorem 1 ([8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let c be a configuration in any dimension that has a non-trivial periodizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then there exist pairwise linearly independent vectors t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , tm with m ≥ 1 such that (Xt1 − 1) · · · (Xtm − 1) ∈ Ann(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' 3 Line polynomials The support of a power series c = � u∈Zd cuXu is the set supp(c) = {u ∈ Zd | cu ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Thus a polynomial is a power series with a finite support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A line polynomial is a polynomial whose support contains at least two points and the points of the support lie on a unique line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In other words, a polynomial f is a line polynomial if it is not a monomial and there exist vectors u, v ∈ Zd such that supp(f) ⊆ u+Qv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In this case we say that f is a line polynomial in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We say that non-zero vectors v, v′ ∈ Zd are parallel if v′ ∈ Qv, and clearly then a line polynomial in direction v is also a line polynomial in any parallel direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A vector v ∈ Zd is primitive if its components are pairwise relatively prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If v is primitive then Qv∩Zd = Zv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For any non-zero v ∈ Zd there exists a parallel primitive vector v′ ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' It follows that we may assume the vector v in the definition of a line polynomial f to be primitive so that supp(f) ⊆ u+Zv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In the following our preferred presentations of directions are in terms of primitive vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Any line polynomial φ in a (primitive) direction v can be written uniquely in the form φ = Xu(a0 + a1Xv + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' + anXnv) = Xu(a0 + a1t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' + antn) where u ∈ Zd, n ≥ 1, a0 ̸= 0, an ̸= 0 and t = Xv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let us call the single variable proper polynomial a0 + a1t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' + antn ∈ C[t] the normal form of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Moreover, for a monomial aXu we define its normal form to be a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Thus two line polynomials in the direction v have the same normal form if and only if they are the same polynomial up to multiplication by Xu, for some u ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Difference polynomials are line polynomials and hence the annihilator provided by The- orem 1 is a product of line polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Annihilation by a difference polynomial means periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' More generally, annihilation of a configuration c by a line polynomial in a primitive direction v can be understood as the annihilation of the one-dimensional v-fibers � k∈Z cu+kvXu+kv of c in direction v, and since annihilation in the one-dimensional setting implies periodicity we conclude that a configuration is periodic if and only if it is annihilated by a line polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' It is known that if c has a periodizer with line polynomial factors in at most one direction then c is periodic: Theorem 2 ([9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let c be a two-dimensional configuration and f ∈ Per(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then the following conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If f does not have any line polynomial factors then c is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If all line polynomial factors of f are in the same direction then c is periodic in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The periodizer ideal Per(c) is a principal ideal generated by a polynomial g = φ1 · · · φm where φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , φm are line polynomials in pairwise non-parallel directions [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Because f ∈ Per(c) we know that g divides f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If f does not have any line polynomial factors then g = 1 and thus c = gc is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If f has line polynomial factors and they are in the same primitive direction v then g is a line polynomial in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Since gc is 4 two-periodic it is annihilated by (Xkv − 1) for some k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then the configuration c is annihilated by the line polynomial (Xkv − 1)g in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We conclude that c is periodic in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' (See the Appendix for an alternative proof that mimics the usage of resultants in [7], instead of relying on the structure of the ideal Per(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=') 2 Line polynomial factors The open and closed discrete half planes determined by a non-zero vector v ∈ Z2 are the sets Hv = {u ∈ Z2 | ⟨u, v⊥⟩ > 0} and Hv = {u ∈ Z2 | ⟨u, v⊥⟩ ≥ 0}, respectively, where v⊥ = (v2, −v1) is orthogonal to v = (v1, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let us also denote by lv = Hv \\ Hv the discrete line parallel to v that goes through the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In other words, the half plane determined by v is the half plane “to the right” of the line lv when moving along the line in the direction of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We say that a finite set D ⊆ Z2 has an outer edge in direction v if there exists a vector t ∈ Z2 such that D ⊆ Hv + t and |D ∩ (lv + t)| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We then call D ∩ (lv + t) an outer edge of D in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' An outer edge corresponding to v means that the convex hull of D has an edge in direction v in the clockwise orientation around D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If a finite non-empty set D does not have an outer edge in direction v then there exists a vector t ∈ Z2 such that D ⊆ Hv + t and |D ∩ (lv + t)| = 1 and then we say that D has a vertex in direction v and we call D ∩ (lv + t) a vertex of D in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We say that a polynomial f has an outer edge or a vertex in direction v if its support has an outer edge or a vertex in direction v, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Note that every finite shape D has either an edge or a vertex in any non-zero direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Note also that in this context directions v and −v are not the same: a shape may have an outer edge in direction v but no outer edge in direction −v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The following lemma shows that a polynomial can have line polynomial factors only in the directions of its outer edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Lemma 3 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let f be a non-zero polynomial with a line polynomial factor in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then f has outer edges in directions v and −v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let v ∈ Z2 \\ {0} be any non-zero primitive vector and let f = � fuXu be a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Recall that a v-fiber of f is a polynomial of the form � k∈Z fu+kvXu+kv for some u ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Thus a non-zero v-fiber of a polynomial is either a line polynomial or a monomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let us denote by Fv(f) the set of different normal forms of all non-zero v-fibers of a polynomial f, which is thus a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The following simple example illustrates the concept of fibers and their normal forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let us determine the set Fv(f) for f = f(X) = f(x, y) = 3x + y + xy2 + xy + x3y3 + x4y4 and v = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' By grouping the terms we can write f = 3x + y(1 + xy) + xy(1 + x2y2 + x3y3) = X(1,0) · 3 + X(0,1)(1 + t) + X(1,1)(1 + t2 + t3) where t = X(1,1) = xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Hence Fv(f) = {3, 1 + t, 1 + t2 + t3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' See Figure 1 for a pictorial illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' 5 3x y xy2 xy x3y3 x4y4 Figure 1: The support of f = 3x + y + xy2 + xy + x3y3 + x4y4 and its different (1, 1)-fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' As noticed in the example above, polynomials are linear combinations of their fibers: for any polynomial f and any non-zero primitive vector v we can write f = Xu1ψ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' + Xunψn for some u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , un ∈ Z2 where ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , ψn ∈ Fv(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We use this in the proof of the next theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A polynomial f has a line polynomial factor in direction v if and only if the polynomials in Fv(f) have a common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For any line polynomial φ in direction v, and for any polynomial g, the v-fibers of the product φg have a common factor φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In other words, if a polynomial f has a line polynomial factor φ in direction v then the polynomials in Fv(f) have the normal form of φ as a common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For the converse direction, assume that the polynomials in Fv(f) have a common factor φ which is thus a line polynomial in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then there exist vectors u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , un ∈ Z2 and polynomials φψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , φψn ∈ Fv(f) such that f = Xu1φψ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' + Xunφψn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Hence φ is a line polynomial factor of f in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Note that Lemma 3 actually follows immediately from Theorem 5: A vertex instead of an outer edge in direction v or −v provides a non-zero monomial v-fiber, which implies that the polynomials in Fv(f) have no common factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Thus to find out the line polynomial factors of f we first need to find out the possible directions of the line polynomials, that is, the directions of the (finitely many) outer edges of f, and then we need to check for which of these possible directions v the polynomials in Fv(f) have a common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' There are clearly algorithms to find the outer edges of a given polynomial and to determine whether finitely many line polynomials have a common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If such a factor exists then f has a line polynomial factor in this direction by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Thus we have proved the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' There is an algorithm to find the line polynomial factors of a given (Laurent) polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' 6 (a) The square grid (b) The king grid (c) The triangular grid Figure 2: The 1-neighborhoods of the black vertex in (a) the square grid, (b) the king grid, and (c) the triangular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' 3 Perfect coverings In this paper a graph is a tuple G = (V, E) where V is the (possibly infinite) vertex set of G and E ⊆ {{u, v} | u, v ∈ V, u ̸= v} is the edge set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Thus the graphs we consider are simple and undirected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We also assume that all vertices have only finitely many neighbors in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For a graph G = (V, E) we call any subset S ⊆ V of the vertex set a code in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The distance d(u, v) of two vertices u, v ∈ V is the length of a shortest path between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The (closed) r-neighborhood of a vertex u ∈ V is the set Nr(u) = {v ∈ V | d(v, u) ≤ r}, that is, the ball of radius r centered at u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let us now give the definition of the family of codes we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let G = (V, E) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A code S ⊆ V is an (r, b, a)-covering in G for non-negative integers b and a if the r-neighborhood of every vertex in S contains exactly b elements of S and the r-neighborhood of every vertex not in S contains exactly a elements of S, that is, if for every u ∈ V |Nr(u) ∩ S| = � b if u ∈ S a if u ̸∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' By a perfect (multiple) covering we mean any (r, b, a)-covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content='1 Infinite grids An infinite grid is a translation invariant graph with the vertex set Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In other words, in infinite grids we have Nr(u) = u + Nr(0) for all u ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The square grid is the graph (Z2, ES) with ES = {{u, v} | u−v ∈ {(±1, 0), (0, ±1)}}, the king grid is the graph (Z2, EK) with EK = {{u, v} | u − v ∈ {(±1, 0), (0, ±1), (±1, ±1)}} and the triangular grid is the graph (Z2, ET ) with ET = {{u, v} | u − v ∈ {(±1, 0), (0, ±1), (1, 1), (−1, −1)}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' See Figure 2 for the 1-neighborhoods of a vertex in these graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A code S ⊆ Z2 is periodic if S = S +t for some non-zero t ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' It is two-periodic if S = S + t1 and S = S + t2 where t1 and t2 are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The following result is by Axenovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Theorem 8 ([1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If b − a ̸= 1 then any (1, b, a)-covering in the square grid is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A code S ⊆ Z2 in any infinite grid can be presented as a configuration c ∈ {0, 1}Z2 which is defined such that cu = 1 if u ∈ S and cu = 0 if u ̸∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The positioning of the codewords in the r-neighborhood of any vertex u ∈ Z2 is then presented as a finite pattern c|u+Nr(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' 7 Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A configuration c ∈ {0, 1}Z2 is a (D, b, a)-covering for a finite shape D ⊆ Z2 (the neighborhood) and non-negative integers b and a (the covering constants) if for all u ∈ Z2 the pattern c|u+D contains exactly b symbols 1 if cu = 1 and exactly a symbols 1 if cu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We call also any (D, b, a)-covering perfect and hence a perfect covering is either a code in a graph or a two-dimensional binary configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Definitions 7 and 9 are consistent in infinite grids: a code S in an infinite grid G is an (r, b, a)-covering if and only if the configuration c ∈ {0, 1}Z2 presenting S is a (D, b, a)- covering where D is the r-neighborhood of 0 in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For a set D ⊆ Z2 we define its char- acteristic polynomial to be fD(X) = � u∈D X−u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let us denote by 1(X) the constant power series � u∈Z2 Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If c is a (D, b, a)-covering then from the definition we get that fD(X)c(X) = (b − a)c(X) + a1(X) which is equivalent to (fD(X) − (b − a)) c(X) = a1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Thus if c is a (D, b, a)-covering then fD(X) − (b − a) ∈ Per(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Using our formulation we get a simple proof for Theorem 8: Reformulation of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let D be the 1-neighborhood of 0 in the square grid and assume that b − a ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then every (D, b, a)-covering is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let c be an arbitrary (D, b, a)-covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We show that g = fD − (b − a) = x−1 + y−1 + 1 − (b − a) + x + y ∈ Per(c) has no line polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then c is two-periodic by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The outer edges of g are in directions (1, 1), (−1, −1), (1, −1) and (−1, 1) and hence by Lemma 3 any line polynomial factor of g is either in direction (1, 1) or (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For v ∈ {(1, 1), (1, −1)} we have Fv(g) = {1 + t, 1 − (b − a)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' See Figure 3 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Since 1 − (b − a) is a non-trivial monomial, by Theorem 5 the periodizer g ∈ Per(c) has no line polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The following result was already proved in a more general form in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We give a short proof using our algebraic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Theorem 10 ([12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let r ≥ 2 and let D be the r-neighborhood of 0 in the square grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then every (D, b, a)-covering is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In other words, all (r, b, a)-coverings in the square grid are two-periodic for all r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let c be an arbitrary (D, b, a)-covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Again, by Theorem 2, it is enough to show that g = fD − (b − a) ∈ Per(c) has no line polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' By Lemma 3 any line polynomial factor of g has direction (1, 1) or (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' So assume that v ∈ {(1, 1), (1, −1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We have φ1 = 1+t+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content='+tr ∈ Fv(g) and φ2 = 1+t+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content='+tr−1 ∈ Fv(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' See Figure 3 for an illustration in the case r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Since φ1−φ2 = tr, the polynomials φ1 and φ2 have no common factors, and hence by Theorem 5 the periodizer g has no line polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If a ̸= b then for all r ≥ 1 any (r, b, a)-covering in the king grid is two-periodic: Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let r ≥ 1 be arbitrary and let D be the r-neighborhood of 0 in the king grid and assume that a ̸= b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then any (D, b, a)-covering is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In other words, all (r, b, a)-coverings in the king grid are two-periodic whenever a ̸= b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' 8 1 + t 1 − (b − a) 1 + t + t2 1 + t 1 + t + t2 + t3 + t4 1 + t + (1 − (b − a))t2 + t3 + t4 Figure 3: The constellation on the left illustrates the proof of Theorem 8, the constellation on the center illustrates the proof of Theorem 10 with r = 2 and the constellation on the right illustrates the proof of Theorem 11 with r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let c be an arbitrary (D, b, a)-covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' By Theorem 2 it is sufficient to show that g = fD − (b − a) has no line polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The outer edges of g are in directions (1, 0), (−1, 0), (0, 1) and (0, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Hence by Lemma 3 any line polynomial factor of g has direction (1, 0) or (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let v ∈ {(1, 0), (0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We have φ1 = 1 + t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' + tr−1 + (1 − (b − a))tr + tr+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' + t2r ∈ Fv(g) and φ2 = 1 + t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' + t2r ∈ Fv(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' See Figure 3 for an illustration with r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Since φ2 − φ1 = (b − a)tr is a non-trivial monomial, φ1 and φ2 have no common factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Thus g has no line polynomial factors by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Similarly as in the square grid we can give simple proofs for known results from [12] con- cerning forced periodicity in the triangular grid: Theorem 12 ([12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let D be the 1-neighborhood of 0 in the triangular grid and assume that b − a ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then every (D, b, a)-covering in the triangular grid is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In other words, all (1, b, a)-coverings in the triangular grid are two-periodic whenever b − a ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Theorem 13 ([12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let r ≥ 2 and let D be the r-neighborhood of 0 in the triangular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then every (D, b, a)-covering is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In other words, all (r, b, a)-coverings in the triangular grid are two-periodic for r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content='2 General convex neighborhoods A shape D ⊆ Z2 is convex if it is the intersection D = conv(D) ∩ Z2 where conv(D) ⊆ R2 is the real convex hull of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let D ⊆ Z2 be a finite convex shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Any (D, b, a)-covering has a periodizer g = fD − (b − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' As earlier, we study whether g has any line polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For any v ̸= 0 the set Fv(fD) contains only polynomials φn = 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' + tn−1 for different n ≥ 1 since D is convex: if D contains two points then D contains every point between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Thus Fv(g) contains only polynomials φn for different n ≥ 1 and, if b − a ̸= 0, also a polynomial φn0 − (b − a)tm0 for some n0 ≥ 1 such that φn0 ∈ Fv(fD) and for some m0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If b − a = 0 then g = fD and thus Fv(g) = Fv(fD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Two polynomials φm and φn have a common factor if and only if gcd(m, n) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' More gen- erally, the polynomials φn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , φnr have a common factor if and only if d = gcd(n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , nr) > 9 1 and, in fact, their greatest common factor is the dth cyclotomic polynomial � 1≤k≤d gcd(k,d)=1 (t − ei· 2πk d ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let us introduce the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For any polynomial f, we denote by F ′ v(f) the set of normal forms of the non-zero fibers � k∈Z fu+kvXu+kv for all u ̸∈ Zv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In other words, we exclude the fiber through the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let us also denote fibv(f) for the normal form of the fiber � k∈Z fkvXkv through the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We have Fv(f) = F ′ v(f)∪{fibv(f)} if fibv(f) ̸= 0 and Fv(f) = F ′ v(f) if fibv(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Applying Theorems 2 and 5 we have the following theorem that gives sufficient conditions for every (D, b, a)-covering to be periodic for a finite and convex D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The first part of the theorem was also mentioned in [4] in a more general form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let D be a finite convex shape, g = fD − (b − a) and let E be the set of the outer edge directions of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Assume that b − a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For any v ∈ E denote dv = gcd(n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , nr) where Fv(g) = {φn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , φnr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If dv = 1 holds for all v ∈ E then every (D, b, a)-covering is two- periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If dv = 1 holds for all but some parallel v ∈ E then every (D, b, a)-covering is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Assume that b − a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For any v ∈ E denote dv = gcd(n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , nr) where F ′ v(g) = {φn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , φnr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If the dv’th cyclotomic polynomial and fibv(g) have no common factors for any v ∈ E then every (D, b, a)-covering is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If the condition holds for all but some parallel v ∈ E then every (D, b, a)-covering is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' (Note that the condition is satisfied, in particular, if dv = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=') 4 Algorithmic aspects All coverings are periodic, in particular, if there are no coverings at all!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' It is useful to be able to detect such trivial cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The set S(D, b, a) = {c ∈ {0, 1}Z2 | (fD − (b − a))c = a1(X)} of all (D, b, a)-coverings is an SFT for any given finite shape D and non-negative integers b and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Hence the question whether there exist any (D, b, a)-coverings for given neighborhood D and covering constants b and a is equivalent to the question whether the SFT S = S(D, b, a) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The question of emptiness of a given SFT is in general undecidable, but if the SFT is known to be not aperiodic then the problem becomes decidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In particular, if g = fD − (b − a) has line polynomial factors in at most one direction then this question is decidable: 10 Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let finite D ⊆ Z2 and non-negative integers b and a be given such that the polynomial g = fD − (b − a) has line polynomial factors in at most one parallel direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then there exists an algorithm to determine whether there exist any (D, b, a)-coverings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let S = S(D, b, a) be the SFT of all (D, b, a)-coverings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Since g has line polyno- mial factors in at most one direction, by Theorem 2 every element of S is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Any two-dimensional SFT that contains periodic configurations contains also two-periodic con- figurations, so S is either empty or contains a two-periodic configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' By a standard argumentation by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Wang [13] there exist semi-algorithms to determine whether a given SFT is empty and whether a given SFT contains a two-periodic configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Running these two semi-algorithms in parallel gives us an algorithm to test whether S ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' One may also want to design a perfect (D, b, a)-covering for given D, b and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' This can be effectively done under the assumptions of Theorem 15: As we have seen, if S = S(D, b, a) is non-empty it contains a two-periodic configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For any two-periodic configuration c it is easy to check if c contains a forbidden pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' By enumerating two-periodic configurations one-by-one one is guaranteed to find eventually one that is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If the polynomial g has no line polynomial factors then the following stronger result holds: Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If the polynomial g = fD − (b − a) has no line polynomial factors for given finite shape D ⊆ Z2 and non-negative integers b and a then the SFT S = S(D, b, a) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' One can then effectively construct all the finitely many elements of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The proof of the first part of above theorem relies on the fact that a two-dimensional subshift is finite if and only if it contains only two-periodic cofigurations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If g has no line polynomial factors then every configuration it periodizes (including every configuration in S) is two- periodic by Theorem 2, and hence S is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The “moreover” part of the theorem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=', the fact that one can effectively produce all the finitely many elements of S holds generally for finite SFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' (The proof is provided in the Appendix for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=') References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A.' metadata={'source': 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algebraic geometric approach to Nivat’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In- formation and Computation, 271, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Kurka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Topological and Symbolic Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Collection SMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Soci´et´e math´ematique de France, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' [11] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Lind and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Marcus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' An Introduction to Symbolic Dynamics and Coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Cambridge University Press, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Puzynina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' On periodicity of generalized two-dimensional infinite words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Informa- tion and Computation, 207(11):1315–1328, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Proving theorems by pattern recognition – II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The Bell System Technical Journal, 40(1):1–41, 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' 12 1 + t 1 + (1 − (b − a))t + t2 1 + t + t2 1 + t + t2 + t3 Figure 4: The constellation on the left illustrates the proof of Theorem 12 and the constel- lation on the right illustrates the proof of Theorem 13 with r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Appendix Proofs of Theorems 12 and 13 Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content='. Let D be the 1-neighborhood of 0 in the triangular grid and assume that b − a ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then every (D, b, a)-covering in the triangular grid is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In other words, all (1, b, a)-coverings in the triangular grid are two-periodic whenever b − a ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let c be an arbitrary (D, b, a)-covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Once again, we show that g = fD −(b−a) = x−1y−1 + x−1 + y−1 + 1 − (b − a) + x + y + xy has no line polynomial factors, so that by Theorem 2 the configuration c is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The outer edges of g have directions (1, 1), (−1, −1), (1, 0), (−1, 0), (0, 1) and (0, −1) and hence by Lemma 3 any line polynomial factor of g has direction (1, 1), (1, 0) or (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' So, let v ∈ {(1, 1), (1, 0), (0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We have Fv(g) = {1+t, 1+(1−(b−a))t+t2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' See Figure 4 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Polynomials φ1 = 1+t and φ2 = 1 + (1 − (b − a))t + t2 satisfy φ2 1 − φ2 = (1 + b − a)t so that they do not have any common factors if b − a ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Thus g has no line polynomial factors by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let r ≥ 2 and let D be the r-neighborhood of 0 in the triangular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then every (D, b, a)-covering is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' In other words, all (r, b, a)-coverings in the triangular grid are two-periodic for r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Let c be an arbitrary (D, b, a)-covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We show that g = fD − (b − a) ∈ Per(c) has no line polynomial factors, which by Theorem 2 implies that the configuration c is two- periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The outer edges of g have directions (1, 1), (−1, −1), (1, 0), (−1, 0), (0, 1) and (0, −1), and hence by Lemma 3 any line polynomial factor of g has direction (1, 1), (1, 0) or (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' So, let v ∈ {(1, 1), (1, 0), (0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' There exists n ≥ 1 such that 1 + t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' + tn ∈ Fv(g) and 1 + t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' + tn+1 ∈ Fv(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' See Figure 4 for an illustration with r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Since these two polynomials have no common factors g has no line polynomial factors by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' An alternative proof of Theorem 2 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content='. Let c be a two-dimensional configuration and f ∈ Per(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Then the following conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' 13 If f does not have any line polynomial factors then c is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If all line polynomial factors of f are in the same direction then c is periodic in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Second proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The existence of a non-trivial periodizer f implies by Theorem 1 that c has a special annihilator g = φ1 · · · φm that is a product of (difference) line polynomials φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' , φm in pairwise different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' All irreducible factors of g are line polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' If f does not have any line polynomial factors then the periodizers f and g do not have common factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We can assume that both are proper polynomials as they can be multiplied by a monomial if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The x-resultant of f, g ∈ C[x, y] is a polynomial Resx(f, g) = αf + βg for some α, β ∈ C[x, y] such that the variable x is eliminated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=', Resx(f, g) is a polynomial in variable y only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Moreover, since f and g do not have common factors, Resx(f, g) is not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Because f, g ∈ Per(c) also Resx(f, g) ∈ Per(c), implying that c has a non- trivial annihilator containing only variable y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' This means that c is periodic in the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Analogously, the y-resultant Resy(f, g) shows that c is horizontally periodic, and hence two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' The proof for the case that f has line polynomial factors only in one direction v goes analogously by considering φc instead of c, where φ is the greatest common line polynomial factor of f and g in the direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' We get that φc is two-periodic, implying that c is periodic in the direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' An algorithm to find all elements of a given finite SFT Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Given a finite F ⊆ A∗ such that XF is finite, one can effectively construct the elements of XF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Given a finite F ⊆ A∗ and a pattern p ∈ AD, assuming that strongly periodic configurations are dense in XF, one can effectively check whether p ∈ L(XF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Indeed, we have a semi-algorithm for the positive instances that guesses a strongly periodic configuration c and verifies that c ∈ XF and p ∈ L(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' A semi-algorithm for the negative instances exists for any SFT XF and is a standard compactness argument: guess a finite E ⊆ Zd such that D ⊆ E and verify that every q ∈ AE such that q|D = p contains a forbidden subpattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Consequently, given finite F, G ⊆ A∗, assuming that strongly periodic configurations are dense in XF and XG, one can effectively determine whether XF = XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Indeed, XF ⊆ XG if and only if no p ∈ G is in L(XF), a condition that we have shown above to be decidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Analogously we can test XG ⊆ XF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Finally, let a finite F ⊆ A∗ be given such that XF is known to be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' All elements of XF are strongly periodic so that strongly periodic configurations are certainly dense in XF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' One can effectively enumerate all finite sets P of strongly periodic configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' For each P that is translation invariant (and hence a finite SFT) one can construct a finite set G ⊆ A∗ of forbidden patterns such that XG = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' As shown above, there is an algorithm to test whether XF = XG = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' Since XF is finite, a set P is eventually found such that XF = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} +page_content=' 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE4T4oBgHgl3EQfRAzN/content/2301.04987v1.pdf'} diff --git a/ftE3T4oBgHgl3EQfHwlI/content/tmp_files/2301.04326v1.pdf.txt b/ftE3T4oBgHgl3EQfHwlI/content/tmp_files/2301.04326v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8fd244f1aaa5cdbba37ea9e8d23a1ede258a6c1 --- /dev/null +++ b/ftE3T4oBgHgl3EQfHwlI/content/tmp_files/2301.04326v1.pdf.txt @@ -0,0 +1,1226 @@ +Realization of Wilson fermions in topolectrical circuits +Huanhuan Yang, Lingling Song, Yunshan Cao, and Peng Yan∗ +School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices, +University of Electronic Science and Technology of China, Chengdu 610054, China +Wilson fermion (WF) is a fundamental particle in the theory of quantum chromodynamics, originally pro- +posed by Kenneth Wilson to solve the fermion doubling problem, i.e., more fermions than expected when one +puts fermionic fields on a lattice. In this Letter, we report a direct observation of the WF in circuit systems. +It is found that WFs manifest as topological spin textures analogous to the half skyrmion, half-skyrmion pair, +and N´eel skyrmion structures, depending on their mass. Transformations of different WF states are realized by +merely tuning the electric elements. Interestingly enough, we show that the WF with a half-skyrmion profile +represents a novel quantum anomalous semimetal phase supporting a chiral edge current. We experimentally +observe the propagation of chiral edge current along the domain-wall separating two circuits with contrast frac- +tional Chern numbers. Our work presents the first experimental evidence for WFs in topolectrical circuits. The +nontrivial analogy between the WF state and the skyrmionic structure builds an intimate connection between +the two burgeoning fields. +Lattice quantum chromodynamics (QCD) is an effective +method to study the strong interactions of quarks mediated +by gluons [1, 2]. In lattice QCD calculation, quarks are rep- +resented by fermionic fields and placed at lattice sites, and +gluons play the role of interactions between neighboring sites +[3]. However, when naively putting the fermionic fields on a +lattice, we will meet the fermion doubling problem [4], i.e., +the emergence of 2d − 1 spurious fermionic particles for each +original fermion (d is the dimension of the spacetime). The +origin of the doubling problem is deeply connected with chi- +ral symmetry and can be traced back to the axial anomaly +[5]. To remove the ambiguity, Kenneth Wilson developed a +technique by introducing wave-vector-dependent mass, which +modifies the Dirac fermions to Wilson ones [6]. The fermion +doubling issue exists in condensed matter physics as well [7– +11]. It prevents the occurrence of quantum anomalies in lat- +tices, such as the quantum anomalous Hall insulator [10] and +Weyl semimetal with single node [11]. It is known that Dirac +fermions manifest as the low-energy excitations of topologi- +cal semimetals/insulators (e.g. graphene) [12–15]. However, +the observation of Wilson fermion (WF) is still lacking. +In this Letter, we propose a lattice model to realize the +WF and probe it in topolectrical circuit experiments [16–35]. +Interestingly enough, we find that the nontrivial state of the +WF strongly depends on its mass and can be classified into +three categories characterized by different Chern numbers of +0, ±1/2, and ±1, corresponding to the half-skyrmion pair, half +skyrmion, and N´eel skyrmion, respectively. We propose a cir- +cuit method to efficiently manipulate the transport and trans- +formation of the WF states. Furthermore, we identify that the +fractional Chern number dictates a novel quantum anomalous +semimetal (QASM) phase with a chiral edge current. We re- +port a direct observation of the chiral current along the do- +main wall (DW) separating two circuits with contrast frac- +tional Chern numbers being 1/2 and −1/2. WFs in a three- +dimensional (3D) circuit system are constructed as well. They +are characterized by 3D winding numbers and accompanied +by the emergence of the surface states and DW states at the +boundaries. Our work opens the door for realizing the exotic +WFs in solid-state systems. +We begin from the Dirac Hamiltonian H = ck · α + mc2β +with c the light speed, k the wave vector, and α, β being the +Dirac matrices, which describes a Dirac fermion with the mass +m [36]. Expressing this Dirac Hamiltonian on a lattice of the +tight-binding form, we obtain HD = �d +i=1 +ℏv +a sin(kia)αi + mv2β +with a the lattice constant and ℏv the hopping strength (d +is the space dimension). It is straightforward to verify that +2d − 1 non-physics fermion doublers appear at the Brillouin +zone (BZ) boundaries ki = π/a. Following Wilson’s method, +we derive the WF Hamiltonian of the following form H = +HD + HW with HW = +4b +a2 sin2 kia +2 β being the k-dependent +Wilson mass term [37]. Here, the k-independent mass m in +Hamiltonian HD is referred to as the dispersionless mass of +WFs. It is noted that the HW term breaks the parity symme- +try in two-dimensional (2D) and chiral symmetry in 3D cases, +which can circumvent the fermion doubling problems [4] and +reproduce the quantum anomaly in the continuum limit. In +Supplemental Material [38], we show the details how the dou- +blers from Dirac Hamiltonian are removed by introducing the +Wilson mass term. Next, we report the realization of the Wil- +son Hamiltonian in electrical circuits. +We consider a 2D spinful square lattice in Fig. 1(a). The +circuit is constituted by four types of capacitors ±C1,2 and the +negative impedance converters with current inversion (INICs) +in Fig. 1(b), where A and B parts correspond to the mass- +less Dirac and Wilson mass Hamiltonians, respectively. It is +noted that one can utilize inductors to replace negative capac- +itors because the admittance of the negative capacitor −iωC is +equivalent to the inductor −i 1 +ωL for L = 1/(Cω2) [39]. Here, ω +is the working frequency. We implement two sites to imitate a +(pseudo-) spin, indicated by the red rectangle in Figs. 1(a) and +1(b). The INIC is set up by an operational amplifier and three +identical resistors R, as shown in Fig. 1(c). In Fig. 1(d), we +show the realization of the staggered on-site potential, which +models the dispersionless mass of WF. +The circuit response is governed by Kirchhoff’s law I(ω) = +J(ω)V(ω) with I the input current and V the node voltage. +arXiv:2301.04326v1 [cond-mat.mes-hall] 11 Jan 2023 + +2 +4∆C2 +-4∆C2 ++ + R + R + R + OP +INIC +On-site potential: ++ - +C2 +-C2 +A +B +(a) +(c) +(d) +(b) +x +y +C1 +-C1 +1 +2 +INIC +spin +FIG. 1: (a) Illustration of a 2D spinful square lattice. (b) The circuit +realization of the hopping terms by A+B parts. Part A consists of +two kinds of capacitors ±C1 and the INICs. Part B is composed of +two types of capacitors ±C2. The red rectangle indicates the corre- +spondence between spin and circuit nodes. (c) The details of INIC. +The INIC is composed of an operational amplifier (OP) and three re- +sistors R, acting as a positive (negative) resistor from right to left (left +to right). (d) The realization of the staggered on-site potentials. +The circuit Laplacian reads +J(ω, k) = +� j11 +j12 +j21 +j22 +� +, +(1) +where j11 = 4iωC2 − 2iωC2(cos kx + cos ky) + 4iω∆C2, j12 = +2iG sin kx + 2ωC1 sin ky, j21 = 2iG sin kx − 2ωC1 sin ky, and +j22 = −4iωC2+2iωC2(cos kx+cos ky)−4iω∆C2, with G = 1/R +the conductance and ∆ being the mass coefficient of WFs. In +the presence of conductance, the time-reversal symmetry (T ) +of the system is broken because of J(ω, k)∗ � −J(ω, −k) +[22]. +By expressing J(ω) = iH(ω) with the Dirac matrices, we +obtain +H(ω) =2G sin kxαx + 2ωC1 sin kyαy ++ 4ωC2(sin2 kx +2 + sin2 ky +2 )β + 4ω∆C2β, +(2) +where αx, αy and β represent the Pauli matrices σx, σy, and +σz, respectively. It is noted that the above Hamiltonian fully +simulates the lattice model of WFs. The first three terms in +Eq. (2) represent the Hamiltonian of WF, and the last term +is the dispersionless mass of WF. Meanwhile, by tuning the +electric elements parameters G, C1, and C2, one can conve- +niently manipulate the shape of Wilson cones (similar to the +Dirac cones). +In what follows, we analyze the topological properties of +the lattice model. +For a T -broken 2D two-band system, one can evaluate the +Chern number [40] +C = − 1 +2π +� +BZ +�∂Ay +∂kx +− ∂Ax +∂ky +� +dkxdky, +(3) +to judge its topological properties. Here A(k) = i ⟨uk|∇k|uk⟩ is +the Berry connection with |uk⟩ the eigenstate of lower band. +In following calculations, we adopt Ci = C = 1 nF +(i = 1, 2), f = ω/(2π) = 806 kHz (In experiments, we will +use L = 39 µH to replace −C, so we choose ω = 1/ +√ +LC), +and G = ωC = 0.005 Ω−1 (R = 200 Ω). Calculations of +Chern number as a function of the dispersionless mass pa- +rameter ∆ are plotted in Fig. 2(a). We find the Chern num- +bers are quantized to five values ±1, ± 1 +2, and 0. It is noted +that the Chern number is irrelevant to the value of C1 but be- +comes opposite if C2 changes its sign. We show the first BZ +and typical band structures in Fig. 2(b). Combined with the +topological index, we classify these topological phases as fol- +lows. The band gaps open for the parameter intervals x and +~, where the Chern number is zero. It gives the trivial insu- +lator phase. For parameters in y and }, the band gaps close +at M and Γ points, respectively. Surprisingly, the Chern num- +bers are quantized to ∓1/2 , respectively. This novel phase +is dubbed as the QASM [37]. For parameter zones z and +|, the band gaps open with the topological number C = ∓1, +both of which represent Chern insulators with opposite chiral- +ities. For the parameter region {, the band structure closes at +X point with a vanishing Chern number, indicating a normal +semimetal phase. +By expressing Eq. (2) as H = f(k)·σ, one can define a unit +spin vector ˆf(k) as ˆf(k) = +f(k) +|f(k)|, where f(k) = ( fx, fy, fz) is the +coefficient of Pauli matrices and |f(k)| = +� +f 2x + f 2y + f 2z . Fig- +ure 2(c) displays the spin textures of 2D WFs, which evolve +as the increasing of dispersionless mass. The spin textures are +reminiscent of the magnetic solitons in the condensed matter +system [41–43]. It is observed that the spin textures of triv- +ial insulator, Chern insulator, QASM, and normal semimetal +correspond to the ferromagnetic ground state, skyrmion, half +skyrmion, and half-skyrmion pair, respectively. By evaluating +the topological charge Q = +1 +4π +� +BZ ˆf · ( ∂ˆf +∂kx × ∂ˆf +∂ky )dkxdky in Fig. +2(c), we identify an intimate connection with the Chern num- +ber as Q + C = 0. This finding thus establishes an interesting +map between WFs and magnetic solitons in electrical circuits. +In the broad spintronics community, the manipulation of +skyrmion motion is crucial for the next-generation informa- +tion industry [44]. +Here, we propose a method to control +the circuit skyrmion motion in momentum space. We first +consider a skyrmion configuration with ∆ = −0.5. To gen- +erate a skyrmion propagation along kx direction over a dis- +tance k0, one can modify Eq. (2) to H(ω) = 2G sin(kx − +k0)σx + 2ωC1 sin kyσy − 4ωC2[cos(kx − k0) + cos ky + 1 +2]σz, +which can be recast as H(ω) = 2G′ sin kxσx −2G′′ cos kxσx + +2ωC1 sin kyσy − [4ωC′ +2 cos kx + 4ωC′′ +2 sin kx + 4ωC2(cos ky + +1 +2)]σz. Compared with the original Eq. (2), one merely needs +to modify two hopping strengths (G′ = G cos k0 and C′ +2 = + +3 +kx +ky +X +M +① +① +① +② +② +② +③ +③ +③ +④ +④ +④ +⑤ +⑤ +⑤ +⑥ +⑥ +⑥ +⑦ +⑦ +⑦ +Q=1/2 +Q=-1/2 +Q=-1 +Q=1 +Q=0 +Q=0 +fz +Q=0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +11 +-1 +0 +Г +(a) +(c) +(b) +FIG. 2: (a) The Chern number steps as a function of the dispersionless mass ∆. +(b) The first BZ and the typical band structures for +∆ = −2.5, −2, −1.5, −1, −0.5, 0, 0.5, respectively, corresponding to different parameter intervals in (a). (c) The spin textures of WFs in the +momentum space. +C2 cos k0) and to add two extra hopping terms (−2G′′ cos kxσx +and 4ωC′′ +2 sin kxσz). Variable resistors and capacitors can be +conveniently adopted to realize these operations in circuit de- +vices. +To show the properties of QASM (∆ = 0 with C = +1 +2), +we consider a ribbon configuration with periodic boundary +condition along ˆx direction and Ny = 50 nodes along ˆy di- +rection. +Figure 3(a) shows the admittance spectra, where +the conduction and valence bands touch at kx = 0. There +is no isolated band in this admittance spectra, so the edge +state is absent. +Considering the Hamiltonian (2), one can +define a velocity operator as v = 1/(iℏ)[x, H] = +∂H +∂kx += +2G cos kxσx + 2ωC2 sin kxσz. The transverse current density +then can be written as +j(y) = +εn<µ +� +n +� � +2G cos kxφ† +n(kx, y)σxφn(kx, y) ++ 2ωC2 sin kxφ† +n(kx, y)σzφn(kx, y) +� +dkx, +(4) +with φn(kx, y) being the wave functions of the n-th band and +the sum index n indicating the bands below the admittance µ. +We plot the current density for different positions in Fig. 3(b). +It is found that the current density decays from the bound- +ary nodes, and its values are opposite for the top and bottom +edges. Consequently, the chiral edge currents constitute the +new bulk-edge correspondence of QASM [37, 45]. +Then, we consider a 10×10 square lattice to study the finite- +size effect. Diagonalizing the corresponding circuit Lapla- +cian, we obtain the admittance spectra shown in Fig. 3(c) +and the wave functions near jn = 0 Ω−1 in the inset of Fig. +3(c). In our circuit, the impedance between the node a and the +ground is computed by Za,ground = � +n +|φn,a|2 +jn +with φn,a the wave +function of node a for nth admittance mode, which reflects +the features of wave functions near jn = 0 Ω−1 and can be +measured readily [20, 28]. By comparing Figs. 3(d) and 3(c) +(inset), we find that the impedance of each node against the +ground exhibits almost the same spatial distribution to wave +functions, and one does not find an edge state as expected. +To demonstrate the bulk-boundary correspondence, we +consider a one-dimensional DW with 10 × 11 “spins” (an +extra column is set up for DW configuration), as shown in +Fig. 3(e), where the capacitor C2 has a kink at the center +of the sample, i.e., C2 > 0 (< 0) in the light blue (green) +region. In this circumstance, the Chern number varies from +1/2 (left) to −1/2 (right). +The eigenvalue and wavefunc- +tion of the bound state can be solved as J = 2ωC1ky and +φ(x, y) = +1 +√ +2πχy +√ +λ exp(−λ|x|+ikyy) with χy = +√ +2 +2 (−i, 1)T and +λ = +C1 +|C2| + +� +(C1 +C2 )2 + k2y. One can obtain the effective velocity +of the bound state veff = ∂J +∂ky = 2ωC1, indicting the DW bound +state propagating along ˆy direction [38]. In Fig. 3(f), we show +the admittance spectrum with the insets displaying the wave +functions and impedances, from which one can clearly see the +bound state confined inside the DW. Here, we use the inverse +participation ratio p = lg(� +i |φn,i|4) of the system to character- +ize the localization properties of the wave functions [46, 47], +and the green dots indicate the localized state. +Then, we prepare a printed circuit board to verify these the- +oretical predictions, as shown in Fig. 3(g) [38]. Figure 3(h) +shows the experimental impedance distribution. It demon- +strates a localized state between two domains, which com- +pares well with the theoretical result in Fig. 3(f). The exis- +tence of one bound state is closely related to the fact that the +topological invariants between the two sides of the DW dif- +fer by 1. In addition, one cannot observe the edge states on +the rest boundaries of the sample, which confirms that there is +indeed no edge state. +To show the chiral propagation of the bound state, we per- +form the circuit simulation with the software LTSPICE [48]. By +inputting a Gauss signal close to the DW, we observe the +bound mode propagating along the ˆy direction of DW. Fi- +nally, the voltage signal becomes a steady bound-state inside + +4 +C2>0 +C2<0 +DW +... +... +50 Ω +220 Ω +55 Ω +57 Ω +OP +INIC +L +C +R +1 cm +... +... +... +... +... +... +... +... +... +... +x +x=0 +y +(a) +(c) +(d) +(b) +(e) +(g) +(h) +(f) +FIG. 3: (a) Admittance spectrum of the ribbon geometry. (b) The current density distribution for two different “Fermi” levels slightly deviating +from the jn = 0 Ω−1 [dashed lines in (a)]. (c) The admittance spectrum with the inset showing the wave functions near jn = 0 Ω−1. (d) The +impedance distribution of the sample. (e) The configuration of the circuit DW with ±1/2 topological charges in light blue and green regions. +(f) The admittance spectrum. Insets: The distribution of the wave functions and impedances. (g) The partial printed circuit board used in the +experiment. (h) Experimentally measured impedance. +the DW [38]. +To characterize Chern insulators (C = ±1), we compute the +admittance for a ribbon configuration [38]. For the param- +eter ∆ = −0.5 and −1.5, we find two crossing bands in the +admittance gaps but with opposite charities. To show the chi- +ral propagation of edge states, we perform the circuit simula- +tion on a finite-size square lattice and observe a chiral voltage +propagation [38]. +Interestingly, we note that at the phase transition point sep- +arating two Chern insulators (∆ = −1), the Chern number +vanishes but the spin texture is still non-trivial. We consider a +finite-size lattice with 10 × 10 “spins”. The admittance spec- +trum is plotted in Fig. 4(a), showing that a series of local- +ized states lie near jn = 0 Ω−1. The wave functions of local- +ized states are displayed in the inset of Fig. 4(a), from which +we identify a corner state. The origin of the emerging corner +states can be interpreted as the convergence of two Chern in- +sulators with opposite chiralities, as shown by the green and +black arrows in Fig. 4(b). Due to the contrast of the chiral- +150 +200 +250 +300 +350 +130 Ω +360 Ω +(a) +(c) +(b) +FIG. 4: (a) The admittance of the finite-size square lattice with 10 × +10 “spins” with the inset showing the wave functions near jn = 0 Ω−1. +(b) The corner state formed by the convergence of Chern insulators +with opposite chiralities. (c) Numerical impedance. +ity, the one-dimensional edge states can only accumulate at +the sample corners, forming the zero-dimensional localized +states, i.e., corner states. These corner states can be detected +by measuring the distributions of impedance, as shown in Fig. +4(c). +As a nontrivial generalization, we extend this model to a 3D +system [38]. The circuit Hamiltonian then can be written as +H(ω) =2G sin kxαx + 2ωC1 sin kyαy + 2G sin kzαz ++ 4ωC2 +� +sin2 kx +2 + sin2 ky +2 + sin2 kz +2 +� +β + 4∆ωC2β, +(5) +where αx = σx ⊗ σx, αy = σx ⊗ σy, αz = σx ⊗ σz, and +β = σz ⊗ σ0 are Dirac matrices. +The topological properties of the 3D system are character- +ized by the winding number w3 [1]. Interestingly, we find that +the topological index w3 can only take five quantized values, +i.e., 0, ± 1 +2, ±1. For the topological insulator phase w3 = 1, +one can observe surface states. At the border of the two topo- +logical insulators with opposite winding numbers, we find the +hinge states induced by the overlap of the surface states. For +the QASM phase w3 = 1/2, we observe the bounded surface +state in a finite-size DW circuit along the ˆx direction [38]. +To summarize, we experimentally observed WFs in cir- +cuit systems. +In addition, we mapped WFs with different +masses or configurations to magnetic solitons with different +skyrmion charges, which will enable us to study the prop- +erties of skyrmions, half skyrmions, and half-skyrmion pairs +in electrical circuit platforms. We showed that the nontrivial +spin-texture of WFs in momentum space is fully character- +ized by Chern numbers and winding number in 2D and 3D +systems, respectively. The chiral edge current associated with + +P38 +P39 +P40 +P41 +P49 +P50 +P51 +P52 +P60 +P61 +P62 +P63 +福 +T5 +the novel QASM state dictated by a fractional Chern number +was directly detected. 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B 101, 094107 (2020). +[48] http://www.linear.com/LTspice. +[49] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, +Classification of topological insulators and superconductors in +three spatial dimensions, Phys. Rev. B 78, 195125 (2008). +Supplemental Material: +Realization of Wilson fermions in topolectrical circuits +Huanhuan Yang, Lingling Song, Yunshan Cao, and Peng Yan +School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of +Electronic Science and Technology of China, Chengdu 610054, China +I. +I. THE SOLUTION TO THE FERMION DOUBLING PROBLEM +In this section, we show how to remove the doublers of Dirac fermions. In the main text, we have expressed the massless +Dirac and Wilson Hamiltonians as HD = �d +i=1 +ℏv +a sin(kia)αi and H = HD + HW with HW = 4b +a2 sin2 kia +2 β, respectively. In Figs. +5(a) and 5(b), we display the band structures of the 2D square and 3D hyper-cubic lattices. The green bands indicate the Dirac +fermions with 3 and 7 doublers (gray dots in the first Brillouin zones), and the blue and red bands represent the Wilson fermions +only appearing at the Γ point. +doubler +original +fermion +kx +kx +ky +ky +kz +X +X +R +M +M +Г +Г +(a) +(b) +FIG. 5: (a)(b) The energy spectra and the first Brillouin zones of 2D and 3D systems. Here, we set a = 1 and ℏv = 1. The green, blue, and +pink curves correspond to the parameters b = 0, 0.1, and 1, respectively. The original fermion and doublers are labeled by red and gray dots in +the first Brillouin zones, respectively. +II. +II. THE PROPAGATION OF DW BOUND STATE +In this section, we show how to obtain the solution of the DW bound state and the propagation of the DW states. Considering +the DW of Fig. 4(e) in the main text with periodic boundary condition in ˆy direction, we can write the secular equation near the +DW as [2, 3] +[2ωC1(−i∂xσx + kyσy) + ωC2(x)(−∂2 +x + k2 +y)σz]φ(x, y) = Jφ(x, y) +(6) + +7 +The solutions of Eq. (6) are given by +J = 2ωC1ky, +φ(x, y) = χy +� +λ2(ky) exp[−λ2(ky)|x| + ikyy], +(7) +with χy = +√ +2 +2 (−i, 1)T and λ2(ky) = +C1 +|C2| + +� +(C1 +C2 )2 + k2y. The effective velocity of the bound state is veff = ∂J +∂ky = 2ωC1, indicting +the DW state propagating along ˆy direction. +To demonstrate the time evolution of the DW bound state, we perform the circuit simulation with LTSPICE. As shown in Fig. +6(a), we consider a sample with 20 × 11 “spins” and input a Gaussian AC signal i(t) = I0 exp[− (t−t0)2 +(∆t)2 ] sin[ω(t − t0)] close to the +DW indicated by the arrow in the first subfigure of Fig. 6(a). Here, we set I0 = 1 mA, ∆t = 20 µs, and t0 = 50 µs. Then, we +plot the voltage propagation at different moments and observe the signal propagation along the (DW channel) ˆy direction. When +the signal arrives at the sample edge, it will leak to bulk nodes and form a loop, as shown in the fourth subfigure of Fig. 6(a). +Finally, the signal becomes a steady bound state. We also calculate the steady-state voltage by the formula V = J−1I, and the +result is plotted in Fig. 6(b), which is consistence with the theoretical calculations. +stable state +t=0 μs +t=20 μs +t=40 μs +t=60 μs +t=80 μs +(a) +(b) +FIG. 6: (a) The propagation of the DW state with the black arrow indicating the position of the signal source. (b) The steady-state voltage. +III. +III. EXPERIMENTAL DETAILS +We implement the circuit experiment on a printed circuit broad shown in Fig. 7(a). The circuit is composed of 10 × 11 +cells, with each cell containing two nodes. The details of the circuit components are shown in the inset of Fig. 7(a). Figure +7(b) displays the experimental instruments: DC power supply (IT6332A) and impedance analyzer (E4990A), which are used to +provide the power for the operational amplifiers and measure the impedance over the sample, respectively. +In Table I, we list all elements used in our experiments, including the product companies, packages, mean values and their +tolerances. +TABLE I: Electric elements used in experiments. +Electric elements +Company +packages mean value tolerance +C1,C2 +Samsung +0805 +1 nF +±5% +L +muRata +1210 +39 µH +±5% +R +Panasonic +0603 +200 Ω +±1% +OP +Texas instruments SOIC-8 +/ +/ +IV. +IV. CHERN INSULATORS +To study the Chern insulator carefully, we consider two insulating phases with ∆ = −1.5 and −0.5. In Figs. 8(a) and 8(b), we +display the admittance spectra for a ribbon with infinite size in ˆx direction and Ny = 50 nodes in ˆy direction. In the band gap, + +8 +INIC +OP +L +R +C +Power supply +Impedance analyzer +1 cm +(a) +(b) +FIG. 7: (a) The full image of the experimental printed circuit board. (b) The photos of the DC power supply and the impedance analyzer. +one can see the crossing of two spectra, manifesting as two chiral edge modes along the two boundaries. Next, we consider a +finite-size square lattice with 10×20 “spins” (20×20 nodes) to study the chiral edge mode. The admittance spectrum is given in +Fig. 8(c) with the wave function near jn = 0 Ω−1 and the impedance distribution plotted in the insets. With the same simulation +method, we obtain the voltage propagation of the Chern insulator at different moments and observe a chiral edge mode [see Fig. +8(d)]. +∆=-1.5 +∆=-0.5 +Z +max +0 +80 +100 +120 +140 +160 +180 +t=0 µs +t=15 µs +t=75 µs +t=60 µs +t=45 µs +t=30 µs +(a) +(d) +(b) +(c) +FIG. 8: (a),(b) The band structures of the ribbon configurations. The magenta and green spectra represent the chiral boundary modes localized +at the top and bottom edges, respectively. (c) The admittance spectrum with the insets showing the wave function near jn = 0 Ω−1 (top left +corner) and the impedance distribution (bottom right corner). (d) The time evolution of the topological boundary modes. The red arrows +indicate the propagation directions of the voltage signals. + +L +L +DO +三 +DO +作 +L +品 +8 +DO +口 +品 +品 +品 +口 +口 +口 +品 +品 +品 +品 +口 +品 +卫 +T +口 +日 +二 +L +DO +可 +马P38 +品 +P39 +P40 +P49 +P50 +P60 +P61 +P62 +日 +-E4990A +Impedance Analyzer +WKEYSIGHTCH2 +F1 +F2 +F3 +F4 +F5 +F6 +ITECHTRIPLEOUTPUTDCPOWERSUPPLY +T6332A +V-SE +I-SET +CHT +Meter +On/Off9 +x +y +x +y +1 2 +3 4 +1 2 +3 4 +z +y +z +x +x +z +y +z +1 2 +3 4 +1 2 +3 4 +INIC +1 +2 +3 +4 +C1 +-C1 +C2 +C2 +C2 +-C2 +-C2 +-C2 +1 +3 +1 +2 +3 +4 +1 +2 +3 +4 +1 +2 +3 +4 +x direction: +y direction: +z direction: +^ +^ +^ +super cell +4∆C2 +-4∆C2 +1 +2 +3 +4 +1 +2 +3 +4 +1 +2 +3 +4 +2 +4 +1 +2 +3 +4 +1 +2 +3 +4 +1 +2 +3 +4 +1 +2 +3 +4 +1 +2 +3 +4 +1 +2 +3 +4 +INIC +1 +2 +3 +4 +1 +2 +3 +4 +1 +2 +3 +4 +1 +2 +3 +4 +2 +3 +4 +1 +(a) +(c) +(b) +FIG. 9: (a) Three-dimensional hyper-cubic lattice model with four sites in each supercell. (b) The interactions between two cells along ˆx, ˆy, +and ˆz directions. (c) The realization of the on-site potentials. +V. +V. THREE-DIMENSIONAL WILSON FERMIONS +We consider a 3D hyper-cubic lattice with four sites in each cell, as shown in Fig. 9a. The hopping terms and on-site potentials +are shown in Fig. 9b and Fig. 9c, respectively. One can write the circuit Laplacian as +ˆx : j14 = j23 = j32 = j41 = −G exp(−ikx) + G exp(ikx) = 2iG sin kx, +j11 = j22 = 2iωC2 − iωC2 exp(−ikx) − iωC2 exp(ikx) = 2iωC2(1 − cos kx), +j33 = j44 = −2iωC2 + iωC2 exp(−ikx) + iωC2 exp(ikx) = −2iωC2(1 − cos kx). +(8) +ˆy : j14 = j32 = iωC exp(−iky) − iωC exp(iky) = iωC1(−2i sin ky), +j23 = j41 = −iωC exp(−iky) + iωC exp(iky) = iωC1(2i sin ky), +j11 = j22 = 2iωC2 − iωC2 exp(−iky) − iωC2 exp(iky) = 2iωC2(1 − cos ky), +j33 = j44 = −2iωC2 + iωC2 exp(−iky) + iωC2 exp(iky) = −2iωC2(1 − cos ky). +(9) +ˆz : j13 = j31 = −G exp(−ikz) + G exp(ikz) = 2iG sin kz, +j24 = j42 = G exp(−ikz) − G exp(ikz) = −2iG sin kz, +j11 = j22 = 2iωC2 − iωC2 exp(−ikz) − iωC2 exp(ikz) = 2iωC2(1 − cos kz), +j33 = j44 = −2iωC2 + iωC2 exp(−ikz) + iωC2 exp(ikz) = −2iωC2(1 − cos kz). +(10) +Summarizing the above equations, we obtain +J(ω) = i +������������������ +j0 +0 +2G sin kz +2G sin kx − 2iωC1 sin ky +0 +j0 +2G sin kx + 2iωC1 sin ky +−2G sin kz +2G sin kz +2G sin kx − 2iωC1 sin ky +− j0 +0 +2G sin kx + 2iωC1 sin ky +−2G sin kz +0 +− j0 +������������������ +, +(11) +with j0 = 4ωC2(sin2 kx +2 + sin2 ky +2 + sin2 kz +2 ). +Similarly, if expressing J(ω) = iH(ω), one can obtain the tight-binding Hamiltonian +H(ω) = 2G sin(kx)αx + 2ωC1 sin(ky)αy + 2G sin(kz)αz + 4ωC2[sin2 kx +2 + sin2 ky +2 + sin2 kz +2 ]β, +(12) + +10 +150 +200 +250 +300 +350 +110 Ω +160 Ω +150 +200 +250 +300 +350 +69 Ω +71 Ω +150 +200 +250 +300 +350 +100 Ω +350 Ω +150 +200 +250 +300 +350 +130 Ω +290 Ω +(a) +(b) +(c) +(d) +FIG. 10: (a)-(d) The distributions of impedance for the quantum anomalous semimetal state, topological insulator state, semimetal state, and +domain wall state. +where αx = σx ⊗ σx = +������������������ +0 0 0 1 +0 0 1 0 +0 1 0 0 +1 0 0 0 +������������������ +, αy = σx ⊗ σy = +������������������ +0 +0 +0 −i +0 +0 +i +0 +0 −i 0 +0 +i +0 +0 +0 +������������������ +, αz = σx ⊗ σz = +������������������ +0 +0 +1 +0 +0 +0 +0 −1 +1 +0 +0 +0 +0 −1 0 +0 +������������������ +, and β = σz ⊗ σ0 = +������������������ +1 0 +0 +0 +0 1 +0 +0 +0 0 −1 +0 +0 0 +0 +−1 +������������������ +. +The energy spectra are given by +jn = ± +� +j2 +0 + j2x + j2y + j2z, +(13) +with j0 = 4ωC2[sin2 kx +2 + sin2 ky +2 + sin2 kz +2 ], jx = 2G sin kx, jy = 2ωC1 sin ky, and jz = 2G sin kz. +One can rewrite the Eq. (12) as +H(ω) = +������ +j0 +j · +j · σ − j0 +������ , +(14) +with j = ( jx, jy, jz). +Due to the presence of global sublattice symmetry ΓHΓ−1 = −H with Γ = exp[−i π +4(d + 1)]β �d +i=1 αi, the above Hamiltonian +can be expressed as the block off-diagonal form +H′(ω) = +������ +0 +q(k) +q(k)† +0 +������ , +(15) +with q(k) = j · σ−ij0. To characterize the topological properties, we evaluate the 3D winding number [1] as +w3 = − +1 +24π2 +� +BZ +trace[(q−1∂kxq)(q−1∂kyq)(q−1∂kzq)]. +(16) + +. +. +.11 +Next, we consider a finite-size sample with 10 × 10 × 10 “spin” (4000 nodes). As shown in Figs. 10(a)-(c), we show the +impedance of the sample for the quantum anomalous semimetal phase, topological insulator phase, semimetal phase, respec- +tively. These results resemble the 2D cases. To show the bulk-boundary correspondence, we also form a 2D domain wall along +ˆx direction (with 11 × 10 × 10 “spin”) and observe the surface state confined inside the DW, as shown in Fig. 10(d). +∗ yan@uestc.edu.cn +[1] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insulators and superconductors in three spatial +dimensions, Phys. Rev. B 78, 195125 (2008). +[2] S. Q. Shen, Topological insulators: Dirac equation in condesed matters, Springer Series of Solid State Science, Vol. 174 (Springer, 2012). +[3] W.-Y. Shan, H.-Z. Lu, and S.-Q. Shen, Effective continuous model for surface states and thin films of three-dimensional topological +insulators, New J. Phys. 12, 043048 (2010). + diff --git a/ftE3T4oBgHgl3EQfHwlI/content/tmp_files/load_file.txt b/ftE3T4oBgHgl3EQfHwlI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4cf8de1bbf7ba14018b8e714ed37800c191a8cd5 --- /dev/null +++ b/ftE3T4oBgHgl3EQfHwlI/content/tmp_files/load_file.txt @@ -0,0 +1,958 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf,len=957 +page_content='Realization of Wilson fermions in topolectrical circuits Huanhuan Yang, Lingling Song, Yunshan Cao, and Peng Yan∗ School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China Wilson fermion (WF) is a fundamental particle in the theory of quantum chromodynamics, originally pro- posed by Kenneth Wilson to solve the fermion doubling problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=', more fermions than expected when one puts fermionic fields on a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In this Letter, we report a direct observation of the WF in circuit systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' It is found that WFs manifest as topological spin textures analogous to the half skyrmion, half-skyrmion pair, and N´eel skyrmion structures, depending on their mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Transformations of different WF states are realized by merely tuning the electric elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Interestingly enough, we show that the WF with a half-skyrmion profile represents a novel quantum anomalous semimetal phase supporting a chiral edge current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' We experimentally observe the propagation of chiral edge current along the domain-wall separating two circuits with contrast frac- tional Chern numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Our work presents the first experimental evidence for WFs in topolectrical circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The nontrivial analogy between the WF state and the skyrmionic structure builds an intimate connection between the two burgeoning fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Lattice quantum chromodynamics (QCD) is an effective method to study the strong interactions of quarks mediated by gluons [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In lattice QCD calculation, quarks are rep- resented by fermionic fields and placed at lattice sites, and gluons play the role of interactions between neighboring sites [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' However, when naively putting the fermionic fields on a lattice, we will meet the fermion doubling problem [4], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=', the emergence of 2d − 1 spurious fermionic particles for each original fermion (d is the dimension of the spacetime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The origin of the doubling problem is deeply connected with chi- ral symmetry and can be traced back to the axial anomaly [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' To remove the ambiguity, Kenneth Wilson developed a technique by introducing wave-vector-dependent mass, which modifies the Dirac fermions to Wilson ones [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The fermion doubling issue exists in condensed matter physics as well [7– 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' It prevents the occurrence of quantum anomalies in lat- tices, such as the quantum anomalous Hall insulator [10] and Weyl semimetal with single node [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' It is known that Dirac fermions manifest as the low-energy excitations of topologi- cal semimetals/insulators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' graphene) [12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' However, the observation of Wilson fermion (WF) is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In this Letter, we propose a lattice model to realize the WF and probe it in topolectrical circuit experiments [16–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Interestingly enough, we find that the nontrivial state of the WF strongly depends on its mass and can be classified into three categories characterized by different Chern numbers of 0, ±1/2, and ±1, corresponding to the half-skyrmion pair, half skyrmion, and N´eel skyrmion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' We propose a cir- cuit method to efficiently manipulate the transport and trans- formation of the WF states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Furthermore, we identify that the fractional Chern number dictates a novel quantum anomalous semimetal (QASM) phase with a chiral edge current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' We re- port a direct observation of the chiral current along the do- main wall (DW) separating two circuits with contrast frac- tional Chern numbers being 1/2 and −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' WFs in a three- dimensional (3D) circuit system are constructed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' They are characterized by 3D winding numbers and accompanied by the emergence of the surface states and DW states at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Our work opens the door for realizing the exotic WFs in solid-state systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' We begin from the Dirac Hamiltonian H = ck · α + mc2β with c the light speed, k the wave vector, and α, β being the Dirac matrices, which describes a Dirac fermion with the mass m [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Expressing this Dirac Hamiltonian on a lattice of the tight-binding form, we obtain HD = �d i=1 ℏv a sin(kia)αi + mv2β with a the lattice constant and ℏv the hopping strength (d is the space dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' It is straightforward to verify that 2d − 1 non-physics fermion doublers appear at the Brillouin zone (BZ) boundaries ki = π/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Following Wilson’s method, we derive the WF Hamiltonian of the following form H = HD + HW with HW = 4b a2 sin2 kia 2 β being the k-dependent Wilson mass term [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Here, the k-independent mass m in Hamiltonian HD is referred to as the dispersionless mass of WFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' It is noted that the HW term breaks the parity symme- try in two-dimensional (2D) and chiral symmetry in 3D cases, which can circumvent the fermion doubling problems [4] and reproduce the quantum anomaly in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In Supplemental Material [38], we show the details how the dou- blers from Dirac Hamiltonian are removed by introducing the Wilson mass term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Next, we report the realization of the Wil- son Hamiltonian in electrical circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' We consider a 2D spinful square lattice in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The circuit is constituted by four types of capacitors ±C1,2 and the negative impedance converters with current inversion (INICs) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 1(b), where A and B parts correspond to the mass- less Dirac and Wilson mass Hamiltonians, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' It is noted that one can utilize inductors to replace negative capac- itors because the admittance of the negative capacitor −iωC is equivalent to the inductor −i 1 ωL for L = 1/(Cω2) [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Here, ω is the working frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' We implement two sites to imitate a (pseudo-) spin, indicated by the red rectangle in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 1(a) and 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The INIC is set up by an operational amplifier and three identical resistors R, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 1(d), we show the realization of the staggered on-site potential, which models the dispersionless mass of WF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The circuit response is governed by Kirchhoff’s law I(ω) = J(ω)V(ω) with I the input current and V the node voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='04326v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='mes-hall] 11 Jan 2023 2 4∆C2 4∆C2 + R R R OP INIC On-site potential: + - C2 C2 A B (a) (c) (d) (b) x y C1 C1 1 2 INIC spin FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 1: (a) Illustration of a 2D spinful square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (b) The circuit realization of the hopping terms by A+B parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Part A consists of two kinds of capacitors ±C1 and the INICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Part B is composed of two types of capacitors ±C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The red rectangle indicates the corre- spondence between spin and circuit nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (c) The details of INIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The INIC is composed of an operational amplifier (OP) and three re- sistors R, acting as a positive (negative) resistor from right to left (left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (d) The realization of the staggered on-site potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The circuit Laplacian reads J(ω, k) = � j11 j12 j21 j22 � , (1) where j11 = 4iωC2 − 2iωC2(cos kx + cos ky) + 4iω∆C2, j12 = 2iG sin kx + 2ωC1 sin ky, j21 = 2iG sin kx − 2ωC1 sin ky, and j22 = −4iωC2+2iωC2(cos kx+cos ky)−4iω∆C2, with G = 1/R the conductance and ∆ being the mass coefficient of WFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In the presence of conductance, the time-reversal symmetry (T ) of the system is broken because of J(ω, k)∗ � −J(ω, −k) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' By expressing J(ω) = iH(ω) with the Dirac matrices, we obtain H(ω) =2G sin kxαx + 2ωC1 sin kyαy + 4ωC2(sin2 kx 2 + sin2 ky 2 )β + 4ω∆C2β, (2) where αx, αy and β represent the Pauli matrices σx, σy, and σz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' It is noted that the above Hamiltonian fully simulates the lattice model of WFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The first three terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (2) represent the Hamiltonian of WF, and the last term is the dispersionless mass of WF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Meanwhile, by tuning the electric elements parameters G, C1, and C2, one can conve- niently manipulate the shape of Wilson cones (similar to the Dirac cones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In what follows, we analyze the topological properties of the lattice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' For a T -broken 2D two-band system, one can evaluate the Chern number [40] C = − 1 2π � BZ �∂Ay ∂kx − ∂Ax ∂ky � dkxdky, (3) to judge its topological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Here A(k) = i ⟨uk|∇k|uk⟩ is the Berry connection with |uk⟩ the eigenstate of lower band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In following calculations, we adopt Ci = C = 1 nF (i = 1, 2), f = ω/(2π) = 806 kHz (In experiments, we will use L = 39 µH to replace −C, so we choose ω = 1/ √ LC), and G = ωC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='005 Ω−1 (R = 200 Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Calculations of Chern number as a function of the dispersionless mass pa- rameter ∆ are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' We find the Chern num- bers are quantized to five values ±1, ± 1 2, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' It is noted that the Chern number is irrelevant to the value of C1 but be- comes opposite if C2 changes its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' We show the first BZ and typical band structures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Combined with the topological index, we classify these topological phases as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The band gaps open for the parameter intervals x and ~, where the Chern number is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' It gives the trivial insu- lator phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' For parameters in y and }, the band gaps close at M and Γ points, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Surprisingly, the Chern num- bers are quantized to ∓1/2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' This novel phase is dubbed as the QASM [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' For parameter zones z and |, the band gaps open with the topological number C = ∓1, both of which represent Chern insulators with opposite chiral- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' For the parameter region {, the band structure closes at X point with a vanishing Chern number, indicating a normal semimetal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' By expressing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (2) as H = f(k)·σ, one can define a unit spin vector ˆf(k) as ˆf(k) = f(k) |f(k)|, where f(k) = ( fx, fy, fz) is the coefficient of Pauli matrices and |f(k)| = � f 2x + f 2y + f 2z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Fig- ure 2(c) displays the spin textures of 2D WFs, which evolve as the increasing of dispersionless mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The spin textures are reminiscent of the magnetic solitons in the condensed matter system [41–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' It is observed that the spin textures of triv- ial insulator, Chern insulator, QASM, and normal semimetal correspond to the ferromagnetic ground state, skyrmion, half skyrmion, and half-skyrmion pair, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' By evaluating the topological charge Q = 1 4π � BZ ˆf · ( ∂ˆf ∂kx × ∂ˆf ∂ky )dkxdky in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 2(c), we identify an intimate connection with the Chern num- ber as Q + C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' This finding thus establishes an interesting map between WFs and magnetic solitons in electrical circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In the broad spintronics community, the manipulation of skyrmion motion is crucial for the next-generation informa- tion industry [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Here, we propose a method to control the circuit skyrmion motion in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' We first consider a skyrmion configuration with ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' To gen- erate a skyrmion propagation along kx direction over a dis- tance k0, one can modify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (2) to H(ω) = 2G sin(kx − k0)σx + 2ωC1 sin kyσy − 4ωC2[cos(kx − k0) + cos ky + 1 2]σz, which can be recast as H(ω) = 2G′ sin kxσx −2G′′ cos kxσx + 2ωC1 sin kyσy − [4ωC′ 2 cos kx + 4ωC′′ 2 sin kx + 4ωC2(cos ky + 1 2)]σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Compared with the original Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (2), one merely needs to modify two hopping strengths (G′ = G cos k0 and C′ 2 = 3 kx ky X M ① ① ① ② ② ② ③ ③ ③ ④ ④ ④ ⑤ ⑤ ⑤ ⑥ ⑥ ⑥ ⑦ ⑦ ⑦ Q=1/2 Q=-1/2 Q=-1 Q=1 Q=0 Q=0 fz Q=0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='9 11 1 0 Г (a) (c) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 2: (a) The Chern number steps as a function of the dispersionless mass ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (b) The first BZ and the typical band structures for ∆ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='5, −2, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='5, −1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='5, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='5, respectively, corresponding to different parameter intervals in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (c) The spin textures of WFs in the momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' C2 cos k0) and to add two extra hopping terms (−2G′′ cos kxσx and 4ωC′′ 2 sin kxσz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Variable resistors and capacitors can be conveniently adopted to realize these operations in circuit de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' To show the properties of QASM (∆ = 0 with C = 1 2), we consider a ribbon configuration with periodic boundary condition along ˆx direction and Ny = 50 nodes along ˆy di- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Figure 3(a) shows the admittance spectra, where the conduction and valence bands touch at kx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' There is no isolated band in this admittance spectra, so the edge state is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Considering the Hamiltonian (2), one can define a velocity operator as v = 1/(iℏ)[x, H] = ∂H ∂kx = 2G cos kxσx + 2ωC2 sin kxσz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The transverse current density then can be written as j(y) = εn<µ � n � � 2G cos kxφ† n(kx, y)σxφn(kx, y) + 2ωC2 sin kxφ† n(kx, y)σzφn(kx, y) � dkx, (4) with φn(kx, y) being the wave functions of the n-th band and the sum index n indicating the bands below the admittance µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' We plot the current density for different positions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' It is found that the current density decays from the bound- ary nodes, and its values are opposite for the top and bottom edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Consequently, the chiral edge currents constitute the new bulk-edge correspondence of QASM [37, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Then, we consider a 10×10 square lattice to study the finite- size effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Diagonalizing the corresponding circuit Lapla- cian, we obtain the admittance spectra shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 3(c) and the wave functions near jn = 0 Ω−1 in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In our circuit, the impedance between the node a and the ground is computed by Za,ground = � n |φn,a|2 jn with φn,a the wave function of node a for nth admittance mode, which reflects the features of wave functions near jn = 0 Ω−1 and can be measured readily [20, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' By comparing Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 3(d) and 3(c) (inset), we find that the impedance of each node against the ground exhibits almost the same spatial distribution to wave functions, and one does not find an edge state as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' To demonstrate the bulk-boundary correspondence, we consider a one-dimensional DW with 10 × 11 “spins” (an extra column is set up for DW configuration), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 3(e), where the capacitor C2 has a kink at the center of the sample, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=', C2 > 0 (< 0) in the light blue (green) region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In this circumstance, the Chern number varies from 1/2 (left) to −1/2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The eigenvalue and wavefunc- tion of the bound state can be solved as J = 2ωC1ky and φ(x, y) = 1 √ 2πχy √ λ exp(−λ|x|+ikyy) with χy = √ 2 2 (−i, 1)T and λ = C1 |C2| + � (C1 C2 )2 + k2y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' One can obtain the effective velocity of the bound state veff = ∂J ∂ky = 2ωC1, indicting the DW bound state propagating along ˆy direction [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 3(f), we show the admittance spectrum with the insets displaying the wave functions and impedances, from which one can clearly see the bound state confined inside the DW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Here, we use the inverse participation ratio p = lg(� i |φn,i|4) of the system to character- ize the localization properties of the wave functions [46, 47], and the green dots indicate the localized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Then, we prepare a printed circuit board to verify these the- oretical predictions, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 3(g) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Figure 3(h) shows the experimental impedance distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' It demon- strates a localized state between two domains, which com- pares well with the theoretical result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 3(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The exis- tence of one bound state is closely related to the fact that the topological invariants between the two sides of the DW dif- fer by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In addition, one cannot observe the edge states on the rest boundaries of the sample, which confirms that there is indeed no edge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' To show the chiral propagation of the bound state, we per- form the circuit simulation with the software LTSPICE [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' By inputting a Gauss signal close to the DW, we observe the bound mode propagating along the ˆy direction of DW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Fi- nally, the voltage signal becomes a steady bound-state inside 4 C2>0 C2<0 DW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 50 Ω 220 Ω 55 Ω 57 Ω OP INIC L C R 1 cm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' x x=0 y (a) (c) (d) (b) (e) (g) (h) (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 3: (a) Admittance spectrum of the ribbon geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (b) The current density distribution for two different “Fermi” levels slightly deviating from the jn = 0 Ω−1 [dashed lines in (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (c) The admittance spectrum with the inset showing the wave functions near jn = 0 Ω−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (d) The impedance distribution of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (e) The configuration of the circuit DW with ±1/2 topological charges in light blue and green regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (f) The admittance spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Insets: The distribution of the wave functions and impedances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (g) The partial printed circuit board used in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (h) Experimentally measured impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' the DW [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' To characterize Chern insulators (C = ±1), we compute the admittance for a ribbon configuration [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' For the param- eter ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='5 and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='5, we find two crossing bands in the admittance gaps but with opposite charities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' To show the chi- ral propagation of edge states, we perform the circuit simula- tion on a finite-size square lattice and observe a chiral voltage propagation [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Interestingly, we note that at the phase transition point sep- arating two Chern insulators (∆ = −1), the Chern number vanishes but the spin texture is still non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' We consider a finite-size lattice with 10 × 10 “spins”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The admittance spec- trum is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 4(a), showing that a series of local- ized states lie near jn = 0 Ω−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The wave functions of local- ized states are displayed in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 4(a), from which we identify a corner state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The origin of the emerging corner states can be interpreted as the convergence of two Chern in- sulators with opposite chiralities, as shown by the green and black arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Due to the contrast of the chiral- 150 200 250 300 350 130 Ω 360 Ω (a) (c) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 4: (a) The admittance of the finite-size square lattice with 10 × 10 “spins” with the inset showing the wave functions near jn = 0 Ω−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (b) The corner state formed by the convergence of Chern insulators with opposite chiralities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (c) Numerical impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' ity, the one-dimensional edge states can only accumulate at the sample corners, forming the zero-dimensional localized states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=', corner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' These corner states can be detected by measuring the distributions of impedance, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' As a nontrivial generalization, we extend this model to a 3D system [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The circuit Hamiltonian then can be written as H(ω) =2G sin kxαx + 2ωC1 sin kyαy + 2G sin kzαz + 4ωC2 � sin2 kx 2 + sin2 ky 2 + sin2 kz 2 � β + 4∆ωC2β, (5) where αx = σx ⊗ σx, αy = σx ⊗ σy, αz = σx ⊗ σz, and β = σz ⊗ σ0 are Dirac matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The topological properties of the 3D system are character- ized by the winding number w3 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Interestingly, we find that the topological index w3 can only take five quantized values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=', 0, ± 1 2, ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' For the topological insulator phase w3 = 1, one can observe surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' At the border of the two topo- logical insulators with opposite winding numbers, we find the hinge states induced by the overlap of the surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' For the QASM phase w3 = 1/2, we observe the bounded surface state in a finite-size DW circuit along the ˆx direction [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' To summarize, we experimentally observed WFs in cir- cuit systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In addition, we mapped WFs with different masses or configurations to magnetic solitons with different skyrmion charges, which will enable us to study the prop- erties of skyrmions, half skyrmions, and half-skyrmion pairs in electrical circuit platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' We showed that the nontrivial spin-texture of WFs in momentum space is fully character- ized by Chern numbers and winding number in 2D and 3D systems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The chiral edge current associated with P38 P39 P40 P41 P49 P50 P51 P52 P60 P61 P62 P63 福 T5 the novel QASM state dictated by a fractional Chern number was directly detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Our work presents the first circuit re- alization of WFs, which sets a paradigm for other platforms, such as cold atoms, photonic, and phononic metamaterials, to further explore these fascinating phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' This work was supported by the National Key Research De- velopment Program under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 2022YFA1402802 and the National Natural Science Foundation of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 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topological insulators and superconductors in three spatial dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' B 78, 195125 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Supplemental Material: Realization of Wilson fermions in topolectrical circuits Huanhuan Yang, Lingling Song, Yunshan Cao, and Peng Yan School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' THE SOLUTION TO THE FERMION DOUBLING PROBLEM In this section, we show how to remove the doublers of Dirac fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In the main text, we have expressed the massless Dirac and Wilson Hamiltonians as HD = �d i=1 ℏv a sin(kia)αi and H = HD + HW with HW = 4b a2 sin2 kia 2 β, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 5(a) and 5(b), we display the band structures of the 2D square and 3D hyper-cubic lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The green bands indicate the Dirac fermions with 3 and 7 doublers (gray dots in the first Brillouin zones), and the blue and red bands represent the Wilson fermions only appearing at the Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' doubler original fermion kx kx ky ky kz X X R M M Г Г (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 5: (a)(b) The energy spectra and the first Brillouin zones of 2D and 3D systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Here, we set a = 1 and ℏv = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The green, blue, and pink curves correspond to the parameters b = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='1, and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The original fermion and doublers are labeled by red and gray dots in the first Brillouin zones, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' THE PROPAGATION OF DW BOUND STATE In this section, we show how to obtain the solution of the DW bound state and the propagation of the DW states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Considering the DW of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 4(e) in the main text with periodic boundary condition in ˆy direction, we can write the secular equation near the DW as [2, 3] [2ωC1(−i∂xσx + kyσy) + ωC2(x)(−∂2 x + k2 y)σz]φ(x, y) = Jφ(x, y) (6) 7 The solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (6) are given by J = 2ωC1ky, φ(x, y) = χy � λ2(ky) exp[−λ2(ky)|x| + ikyy], (7) with χy = √ 2 2 (−i, 1)T and λ2(ky) = C1 |C2| + � (C1 C2 )2 + k2y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The effective velocity of the bound state is veff = ∂J ∂ky = 2ωC1, indicting the DW state propagating along ˆy direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' To demonstrate the time evolution of the DW bound state, we perform the circuit simulation with LTSPICE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 6(a), we consider a sample with 20 × 11 “spins” and input a Gaussian AC signal i(t) = I0 exp[− (t−t0)2 (∆t)2 ] sin[ω(t − t0)] close to the DW indicated by the arrow in the first subfigure of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Here, we set I0 = 1 mA, ∆t = 20 µs, and t0 = 50 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Then, we plot the voltage propagation at different moments and observe the signal propagation along the (DW channel) ˆy direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' When the signal arrives at the sample edge, it will leak to bulk nodes and form a loop, as shown in the fourth subfigure of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Finally, the signal becomes a steady bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' We also calculate the steady-state voltage by the formula V = J−1I, and the result is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 6(b), which is consistence with the theoretical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' stable state t=0 μs t=20 μs t=40 μs t=60 μs t=80 μs (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 6: (a) The propagation of the DW state with the black arrow indicating the position of the signal source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (b) The steady-state voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' EXPERIMENTAL DETAILS We implement the circuit experiment on a printed circuit broad shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The circuit is composed of 10 × 11 cells, with each cell containing two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The details of the circuit components are shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Figure 7(b) displays the experimental instruments: DC power supply (IT6332A) and impedance analyzer (E4990A), which are used to provide the power for the operational amplifiers and measure the impedance over the sample, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In Table I, we list all elements used in our experiments, including the product companies, packages, mean values and their tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' TABLE I: Electric elements used in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Electric elements Company packages mean value tolerance C1,C2 Samsung 0805 1 nF ±5% L muRata 1210 39 µH ±5% R Panasonic 0603 200 Ω ±1% OP Texas instruments SOIC-8 / / IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' CHERN INSULATORS To study the Chern insulator carefully, we consider two insulating phases with ∆ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='5 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 8(a) and 8(b), we display the admittance spectra for a ribbon with infinite size in ˆx direction and Ny = 50 nodes in ˆy direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' In the band gap, 8 INIC OP L R C Power supply Impedance analyzer 1 cm (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 7: (a) The full image of the experimental printed circuit board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (b) The photos of the DC power supply and the impedance analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' one can see the crossing of two spectra, manifesting as two chiral edge modes along the two boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Next, we consider a finite-size square lattice with 10×20 “spins” (20×20 nodes) to study the chiral edge mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The admittance spectrum is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 8(c) with the wave function near jn = 0 Ω−1 and the impedance distribution plotted in the insets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' With the same simulation method, we obtain the voltage propagation of the Chern insulator at different moments and observe a chiral edge mode [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 8(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' ∆=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='5 ∆=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='5 Z max 0 80 100 120 140 160 180 t=0 µs t=15 µs t=75 µs t=60 µs t=45 µs t=30 µs (a) (d) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 8: (a),(b) The band structures of the ribbon configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The magenta and green spectra represent the chiral boundary modes localized at the top and bottom edges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (c) The admittance spectrum with the insets showing the wave function near jn = 0 Ω−1 (top left corner) and the impedance distribution (bottom right corner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (d) The time evolution of the topological boundary modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The red arrows indicate the propagation directions of the voltage signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='DO ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 9: (a) Three-dimensional hyper-cubic lattice model with four sites in each supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (b) The interactions between two cells along ˆx, ˆy, and ˆz directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (c) The realization of the on-site potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' THREE-DIMENSIONAL WILSON FERMIONS We consider a 3D hyper-cubic lattice with four sites in each cell, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The hopping terms and on-site potentials are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 9b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 9c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' One can write the circuit Laplacian as ˆx : j14 = j23 = j32 = j41 = −G exp(−ikx) + G exp(ikx) = 2iG sin kx, j11 = j22 = 2iωC2 − iωC2 exp(−ikx) − iωC2 exp(ikx) = 2iωC2(1 − cos kx), j33 = j44 = −2iωC2 + iωC2 exp(−ikx) + iωC2 exp(ikx) = −2iωC2(1 − cos kx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (8) ˆy : j14 = j32 = iωC exp(−iky) − iωC exp(iky) = iωC1(−2i sin ky), j23 = j41 = −iωC exp(−iky) + iωC exp(iky) = iωC1(2i sin ky), j11 = j22 = 2iωC2 − iωC2 exp(−iky) − iωC2 exp(iky) = 2iωC2(1 − cos ky), j33 = j44 = −2iωC2 + iωC2 exp(−iky) + iωC2 exp(iky) = −2iωC2(1 − cos ky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (9) ˆz : j13 = j31 = −G exp(−ikz) + G exp(ikz) = 2iG sin kz, j24 = j42 = G exp(−ikz) − G exp(ikz) = −2iG sin kz, j11 = j22 = 2iωC2 − iωC2 exp(−ikz) − iωC2 exp(ikz) = 2iωC2(1 − cos kz), j33 = j44 = −2iωC2 + iωC2 exp(−ikz) + iωC2 exp(ikz) = −2iωC2(1 − cos kz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (10) Summarizing the above equations, we obtain J(ω) = i ������������������ j0 0 2G sin kz 2G sin kx − 2iωC1 sin ky 0 j0 2G sin kx + 2iωC1 sin ky −2G sin kz 2G sin kz 2G sin kx − 2iωC1 sin ky − j0 0 2G sin kx + 2iωC1 sin ky −2G sin kz 0 − j0 ������������������ , (11) with j0 = 4ωC2(sin2 kx 2 + sin2 ky 2 + sin2 kz 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Similarly, if expressing J(ω) = iH(ω), one can obtain the tight-binding Hamiltonian H(ω) = 2G sin(kx)αx + 2ωC1 sin(ky)αy + 2G sin(kz)αz + 4ωC2[sin2 kx 2 + sin2 ky 2 + sin2 kz 2 ]β, (12) 10 150 200 250 300 350 110 Ω 160 Ω 150 200 250 300 350 69 Ω 71 Ω 150 200 250 300 350 100 Ω 350 Ω 150 200 250 300 350 130 Ω 290 Ω (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 10: (a)-(d) The distributions of impedance for the quantum anomalous semimetal state, topological insulator state, semimetal state, and domain wall state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' where αx = σx ⊗ σx = ������������������ 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 ������������������ , αy = σx ⊗ σy = ������������������ 0 0 0 −i 0 0 i 0 0 −i 0 0 i 0 0 0 ������������������ , αz = σx ⊗ σz = ������������������ 0 0 1 0 0 0 0 −1 1 0 0 0 0 −1 0 0 ������������������ , and β = σz ⊗ σ0 = ������������������ 1 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 −1 ������������������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' The energy spectra are given by jn = ± � j2 0 + j2x + j2y + j2z, (13) with j0 = 4ωC2[sin2 kx 2 + sin2 ky 2 + sin2 kz 2 ], jx = 2G sin kx, jy = 2ωC1 sin ky, and jz = 2G sin kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' One can rewrite the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (12) as H(ω) = ������ j0 j · j · σ − j0 ������ , (14) with j = ( jx, jy, jz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Due to the presence of global sublattice symmetry ΓHΓ−1 = −H with Γ = exp[−i π 4(d + 1)]β �d i=1 αi, the above Hamiltonian can be expressed as the block off-diagonal form H′(ω) = ������ 0 q(k) q(k)† 0 ������ , (15) with q(k) = j · σ−ij0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' To characterize the topological properties, we evaluate the 3D winding number [1] as w3 = − 1 24π2 � BZ trace[(q−1∂kxq)(q−1∂kyq)(q−1∂kzq)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' (16) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='11 Next, we consider a finite-size sample with 10 × 10 × 10 “spin” (4000 nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 10(a)-(c), we show the impedance of the sample for the quantum anomalous semimetal phase, topological insulator phase, semimetal phase, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' These results resemble the 2D cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' To show the bulk-boundary correspondence, we also form a 2D domain wall along ˆx direction (with 11 × 10 × 10 “spin”) and observe the surface state confined inside the DW, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' 10(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' ∗ yan@uestc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content='cn [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Schnyder, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Ryu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' Furusaki, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQfHwlI/content/2301.04326v1.pdf'} +page_content=' W.' metadata={'source': 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SCHUBART ORBITS ON THE CIRCLE +Shuqiang Zhu +School of Mathematics, Southwestern University of Finance and Economics, +Chengdu 611130, China +zhusq@swufe.edu.cn +Abstract. We consider the three body problem on S1 under the +cotangent potential. We first construct homothetic orbits ending +in singularities, including total collision singularity and collision- +antipodal singularity. +Then certain symmetrical periodic orbits +with two equal masses, called Schubart orbits, are shown to exist. +The proof is based on the construction of a Wazewski set in the +phase space. +1. Introduction +The Newtonian n-body problem has been generalized in many ways, +for instance, under the general homogeneous potentials, or in higher +dimensional Euclidean spaces. Among them, the curved n-body prob- +lem, which studies n-body problem on surfaces of constant curvature +under the cotangent potential, has received lot of attentions in the last +decade (cf [1, 6, 8] and the references therein ). +In the Newtonian n-body problem, the two-body case is a Hamilton- +ian system with one degree of freedom, so is integrable. The cases with +two degrees of freedom, namely, the restricted three-body problem and +the collinear three-body problem, remains to be largely unsolved. How- +ever, the ideas emerged in attacking them shed light on more general +problems (cf [10, 12, 17] and the references therein). +We consider one case of the curved n-body problem with two degrees +of freedom, namely, the three-body problem on the circle. As a pre- +liminary study on the problem, in this manuscript we construct some +interesting orbits, the Eulerian homothetic orbits and Schubart orbits. +The Eulerian homothetic orbits are connected with the singularities. +In the curved three-body problem, there are collision, antipodal and +collision-antipodal singularities. The antipodal singularities turn out to +be impossible by consideration of Hill’s region. The collision-antipodal +singularities are more interesting. In fact, our study indicates that it +might be sensitive to the choice of masses. +1 +arXiv:2301.00193v1 [math.DS] 31 Dec 2022 + +2 +Shuqiang Zhu +Intuitively, if all the masses stay on a semi-circle, they would attract +each other and end in a triple collision. However, if two masses are +equal, there exist certain symmetric periodic orbits, provided collisions +are regularized. The behavior of these orbits is as follows. Initially, the +unequal mass, m3, is at the midpoint of the equal ones. If the masses +were released with zero velocity, it would be a homothetic collapse to +triple collision. However, we set the initial velocities such that m2 and +m3 move towards each other and m1 leaves them. Then m1 slows and +stops exactly when m2 and m3 collide. This is the first quarter of the +orbit. The second quarter of the orbit is the time-reverse of the first, +and the second half is the reflection of the first half with the roles of +m1 and m2 reversed. +Such orbits are called Schubart orbits. +They +were found numerically by Schubart [13] for the Newtonian three-body +problem and the analytic existence proof, was given by in [11, 14], +among others. +For the existence proof of the Schubart orbits, we follow that of +Moeckel in [11]. It is a topological argument and it is a variation of an +idea used by Conley [3] in the Newtonian restricted three-body prob- +lem. More precisely, it is based on the construction of a Wazewski set. +Unlike that of the Newtonian case, where the potential is almost a func- +tion of the shape variable, the cotangent potential depends essentially +on the two variables. Some computations are relatively lengthy. +The paper is organized as follows. In Sect. 2, we discuss the basic +setting of the three-body problem on S1 and the Eulerian homothetic +orbits. In Sect. 3, we regularize the collision singularities. In Sect. +4, we apply a topological argument to show the existence of Schubart +orbits. Some technical computations are presented in the Appendix. +2. Settings and Eulerian homothetic orbits +The configuration space is (S1)3. +The coordinates are ϕ1,ϕ2,ϕ3, +with ϕi ∈ R/2πZ. The Lagrangian is +L = ∑ +i +1 +2mi ˙ϕ2 +i + ∑ +j≠i +mimj cotdij, +where dij = min{∣ϕi −ϕj∣,2π −∣ϕi −ϕj∣}. The system is undefined in the +set ∆ = ⋃i≠j ∆ij, with +∆ij = {(ϕ1,ϕ2,ϕ3) ∶ dij = 0}⋃{(ϕ1,ϕ2,ϕ3) ∶ dij = π}. +The cases dij = 0 are collisions, whereas the cases dij = π are antipodal +configurations, when some bodies are at the opposite ends of a diame- +ter. In both cases, the forces are infinite. There are other possibilities. +For instance, consider the case d12 = 0,d23 = π, which corresponds to a + +Schubart orbits on S1 +3 +configuration with m1,m2 at collision and m3 lies at the opposite end. +This will be called collision-antipodal singularity, [5, 7]. +The rotation group SO(2) acts on the configuration space by +(ϕ1,ϕ2,ϕ3) ↦ (ϕ1 + s,ϕ2 + s,ϕ3 + s),s ∈ R. +This action keeps the potential function, and actually keeps the sys- +tem by tangent lift. The corresponding first integral is the angular +momentum, i.e., J = ∑n +i=1 mi ˙ϕi. Thus, we have two first integrals +(1) +∑miϕi = αt + α′, K − U = h. +Note that is ϕi(t) is a solution, so is ϕi(t) + at + a′. We may assume, +by changing the coordinates by linear functions of the variable t, that +∑miϕi = 0 ( +mod 2π). +We further assume that ∑mi = 1. +2.1. Jacobi Coordinates and the Singularities. Let +1 = m1 + m2 + m3, +α1 = +m1 +m1 + m2 +, +α2 = +m2 +m1 + m2 +. +Introduce Jacobi variables as +x1 = ϕ2 − ϕ1, +x2 = ϕ3 − α1ϕ1 − α2ϕ2 +and their velocities ui = ˙xi. Then the inverse is +ϕ1 = −α2x1 − m3x2, +ϕ2 = α1x1 − m3x2, +ϕ3 = (m1 + m2)x2, +ϕ2 − ϕ1 = x1, +ϕ3 − ϕ1 = α2x1 + x2, +ϕ2 − ϕ3 = α1x1 − x2. +Then the kinetic energy is +2K = ∑mi ˙ϕ2 +i = m1(−α2 ˙x1 − m3 ˙x2)2 + m2(α1 ˙x1 − m3 ˙x2)2 + m3(m1 + m2)2 ˙x2 +2 += µ1u2 +1 + µ2u2 +2, +where µ1 = +m1m2 +m1+m2 and µ2 = (m1 + m2)m3. The potential is +U(x1,x2) = m1m2 cotd12 + m1m3 cotd13 + m2m3 cotd23. +Now we assume that the three bodies are ordered on the circle anti- +clockwise as +−π + 2kπ ≤ ϕ1 ≤ ϕ3 ≤ ϕ2 ≤ π + 2kπ. +Then +d12 = min{x1,2π − x1}, d13 = min{x2 + α2x1,2π − x2 − α2x1}, +d23 = min{α1x1 − x2,2π − α1x1 + x2}. +For the Newtonian collinear three-body problem, a similar ordering +is q1 < q3 < q2, where qi is the coordinates of mi, i = 1,2,3. For such +an ordering, possible singularities are total collision, collision between + +4 +Shuqiang Zhu +m2,m3, collision between m1,m3. +With the Jacobi coordinates, the +configuration space is a region between two half lines, [11]. +For our three-body problem on S1, obviously, we have +ϕ2−ϕ1 = x1 ∈ [0,2π], ϕ2−ϕ3 = α1x1−x2 ∈ [0,2π], ϕ3−ϕ1 = α2x1+x2 ∈ [0,2π], +Then the configuration space is the triangular region bounded by +α1x1 − x2 = 0, α2x1 + x2 = 0, x1 = 2π. +as shown in Figure 1. The singularities are +Figure 1. The configuration space, and some zero ve- +locity curves. The masses are 1 +3, 1 +3, 1 +3. Zero velocity curves +in region I for h = −100 (orange), 0 (blue), 100 (green) +are shown. +ϕ2 − ϕ1,ϕ2 − ϕ3,ϕ3 − ϕ1 = 0,π,2π. +That is, +x1 = 0,π,2π, α1x1 − x2 = 0,π,2π, α2x1 + x2 = 0,π,2π, +i.e., the vertices, boundary and the three mid-segments of the triangular +region. These singularities divide the configuration space into four sub- +triangles. Let us denote the four sub-triangular regions as I, II, III, and +IV, as in Figure 1. +The three vertices correspond to the total collisions, the three sides +correspond to the double collisions, the three mid-segments correspond +to antipodal singularities, and the intersections of the three mid-segments +correspond to three collision-antipodal singularities. In Figure 2, we +sketch the real configurations corresponding to typical points of the +configuration space. + +m3 +m2 +0 +m1C2 = α1 +III +22 +I +II +1 +IV +2 = α2X1Schubart orbits on S1 +5 +Figure 2. The configurations corresponding to vertices, +sides and interior of the four sub-triangular regions +2.2. Hill’s region. Consider the motion on the energy surface H = h, +then the projection of the energy surface to the configuration space is +called the Hill’s region corresponding to energy h, +H(h) = {x ∶ U(x) + h ≥ 0} +Recall that the energy surface lies over its projection H(h) as a kind +of degenerate circle bundle. The boundary ∂H(h) = {x ∶ U(x) = −h} is +the zero-velocity curve. +We sketched some zero-velocity curves in region I in Figure 1. The +zero-velocity curves in region III and IV are similar to that in region I, +but they are more complex in region II, as we shall see soon. Note that +U is undefined at the three intersections of the mid-segments. Other +than the three points, U(x) → ∞ as x approaches the three sides, and +U(x) → −∞ as x approaches the three mid-segments. Thus we have +the following +Proposition 1. The antipodal singularities are repelling for the three- +body problem on S1. + +6 +Shuqiang Zhu +2.3. Eulerian homothetic orbits. For three-body problem, it is enough +to consider the motion in region I and II. Let us first consider a simple +case. +Consider the isosceles problem on S1. The masses are m1 = m2 = +1−m +2 ,m3 = m. The initial data is +ϕ2 = ϕ ∈ (0,π),ϕ1 = −ϕ,ϕ3 = 0, ˙ϕ1 = − ˙ϕ2, ˙ϕ3 = 0. +By the symmetry, the configuration would stay isosceles with ϕ3 = 0. +That is, the system has just one degree of freedom. More precisely, since +ϕ2 − ϕ3 = ϕ3 − ϕ1, we see x2 = 0, and x1 = 2ϕ ∈ (0,2π), µ1 = m1/2 = 1−m +4 . +Let p1 = µ1u1, then +H = 1 +2 +p2 +1 +µ1 +− U(x1), U = (1 − m)2 +4 +(cotd12 + 4m +1 − m cotd23). +The motion depends on the function, +{cotx1 + 4m +1−m cot x1 +2 , +0 < x1 < π +−cotx1 + 4m +1−m cot x1 +2 . +π < x1 < 2π, +On [0,π], the function is decreasing from ∞ to −∞ for any value of m. +While on [π,2π], the graph depends on the value of m, since +− cot2ϕ + 4m +1 − m cotϕ = sin2 ϕ − cos2 ϕ +2sinϕcosϕ + 4m +1 − m cotϕ += +sin2 ϕ +2sinϕcosϕ + ( 4m +1 − m − 1 +2)cotϕ = 1 +2 tan x1 +2 + 9m − 1 +2(1 − m) cot x1 +2 . +Figure 3. +Graphs of U on [π,2π] for m < 1/9 (orange), +m = 1/9 (blue), m > 1/9 (green) +The motions are obtained by the conservation of energy. +Recall +that x1 = 0,π,2π are singularities. More precisely, x1 = 0 is the total +collision, x1 = π is the antipodal singularity, and x1 = 2π is the collision- +antipodal singularity. On (0,π), all motions would eventually go to +x1 = 0, or, triple collision must happens. At x1 = π, −U is ∞, so it + +4 +2 +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +-2 +6 +8Schubart orbits on S1 +7 +is repelling, or, antipodal singularity is impossible. +The qualitative +feature of motions on (π,2π) depends on the sign of 9m − 1. +● If m < 1/9, all motions would eventually go to x1 = 2π, or, +collision-antipodal singularity must happen, and at that mo- +ment, the velocity is infinite. +● If m = 1/9, −U is decreasing on (π,2π) and −U(2π) = 0. Then +for H = h ≥ 0, collision-antipodal singularity must happen, and +at that moment, the velocity is finite. +● If m > 1/9, there is one critical value h0 of −U. If H = h > h0, the +motions are periodic. If H = h = h0, it is a stable equilibrium. +For later use, let us refer to those configurations as Eulerian central +configurations and the orbits as Eulerian homothetic orbits. +Remark 1. The above example was first considered by Florin et al, +see [7]. +3. Regularization of the collision singularities +We now focus on motions in region I. Intuitively, all motions in this +region seems to end in a total collision. However, we will construct +symmetric periodic orbits in region I, called Schubart orbits, in next +section. In this section, we regularize the double and triple collision +singularities. +Assume that m3 = m ∈ (0,1), m1 = m2 = n, then n = (1 − m)/2, and +α1 = α2 = 1/2, µ1 = 1 − m +4 += n/2, µ2 = (1 − m)m = 2nm. +In region I, the distances are d12 = x1,d13 = 1 +2x1 + x2,d23 = 1 +2x1 − x2, so +L = K + U, K = 1 +2(µ1u2 +1 + µ2u2 +2), +U(x1,x2) = n2 cotx1 + mn(cot(1 +2x1 + x2) + cot(1 +2x1 − x2)) +Recall that region I is a triangular region bounded by 1 +2x1±x2 = 0 and +x1 = π. The three vertices and sides are singularities. We perform first +Mcgehee’s coordinates then another change of variables to eliminate +the singularities corresponding to the collisions, see Figure 1. +Let +x1 = +1 +√µ1 +r cosθ, x2 = +1 +√µ2 +r sinθ. +Then 2K = ˙r2 + r2 ˙θ2, and +U(x1,x2) = n2 cot(rA1 cosθ)+mn(cot[rA2 sin(θ + θ∗)] + cot[rA2 sin(θ∗ − θ)]) + +8 +Shuqiang Zhu +where A1 = +√ +2 +n,A2 = +√ +m+1 +2nm, and +θ∗ = arctan√m, θ∗ < π +4 +The configuration space has been blew up to +θ ∈ (−θ∗,θ∗), 0 ≤ r < +√µ1π +cosθ = +√nπ +√ +2cosθ +. +The corresponding second-order Euler-Lagrange equations are: +¨r = r ˙θ2 + Ur, +˙ +r2 ˙θ = Uθ. +(2) +Next, one can blow-up the triple collision singularity at r = 0 by intro- +ducing the the time rescaling ′ = r +3 +2 ˙ and the variable ν = r′/r. Setting +τ = θ′ gives the following first-order system of differential equations: +r′ = νr, +ν′ = 1 +2ν2 + τ 2 + r2Ur +θ′ = τ +τ ′ = −1 +2τν + rUθ +(3) +with energy equation: +(4) +ν2 + τ 2 +2 +− rU = rh +Explicitly, the functions are +rU = n2r cot(rA1 cosθ) + mn(r cot[rA2 sin(θ + θ∗)] + r cot[rA2 sin(θ∗ − θ)]) +rUθ = n2 +r2A1 sinθ +sin2(rA1 cosθ) + mn(− +r2A2 cos(θ + θ∗) +sin2[rA2 sin(θ + θ∗)] + +r2A2 cos(θ∗ − θ) +sin2[rA2 sin(θ∗ − θ)]) +r2Ur = −n2 +r2A1 cosθ +sin2(rA1 cosθ) − mn( +r2A2 sin(θ + θ∗) +sin2[rA2 sin(θ + θ∗)] + +r2A2 sin(θ∗ − θ) +sin2[rA2 sin(θ∗ − θ)]) +They are well-defined at r = 0. Hence {r = 0} is now an invariant set +for the flow, called the triple collision manifold. +The differential equations are still singular due to the double col- +lisions at θ = ±θ∗. The final coordinate change will eliminate these +singularities. Define new variables u,γ such that +θ = θ∗ sinu, γ = τ cos2 u. +Note that −θ∗ ≤ θ ≤ θ∗ corresponds to −π +2 ≤ u ≤ π +2. After calculating the +differential equations for u,γ, introduce a further rescaling of time by + +Schubart orbits on S1 +9 +multiplying the vector field by θ∗ cos2 u. Retaining the prime to denote +differentiation with respect to the new time variable one finds +r′ = θ∗νr cos2 u +ν′ = θ∗ cos2 u(1 +2ν2 + τ 2 + r2Ur) = θ∗(1 +2ν2 cos2 u + +γ2 +cos2 u + r2Ur cos2 u) += θ∗ cos2 u(−1 +2ν2 + 2rh + 2rU + r2Ur) +u′ = +γ +cosu +γ′ = −1 +2θ∗νγ cos2 u + θ∗rUθ cos4 u − 2τ 2 sinucos2 u, += −1 +2θ∗νγ cos2 u + θ∗rUθ cos4 u − 2sinu +γ2 +cos2 u, +(5) +with energy equation: +(6) +ν2 cos2 u + γ2/cos2 u +2 +− rU cos2 u = rhcos2 u +The configuration space is now +u ∈ R, 0 ≤ r < +√nπ +√ +2cos(θ∗ sinu) +. +Note that there is still one singularity on the boundary of the con- +figuration space, r = +√nπ +√ +2 cos(θ∗),u = π +2. Recall that it is one of the inter- +sections of the mid-segments and that the potential is undefined there. +Denote it by Q, see Figure 5. Except this singularity, The vector field +is smooth and continuous on the boundary. +The differential equations (5) represent the three-body problem on S1 +with the prescribed energy for configurations being an obtuse triangle +and with m3 in the middle, with triple collision blown-up and double +collisions regularized. The shape variable u need not be restricted to +the interval [−π +2, π +2]. As u ranges over the real axis, the configuration +oscillates between the double collisions at ±θ∗ and the mass m3 collides +with m1 and m2 successively. +The equations has some symmetries and a Schubart orbit can be +obtained from an orbit from u = 0,ν = 0 to u = π +2,ν = 0. Suppose that +we have a quarter of the trajectory Γ(t),t ∈ [0,t1] with +Γ(0) = (r0,0,0,γ0), Γ(1) = (r1,0, π +2 ,γ1). + +10 +Shuqiang Zhu +we can construct the second quarter of the trajectory from Γ(1) to +Γ(2) = (r0,0,π,γ0) by +[t1,2t1] → (r(2t1 − t),−ν(2t1 − t),π − u(2t1 − t),γ(2t1 − t)), +(it satisfies the boundary condition and is a solution since Uθ(π − u) = +Uθ(u),Ur(π − u) = Ur(u)) and so the third and the fourth quarters. +Then, using the symmetry of the vector field, it follows that one can +piece together the four of them. Hence, the existence of the required +orbit is reduced to find the first quarter, which will be proved by a +topological shooting argument in the next section. +4. Schubart orbits by the shooting method +Consider the system (5) on the manifold of fixed energy h = −1. We +construct the first quarter of the claimed Schubart orbit in this section. +We will follow the shooting method in [11], where Moeckel used it to +show the existence of Schubart orbit for the Newtonian collinear three- +body problem. The idea is to construct a continuous map in the phase +space and then apply a shooting argument. +The construction of the continuous map in the phase space is based +on the result of Wazewski [15]. +Roughly speaking, a subset, called +a Wazewski set, of the phase space is carefully chosen such that the +amount of time required to leave depends continuously on initial con- +ditions. Then the exit point also depends continuously on initial con- +ditions. This idea were developed by Conley and Easton [2, 4, 9] to +isolating blocks, topological index for invariant sets. +There are several technical computations in this section. +To not +interrupt the flow of the argument, we will just claim them in this +section and give the detail in the appendix. +4.1. The Wazewski set W. Consider a flow φt(x) on a topological +space X and a subset W ⊂ X of X. Let W0 be the set of points in W +which eventually leave W in forward time, and let E the set of points +which exit immediately: +W0 = {x ∈ W ∶ ∃t > 0,φt(x) ∉ W}, +E = {x ∈ W ∶ ∀t > 0,φ[0,t)(x) ⊈ W}. +Clearly, E ⊂ W0. Given x ∈ W0 define the exit time +τ(x) = sup{t ≥ 0 ∶ φ[0,t)(x) ⊆ W}}. +Note that τ(x) = 0 if and only if x ∈ E. Then τ is continuous if +● If x ∈ W0 and φ[0,t](x) ⊆ W, then φ[0,t](x) ⊆ W. +● E is a relatively closed subset of W0. + +Schubart orbits on S1 +11 +In this case, the set W is called a Wazewski set. +Since we are considering the motion on the energy manifold h = −1, +the configuration is in the region, {(r,u) ∶ U ≥ 1}. Note that U = 1 +defines an implicit function r(u) since Ur < 0. Define r∗ = r(0). Let +W = {(r,ν,u,γ) ∶ (6) holds, 0 ≤ r ≤ r∗, ν ≤ 0,0 ≤ u ≤ π +2 ,γ ≥ 0}, +The choice of the set W is motivated by that of Moeckel in [11]. The +major difference is that the value of r is confined to [0,r∗] in our case, +and it is not in that of Moeckel’s proof. Thus, the configuration space +is restricted to a rectangle, see Figure 4. As it turns out, the restriction +0 ≤ r ≤ r∗ is essential for our proof. On one hand, this restriction avoids +the singularity Q so that the system leads to a well-defined flow on W. +On the other hand, the restriction leads to the estimate(8), which is +essential for our proof. Note that the restriction makes no harm since +r is non-increasing in W, so the exit points must have r ≤ r∗. +To visualize W, we use coordinates (r,ν,u) on the energy manifold, +and the value of γ is determined by energy equation. +The energy +manifold projects to the three-dimensional region +ν2 cos2 u +2 +− rU cos2 u + r cos2 u ≤ 0, 0 ≤ r ≤ r∗, +see Figure 4. The south part of the upper surface in the figure, where +equality holds in (6) corresponds to γ = 0. The figure also shows a +sketch of the shooting argument. +4.2. The invariant manifold H = {u = 0,γ = 0}. It is easy to verify +that it is invariant under the flow since u′ = +γ +cos u = 0 and +γ′ = θ∗rUθ∣θ=0 +where we have used that fact that τ = 0,γ = 0 and the following claim. +Claim 1: U,r2Ur are even in θ, rUθ is even in r, and limr→0 −r2Ur = +rU. In words, the last identity implies that the function U is homoge- +neous of degree −1 on r where r is small. +The dynamics on H is thus +r′ = θ∗νr, ν′ = θ∗(1 +2ν2 + r2Ur) +Since u = 0, it is just the homothetic orbits considered in Subsection +2.3, but is regularized. There is one equilibrium point, the intersection +of the collision manifold and H, denote it by P. The exact coordinates +is +P = (0,−ν0,0,0), 1 +2ν2 +0 = rU∣r=0,θ=0. + +12 +Shuqiang Zhu +Figure 4. The Wazewski set W and a sketch of the +existence proof +4.3. The equilibrium point P is hyperbolic. As in the Newto- +nian collinear three-body problem, the equilibrium P is found to be +hyperbolic [10]. +We use the coordinates r,u,γ, and the variable ν is treated as a +function of r,u,γ. The energy relation gives +∂ν +∂r = (rU)r − 1 +ν += 1 +ν0 +, ∂ν +∂u = θ∗ +rUθ cosu − 2γ2 sinu/cos5 u +ν += 0, +∂ν +∂γ = +−γ +ν cos4 u = 0, +at the point P. Then one finds that the linearized differential equations +at P have matrix +⎡⎢⎢⎢⎢⎢⎣ +θ∗ν cos2 u + θ∗r cos2 u ∂ν +∂r +θ∗r cos2 u ∂ν +∂u +θ∗r cos2 u ∂ν +∂γ +0 +γ( +1 +cos u)′ +1 +cos u +−θ∗ +2 γ cos2 u ∂ν +∂r + θ∗(rUθ)r cos4 u +☀ +−2 sin u +cos2 u +⎤⎥⎥⎥⎥⎥⎦ += +⎡⎢⎢⎢⎢⎢⎣ +−θ∗ν0 +0 +0 +0 +0 +1 +0 +θ2 +∗rUθθ +1 +2θ∗ν0 +⎤⎥⎥⎥⎥⎥⎦ +where +☀ = −θ∗ +2 γ(ν cos2 u)u + rUθθθ2 +∗ cos5 u + rUθθ∗(cos4 u)u − 2γ2( sinu +cos2 u)u, + +B +s +A +F(s) +T +Wp) +u +P +1Schubart orbits on S1 +13 +and we use the fact in Claim 1 and the following Claim 2. +Claim 2: At the point r = 0,u = 0, we have rUθθ > 0. +Thus, the equilibrium P is hyperbolic, with eigenvalues λ1 = −θ∗ν0 < +0 and λ2 < 0,λ3 > 0. +Then it has two-dimensional stable manifold +and one-dimensional unstable manifold. The eigenvectors are (1,0,0), +(0,1,λ2), and (0,1,λ3). The first stable eigenvector is tangent to the +homothetic orbit H. Note that the other stable eigenvector(0,1,λ2) +points out of W since γ ≥ 0 in W. It follows that H ⋂W = W s(P)⋂W. +The unstable manifold of P is on the collision manifold, with one branch +in W. +Lemma 1. The branch of W u(P) in W exits W at a point of the form +(0,ν, π +2,γ) with ν < 0. +The following fact will be used in the proof. +Claim 3: Restricted on r = 0, the maximum of 2rU cos2 u is at u = 0. +Proof. Consider the system for u,ν. By the energy relation, the equa- +tions read +ν′ = θ∗ cos2 u(−1 +2ν2 + 2rh + 2rU + r2Ur) = θ∗ cos2 u(rU − 1 +2ν2) +u′ = +γ +cosu = cosu +√ +2rU − ν2. +Then +dν +du = θ∗ +2 +√ +2rU cos2 u − ν2 cos2 u ≤ θ∗ +2 +√ +2rU cos2 u. +So +dν +du ≤ θ∗ +2 ν0 +which implies hat the increment in ν for 0 ≤ u ≤ π +2 satisfies: +∆ν ≤ π +2 +θ∗ +2 ν0 ≤ ν0 +since θ∗ ≤ π +4. Since the branch of W u(P) begins near P, and P has +coordinates u = 0 and ν = −ν0, then it arrives at u = π +2 without crossing +the line ν = 0. +□ +4.4. W is a Wazewski set. In this subsection, we identify the subsets +W0,E and show +Theorem 1. W is a Wazewski set for the flow on the energy manifold. +The first property obviously holds since the set W is closed. For the +second property, we first identify the subsets W0,E. + +14 +Shuqiang Zhu +Lemma 2. W0 = {x ∈ W ∶ ∃t > 0,φt(x) ∉ W} = W ∖ H +The following facts will be used in the proof. +Claim 4: 2rU cos2 u∣u= π +2 has a positive lower bound c2 +2. +Claim 5: The function θ∗rUθ +cos4 u +sin u has a positive lower bound c3. +Proof. Let x0 = (r0,ν0,u0,γ0) ∈ W, It is easy to that the solution begin +from x0 exist as long as it remains in W. Now suppose x0 ∈ W ∖ H. +Our goal is to show that φt(x0) eventually leaves W. If u0 = 0 then +u′(0) = γ0 > 0 since x0 ∉ H. It follows that for every t0 > 0,u(t0) > 0. +Thus it is enough to assume u0 > 0. +Let u0 be a positive constant and Wu0 = {x ∈ W ∶ u ≥ u0}. Since u(t) +is non-decreasing in W, Wu0 is positively invariant relative to W. We +show below that there are two constants c0 > 0 and c1 > 0 such that for +every x ∈ Wu0 +either +γ +cosu ≥ c0, or ( +γ +cosu)′ ≥ c1. +Then it is easy to see that φt(x0) must eventually leave W. Note +that +Wu0 = W+ +u0 ⋃W− +u0, +where +W + +u0 = {x ∈ Wu0, +γ +cosu ≥ c0}, W− +u0 = {x ∈ Wu0,0 ≤ +γ +cosu < c0}. +Since ( +γ +cos u)′ ≥ c1 > 0 in W− +u0 it implies that an orbit segment can stay +in W− +u0 for time at most c0/c1, and then would enter W+ +u0. Note that +W+ +u0 is positively invariant relative to Wu0. Finally, an orbit can remain +in W+ +u0 for time not longer than +π +2c0 since u′ = +γ +cos u ≥ c0. Hence, every +orbit starting in Wu0 must leaves W eventually. +We now construct c0 > 0,c1 > 0 such that either +γ +cos u ≥ c0 or ( +γ +cos u)′ ≥ +c1 for all x ∈ Wu0. For u = π +2, the equation (6) implies +γ +cos u = +√ +2rU cos2 u∣u= π +2 ≥ +c2. We can choose c0 to be less than c2 then +γ +cos u ≥ c0 holds for u = π +2. +For u0 ≤ u < π +2, we have +( +γ +cosu)′ = +γ′ +cosu + tanu( +γ +cosu)2 += −1 +2θ∗νγ cosu + θ∗rUθ cos3 u − 2sinu +γ2 +cos3 u + tanu( +γ +cosu)2 +≥ tanu(θ∗rUθ +cos4 u +sinu − ( +γ +cosu)2) +≥ tanu0 (c3 − ( +γ +cosu)2). + +Schubart orbits on S1 +15 +Then we take c0 such that c0 ≤ c2,c2 +0 ≤ c3 +2 , and take c1 = c3 tan u0 +2 +. Then +on u = π +2, we have +γ +cos u ≥ c2 ≥ c0. For u0 ≤ u < π +2, if +γ +cos u ≤ c0, then +( +γ +cos u)′ ≥ tanu0 (c3 − ( +γ +cos u)2) ≥ c1, as required. +□ +It remains to identify the immediate exit set E. It is useful to dis- +tinguish two subsets of the boundary. Let x = (r,ν,u,γ) and let +B1 = {x ∈ W ∶ u = π +2 }, +B2 = {x ∈ W ∶ ν = 0,0 ≤ u < π +2 ,2rU + r2Ur − 2r ≥ 0}. +Obviously, the two subsets B1 and B2 are relatively closed in W. Hence +Theorem 1 is proved once we show +Lemma 3. The immediate exit set of W is E = B1 ⋃B2. +Figure 5. The configuration space and the rectangle +[0, π +2] × [0,r∗] +The following fact will be used in the proof. +Claim 6: Let F(r,u) = 2rU + r2Ur − 2r. Fu > 0 for 0 ≤ r ≤ r∗,0 < u ≤ +π +2. At u = 0,F = 0, we have Fu = 0,Fuu > 0. + +T +U= +B +0=1 +A +n +0 +216 +Shuqiang Zhu +Proof. As claimed, +√ +2rU cos2 u∣u= π +2 has a positive lower bound, so B1 ⊂ +E. Consider a point x ∈ B2. Note that h = −1,ν = 0, then +ν′ = θ∗( +γ2 +cos2 u + r2Ur cos2 u) = θ∗ cos2 u(2rU + r2Ur − 2r) ≥ 0. +Let F(r,u) = 2rU+r2Ur−2r. Since Fu > 0 in the rectangle (0, π +2]×[0,r∗], +the set F = 0 in this rectangle is a curve bounded by the two points +A,B (see Figure 5). +The curve divides the rectangle into two parts. The bottom r = 0 is +in the set F > 0 since on which F = rU, while the vertex u = 0,r = r∗ is +in the set F < 0 since U = 1 and that r2Ur < 0. +If F > 0, then ν′ > 0 and x is an immediate exit point. If F = 0 +and u ≠ 0, one has ν = ν′ = 0 and one finds that the second derivative +reduces to +ν′′ = θ∗(−sin2uF + cos2 uFu)u′ + cos2 uFrr′ = cosuFuγ > 0, +and x is an immediate exit point. Finally, if u = ν = 0 and F = 0, one +has ν = ν′ = ν′′ = 0, The third derivative at the point A is found to be +ν′′′ = θ∗γ2Fuu > 0. +Again, x is an immediate exit point. +It remains to check that there are no other immediate exit points. +Suppose that x0 ∈ W is an immediate exit point and it is not in B1 ⋃B2. +Following the argument in [11], it is enough the check the following +cases. +First, it may happen that r0 = 0 but r(t) < 0 for small positive times. +This is impossible because the manifold {r = 0} is invariant. +Secondly, it may happen that u0 = 0 but u(t) < 0 for small positive +times. It requires u′(0) = γ0 ≤ 0 and since x0 ∈ W this means γ0 = 0, so +x0 ∈ H and points of H are certainly not leaving W. +Thirdly, it may happen that ν0 = 0 but ν(t) increases for small +positive times. This forces ν′(0) ≥ 0 and then x0 ∈ B2. +Fourthly, it may happen that r0 = r∗ but r(t) increases. This forces +r′(0) = 0 and then ν0 = 0, i.e., the coordinates of the point is r = r∗,0 ≤ +u < α, where (r∗,α) is the coordinates of the point B, see Figure 5. So +ν′ < 0. Then one finds +r′′ = θ∗(ν(r cos2 u)′ + r cosu ν′) = θ∗r∗ν′ cos2 u < 0, +This mode of existing is impossible. +At last, it may happen that γ0 = 0 but γ(t) decreases for small +positive times. If u0 = 0, then x0 ∈ H, and points of H are certainly not +leaving W. If u0 = π +2, then x0 ∈ B1. One may assume 0 < u0 < π +2. In +this case, it follows from the proof of Lemma 2 that there are positive + +Schubart orbits on S1 +17 +constants c0,c1 such that ( +γ +cos u)′ ≥ c1 > 0 whenever +γ +cos u < c0. So this +mode of exiting is also impossible. This completes the proof. +□ +4.5. The shooting argument. Finally, we can complete the con- +struction of the symmetric periodic orbit. Recall that it suffices to con- +struct the first quarter, which is required to be an orbit from u = 0,ν = 0 +to u = π +2,ν = 0. +Since W is a Wazewski set, the time required to reach E depends +continuously on initial conditions and so there is a continuous flow- +defined map from W0 to E. The map is also continuous if we restrict +the domain to +S = {(r,ν,u,γ) ∈ W0,u = ν = 0,0 ≤ r < r∗}. +That is, the flow-defined map F ∶ S → E is continuous. Let +T = {(r,ν,u,γ) ∈ W,u = π +2 ,ν = 0}. +Note that T ⊂ E and that S and T are two of the edges in the boundary +of the three-dimensional Wazewski set W (shown as bold vertical lines +in Figure 4). Then the construction of the first quarter of the orbit +reduces to show that +F(S)⋂T ≠ ∅. +First, note that part of S near r = 0 is contained in B2 ⊂ E. These +points exit W immediately, so the map F is the identity there. Sec- +ondly, points of S with r close to r∗ will enter the interior of W and exit +elsewhere. By continuous dependence of the initial conditions, these +points will follow the homothetic orbit H to a neighborhood of the equi- +librium point P = (0,−ν0,0,0). Then the lambda lemma [16] implies +that they will follow a branch of the unstable manifold W u(P), which +is one-dimensional and is contained entirely in the invariant manifold +r = 0, as shown in Subsection 4.3. Furthermore, by Lemma 1, one of the +two branches lies in W0 and it goes to some point on r = 0,ν < 0,u = π +2. +Then the lambda lemma implies that the image of points near the +upper endpoint of S under the continuous mapping F are on B1 ∖ T . +We can now complete the shooting argument. Recall that there is +continuous map F ∶ S → E, E = B1 ⋃B2 and that B1and B2 are two- +dimensional continuum meeting along the edge T . As we have shown, +the image of points near r = 0 under F are in B2 ∖ T , while the image +of points near r = r∗ under F are in B1 ∖ T . It follows that there must +exist at least one intersection point U ∈ F(S)⋂T . This shows that +F(S)⋂T ≠ ∅ and completes the existence proof for the symmetric +periodic orbits. + +18 +Shuqiang Zhu +Remark 2. The orbit constructed lies in the energy manifold h = −1. +By restricting the configuration to the rectangle 0 ≤ r ≤ r∗,0 ≤ u ≤ π +2, +where u = 0,r = r∗ is the intersection of U = 1 and u = 0, the six Claims +hold. Consider an energy manifold h < −1. Since U is decreasing on +u = 0, the intersection of U = −h and u = 0 is lower than r∗. Then +the six Claims made in this section still hold and all arguments can be +applied as well. Thus, we have the following +Theorem 2. Given three positive masses m1 = m2 and m3 and an +energy h ≤ −1. Then there exists a symmetric periodic solution of the +collinear three-body problem on S1 with energy h and regularized double +collisions. The orbit has the following features. +● The configuration lies in region I. +● In the first quarter of the orbit, the masses move from the Euler- +ian central configuration with m3 in the middle of m1,m2 to a +double collision between m2 and m3. At the moment of the dou- +ble collision the velocity of m1 is zero. +● The second quarter of the orbit is the time-reverse of the first, +and the second half is the reflection of the first half with the +roles of m1 and m2 reversed. +5. Appendix: Proofs of the six Claims +Recall that +rU = n2r cot(rA1 cosθ) + mn(r cot[rA2 sin(θ + θ∗)] + r cot[rA2 sin(θ∗ − θ)]) +rUθ = n2 +r2A1 sinθ +sin2(rA1 cosθ) + mn(− +r2A2 cos(θ + θ∗) +sin2[rA2 sin(θ + θ∗)] + +r2A2 cos(θ∗ − θ) +sin2[rA2 sin(θ∗ − θ)]) +r2Ur = −n2 +r2A1 cosθ +sin2(rA1 cosθ) − mn( +r2A2 sin(θ + θ∗) +sin2[rA2 sin(θ + θ∗)] + +r2A2 sin(θ∗ − θ) +sin2[rA2 sin(θ∗ − θ)]) +5.1. +Claim 1: U,r2Ur are even in θ, rUθ is even in r, and limr→0 −r2Ur = +rU. In words, the last identity implies that the function U is almost +homogeneous of degree −1 on r where r is small. +It is easy to see by the explicit form of the functions. +5.2. +Claim 2: At the point r = 0,u = 0, we have rUθθ > 0. +We show that +rUθ = n2 +r2A1 sinθ +sin2(rA1 cosθ)+mn(− +r2A2 cos(θ + θ∗) +sin2[rA2 sin(θ + θ∗)] + +r2A2 cos(θ∗ − θ) +sin2[rA2 sin(θ∗ − θ)]) +is strictly increasing on θ at r = 0,θ = 0. + +Schubart orbits on S1 +19 +The first term is strictly increasing in θ, at r = θ = 0. Direct compu- +tation gives +( +r2A1 sinθ +sin2(rA1 cosθ))θ = r2A1 +sin2(rA1 cosθ)cosθ − rA1 sin2 θ sin(2rA1 cosθ) +sin4(rA1 cosθ) +→ 1 +A1 +> 0. +The second term, denoted by g(r,θ), is an increasing function on θ +in a neighborhood of r = θ = 0. Indeed, g(r,0) = 0, and g(r,θ) > 0 if +0 < θ ≤ θ∗ and r is small. Note that θ∗ ≤ π +4, then +cos(θ∗ − θ) > cos(θ∗ + θ), rA2 sin(θ∗ − θ) < rA2 sin(θ∗ + θ) < π +2 . +Thus, +g(r,θ) = mnr2A2 ( +cos(θ∗ − θ) +sin2[rA2 sin(θ∗ − θ)] − +cos(θ + θ∗) +sin2[rA2 sin(θ + θ∗)]) > 0, +and the derivative gθ(0,0) ≥ 0. +5.3. +Claim 3: Restricted on r = 0, the maximum of 2rU cos2 u is at +u = 0. +Recall that +rU = n2r cot(rA1 cosθ)+mn(r cot[rA2 sin(θ + θ∗)] + r cot[rA2 sin(θ∗ − θ)]) +Let r → 0, we have +rU cos2 u = n2 +cos2 u +A1 cos(θ∗ sinu) + mncos2 u( +1 +A2 sin(θ∗ + θ) + +1 +A2 sin(θ∗ − θ)) += n2 +cos2 u +A1 cos(θ∗ sinu) + 2mnsinθ∗ +A2 +cos2 ucosθ +cos2 θ − cos2 θ∗ +. +The first term is a decreasing function of u. Since θ∗ ≤ π +4 < 1, we +have +( +cos2 u +cos(θ∗ sinu))′ = +cosu +cos2(θ∗ sinu)[θ∗ cos2 usin(θ∗ sinu) − 2sinucos(θ∗ sinu)] +≤ +cosu +cos2(θ∗ sinu)2[cosusin(θ∗ sinu) − sinucos(θ∗ sinu)] += +cosu +cos2(θ∗ sinu)2[sin(θ∗ sinu − u)] ≤ 0. +It remains to show +(7) +1 +1 − cos2 θ∗ +≥ +cos2 ucosθ +cos2 θ − cos2 θ∗ +, u ∈ [0, π +2 ]. + +20 +Shuqiang Zhu +it is equivalent to +J = cos2 θ − cos2 θ∗ − (1 − cos2 θ∗)cos2 ucosθ ≥ 0. +View J as a function of the two variables (θ∗,θ), on the triangular +region 0 < θ∗ ≤ π +4,0 ≤ θ ≤ θ∗. Note that J (θ∗,θ∗) = 0 and +∂J +∂θ∗ += 2sinθ∗ cosθ∗(1 − cos2 ucosθ) ≥ 0. +We conclude that the function J is non-negative on the triangular +region. +5.4. +Claim 4: 2rU cos2 u∣u= π +2 has a positive lower bound. +Recall that rU = n2r cotd12 + mn(cotd13 + cotd23), and the fact that +we are not at the singularity Q. Hence, when u = π +2, we have d23 = 0, +and the two distance d12,d13 are different from 0,π. +Hence +rU cos2 u = mncotd23 cos2 u = mnr cot[rA2 sin(θ∗ − θ)]cos2 u +Since cos2 u = sin2(π +2 − u), and θ∗ − θ = 2θ∗ sin2(π +2 − u), so we obtain +rU cos2 u = +mn +2θ∗A2 +. +5.5. +Claim 5: The function rUθ +cos4 u +sin u has a positive lower bound. +Recall that +rUθ = n2 +r2A1 sinθ +sin2(rA1 cosθ)+mn(− +r2A2 cos(θ + θ∗) +sin2[rA2 sin(θ + θ∗)] + +r2A2 cos(θ∗ − θ) +sin2[rA2 sin(θ∗ − θ)]). +We first claim that the function rUθ is non-negative. The first term is +non-negative. For the second term, which has been denoted by g(r,θ). +We have showed that g(r,0) = 0, and g(r,θ) > 0 if 0 < θ ≤ θ∗ and r is +small. Now we show that +0 ≤ r ≤ r∗, 0 < θ ≤ θ∗,⇒ g(r,θ) ≥ 0. +For this, it suffices to show that +r∗A2 sin(2θ∗) ≤ π +2 . +Recall that at θ = 0,r = r∗, we have U = 1. Note that rA2 sinθ∗ = +r +√ +m+1 +2nm +√ m +m+1 = rA1/2. Then +n2 cot(r∗A1) + 2mncot(r∗A1/2) = 1. +Let a = cot(r∗A1/2). Note that a > 0 since rA2 sinθ∗ = d23 < π +2. Then +2mna + n2a2 − 1 +2a += 1,⇒ (4mn + n2)a2 − 2a − n2 = 0. + +Schubart orbits on S1 +21 +So +a = cot(r∗A1/2) = +4 +−7m2 + 6m + 1 + +√ +−7m4 + 20m3 − 18m2 + 4m + 17 +−7m2 + 6m + 1 +. +Let g(m) = −7m4+20m3−18m2+4m+17. We have g′ = −4(m−1)2(7m− +1), so g(m) ≥ min{g(0),g(1)} = 16. Since −7m2+6m+1 ≤ 16 +7 , we obtain +the desired estimate +cot(r∗A1/2) ≥ 7 +2,⇒ r∗A1/2 ≤ π +10 +(8) +r∗A2 sin(2θ∗) = r∗A1 cosθ∗ = +r∗A1 +√ +m + 1 +< π +5 . +Now we show that θ∗rUθ +cos4 u +sin u ,0 < u0 < u < π +2,0 ≤ r ≤ r∗, has a positive +lower bound. Obviously, at u = π +2, the second term equals to +lim +u→ π +2 +r2A2 cos(θ∗ − θ) +sin2[rA2 sin(θ∗ − θ)] +cos4 u +sinu += lim +u→ π +2 +mnsin4(π/2 − u) +A2 sin2[2θ∗ sin2(π/2−u +2 +)] += 4mn +A2θ2∗ +. +Then there is some u1 < π +2 such that +θ∗rUθ +cos4 u +sinu ≥ 2mn +A2θ2∗ +, u1 ≤ u ≤ π +2 ,0 ≤ r ≤ r∗. +For the first term, let θ0 = u0 sinu, then +r2A1 sinθ +sin2(rA1 cosθ) +cos4 u +sinu ≥ r2A2 +1 sinθ cos4 u +A1 sin2(rA1) +≥ sinθ0 cos4 u1 +A1 +, u0 ≤ u ≤ u1,0 ≤ r ≤ r∗. +Thus, we conclude that rUθ +cos4 u +sin u ,0 < u0 < u < +π +2,0 ≤ r ≤ r∗, has a +positive lower bound. +5.6. +Claim 6: Let F(r,u) = 2rU + r2Ur − 2r. Fu > 0 for 0 ≤ r ≤ r∗,0 < +u ≤ π +2. At u = 0,F = 0, we have Fu = 0,Fuu > 0. +Recall that +2rU + r2Ur = n2r[2cot(rA1 cosθ) − +rA1 cosθ +sin2(rA1 cosθ)] + mnr{2cot[rA2 sin(θ∗ − θ)] +− +rA2 sin(θ∗ − θ) +sin2[rA2 sin(θ∗ − θ)] + 2cot[rA2 sin(θ + θ∗)] − +rA2 sin(θ + θ∗) +sin2[rA2 sin(θ + θ∗)]} +Introduce new variables +ρ = rA1 cosθ, ξ = rA2 sin(θ∗ − θ), η = rA2 sin(θ∗ + θ), +and define f(x) = 2cotx − +x +sin2 x. Then +2rU + r2Ur = n2rf(ρ) + mnr[f(ξ) + f(η)]. + +22 +Shuqiang Zhu +Let us first study the function f(x). On [0, π +5] +f ′(x) = −3 − 2xcotx +sin2 x +< 0, +f ′′(x) = − +2 +sin4 x(2x − 2sin2x + xcos2x) ≥ +2 +sin4 x(2sin2x − 3x) ≥ 0, +since k(x) = 2sin2x−3x is a concave function on [0, π +2]. Thus its value +on [0, π +5] is at least min{k(0),k(π +5)} = 0. +ρ′ = −rA1 sinθ, +ξ′ = −rA2 cos(θ∗ − θ), η′ = rA2 cos(θ∗ + θ), +ρ′′ = −ρ, +ξ′′ = −ξ, +η′′ = −η, +ρ ≤ π +5 , +ξ ≤ η ≤ r∗A2 sin2θ∗ < π +5 . +For the first derivative, one finds Fu = Fθθ∗ cosu, then it suffices to +show that Fθ = (2rU + r2Ur)θ > 0 for 0 ≤ r ≤ r∗,0 < θ < θ∗. +Fθ = n2rf ′(ρ)ρ′ + mnr(f ′(ξ)ξ′ + f ′(η)η′). +The first term is positive if θ ∈ (0, π +5], and it is zero if θ = 0. The +second term is zero if θ = 0, and it is positive if θ ∈ (0, π +5) since both +−f ′(x) and cos(x) are decreasing. Hence, we have proved that Fu > 0 +for 0 ≤ r ≤ r∗,0 < u < π +2 and Fu = 0 for 0 ≤ r ≤ r∗,u = 0. +For the second derivative at the point A, one finds +Fuu = Fθθ(θ∗ cosu)2 − Fθθ∗ sinu = Fθθθ2 +∗. +and +Fθθ = n2r[f ′′(ρ)(ρ′)2 − f ′(ρ)ρ] + mnr[f ′′(ξ)(ξ′)2 − f ′(ξ)ξ + f ′′(η)(η′)2 − f ′(η)η] += n2r[−f ′(ρ)ρ] + mnr[2f ′′(ξ)(ξ′)2 − 2f ′(ξ)ξ] > 0. +Hence, we have proved that Fuu > 0 at the point A. +Acknowledgments. +The author would like to thank Cristina +Stoica and Jean-Marie Becker for enlightening discussions. +References +[1] AV Borisov, LC Garc´ıa-Naranjo, IS Mamaev, and James Mon- +taldi. Reduction and relative equilibria for the two-body problem +on spaces of constant curvature. Celestial Mechanics and Dynam- +ical Astronomy, 130(6):1–36, 2018. +[2] C. Conley and R. Easton. +Isolated invariant sets and isolating +blocks. Trans. Amer. Math. Soc., 158:35–61, 1971. +[3] C. C. Conley. The retrograde circular solutions of the restricted +three-body problem via a submanifold convex to the flow. SIAM +J. Appl. Math., 16:620–625, 1968. + +Schubart orbits on S1 +23 +[4] Charles Conley. Isolated invariant sets and the Morse index, vol- +ume 38 of CBMS Regional Conference Series in Mathematics. +American Mathematical Society, Providence, R.I., 1978. +[5] Florin Diacu. On the singularities of the curved n-body problem. +Transactions of the American Mathematical Society, 363(4):2249– +2264, 2011. +[6] Florin Diacu. Relative equilibria in the 3-dimensional curved n- +body problem. Mem. Amer. Math. Soc., 228(1071):vi+80, 2014. +[7] Florin Diacu, Ernesto P´erez-Chavela, and Manuele Santoprete. +The n-body problem in spaces of constant curvature. part ii: Sin- +gularities. Journal of nonlinear science, 22(2):267–275, 2012. +[8] Florin Diacu, Cristina Stoica, and Shuqiang Zhu. Central con- +figurations of the curved n-body problem. 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Nachr., 283:17–22, 1956. +[14] Mitsuru Shibayama. +Minimizing periodic orbits with regulariz- +able collisions in the n-body problem. Arch. Ration. Mech. Anal., +199(3):821–841, 2011. +[15] Tadeusz Wa˙zewski. Sur un principe topologique de l’examen de +l’allure asymptotique des int´egrales des ´equations diff´erentielles +ordinaires. Ann. Soc. Polon. Math., 20:279–313 (1948), 1947. +[16] Stephen Wiggins. +Introduction to applied nonlinear dynamical +systems and chaos, volume 2 of Texts in Applied Mathematics. +Springer-Verlag, New York, 1990. +[17] Zhihong Xia. The existence of noncollision singularities in Newto- +nian systems. Ann. of Math. (2), 135(3):411–468, 1992. + diff --git a/i9AyT4oBgHgl3EQfX_dB/content/tmp_files/load_file.txt b/i9AyT4oBgHgl3EQfX_dB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3bc656b47a5b9970d7cffe218a3dac3745eebce9 --- /dev/null +++ b/i9AyT4oBgHgl3EQfX_dB/content/tmp_files/load_file.txt @@ -0,0 +1,546 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf,len=545 +page_content='THE SCHUBART ORBITS ON THE CIRCLE Shuqiang Zhu School of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China zhusq@swufe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='cn Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We consider the three body problem on S1 under the cotangent potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We first construct homothetic orbits ending in singularities, including total collision singularity and collision- antipodal singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then certain symmetrical periodic orbits with two equal masses, called Schubart orbits, are shown to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The proof is based on the construction of a Wazewski set in the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Introduction The Newtonian n-body problem has been generalized in many ways, for instance, under the general homogeneous potentials, or in higher dimensional Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Among them, the curved n-body prob- lem, which studies n-body problem on surfaces of constant curvature under the cotangent potential, has received lot of attentions in the last decade (cf [1, 6, 8] and the references therein ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In the Newtonian n-body problem, the two-body case is a Hamilton- ian system with one degree of freedom, so is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The cases with two degrees of freedom, namely, the restricted three-body problem and the collinear three-body problem, remains to be largely unsolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' How- ever, the ideas emerged in attacking them shed light on more general problems (cf [10, 12, 17] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We consider one case of the curved n-body problem with two degrees of freedom, namely, the three-body problem on the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' As a pre- liminary study on the problem, in this manuscript we construct some interesting orbits, the Eulerian homothetic orbits and Schubart orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The Eulerian homothetic orbits are connected with the singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In the curved three-body problem, there are collision, antipodal and collision-antipodal singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The antipodal singularities turn out to be impossible by consideration of Hill’s region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The collision-antipodal singularities are more interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In fact, our study indicates that it might be sensitive to the choice of masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='00193v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='DS] 31 Dec 2022 2 Shuqiang Zhu Intuitively, if all the masses stay on a semi-circle, they would attract each other and end in a triple collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' However, if two masses are equal, there exist certain symmetric periodic orbits, provided collisions are regularized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The behavior of these orbits is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Initially, the unequal mass, m3, is at the midpoint of the equal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' If the masses were released with zero velocity, it would be a homothetic collapse to triple collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' However, we set the initial velocities such that m2 and m3 move towards each other and m1 leaves them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then m1 slows and stops exactly when m2 and m3 collide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' This is the first quarter of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The second quarter of the orbit is the time-reverse of the first, and the second half is the reflection of the first half with the roles of m1 and m2 reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Such orbits are called Schubart orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' They were found numerically by Schubart [13] for the Newtonian three-body problem and the analytic existence proof, was given by in [11, 14], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For the existence proof of the Schubart orbits, we follow that of Moeckel in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' It is a topological argument and it is a variation of an idea used by Conley [3] in the Newtonian restricted three-body prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' More precisely, it is based on the construction of a Wazewski set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Unlike that of the Newtonian case, where the potential is almost a func- tion of the shape variable, the cotangent potential depends essentially on the two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Some computations are relatively lengthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 2, we discuss the basic setting of the three-body problem on S1 and the Eulerian homothetic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 3, we regularize the collision singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 4, we apply a topological argument to show the existence of Schubart orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Some technical computations are presented in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Settings and Eulerian homothetic orbits The configuration space is (S1)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The coordinates are ϕ1,ϕ2,ϕ3, with ϕi ∈ R/2πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The Lagrangian is L = ∑ i 1 2mi ˙ϕ2 i + ∑ j≠i mimj cotdij, where dij = min{∣ϕi −ϕj∣,2π −∣ϕi −ϕj∣}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The system is undefined in the set ∆ = ⋃i≠j ∆ij, with ∆ij = {(ϕ1,ϕ2,ϕ3) ∶ dij = 0}⋃{(ϕ1,ϕ2,ϕ3) ∶ dij = π}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The cases dij = 0 are collisions, whereas the cases dij = π are antipodal configurations, when some bodies are at the opposite ends of a diame- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In both cases, the forces are infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' There are other possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For instance, consider the case d12 = 0,d23 = π, which corresponds to a Schubart orbits on S1 3 configuration with m1,m2 at collision and m3 lies at the opposite end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' This will be called collision-antipodal singularity, [5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The rotation group SO(2) acts on the configuration space by (ϕ1,ϕ2,ϕ3) ↦ (ϕ1 + s,ϕ2 + s,ϕ3 + s),s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' This action keeps the potential function, and actually keeps the sys- tem by tangent lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The corresponding first integral is the angular momentum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=', J = ∑n i=1 mi ˙ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Thus, we have two first integrals (1) ∑miϕi = αt + α′, K − U = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that is ϕi(t) is a solution, so is ϕi(t) + at + a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We may assume, by changing the coordinates by linear functions of the variable t, that ∑miϕi = 0 ( mod 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We further assume that ∑mi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Jacobi Coordinates and the Singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Let 1 = m1 + m2 + m3, α1 = m1 m1 + m2 , α2 = m2 m1 + m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Introduce Jacobi variables as x1 = ϕ2 − ϕ1, x2 = ϕ3 − α1ϕ1 − α2ϕ2 and their velocities ui = ˙xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then the inverse is ϕ1 = −α2x1 − m3x2, ϕ2 = α1x1 − m3x2, ϕ3 = (m1 + m2)x2, ϕ2 − ϕ1 = x1, ϕ3 − ϕ1 = α2x1 + x2, ϕ2 − ϕ3 = α1x1 − x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then the kinetic energy is 2K = ∑mi ˙ϕ2 i = m1(−α2 ˙x1 − m3 ˙x2)2 + m2(α1 ˙x1 − m3 ˙x2)2 + m3(m1 + m2)2 ˙x2 2 = µ1u2 1 + µ2u2 2, where µ1 = m1m2 m1+m2 and µ2 = (m1 + m2)m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The potential is U(x1,x2) = m1m2 cotd12 + m1m3 cotd13 + m2m3 cotd23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Now we assume that the three bodies are ordered on the circle anti- clockwise as −π + 2kπ ≤ ϕ1 ≤ ϕ3 ≤ ϕ2 ≤ π + 2kπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then d12 = min{x1,2π − x1}, d13 = min{x2 + α2x1,2π − x2 − α2x1}, d23 = min{α1x1 − x2,2π − α1x1 + x2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For the Newtonian collinear three-body problem, a similar ordering is q1 < q3 < q2, where qi is the coordinates of mi, i = 1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For such an ordering, possible singularities are total collision, collision between 4 Shuqiang Zhu m2,m3, collision between m1,m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' With the Jacobi coordinates, the configuration space is a region between two half lines, [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For our three-body problem on S1, obviously, we have ϕ2−ϕ1 = x1 ∈ [0,2π], ϕ2−ϕ3 = α1x1−x2 ∈ [0,2π], ϕ3−ϕ1 = α2x1+x2 ∈ [0,2π], Then the configuration space is the triangular region bounded by α1x1 − x2 = 0, α2x1 + x2 = 0, x1 = 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The singularities are Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The configuration space, and some zero ve- locity curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The masses are 1 3, 1 3, 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Zero velocity curves in region I for h = −100 (orange), 0 (blue), 100 (green) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' ϕ2 − ϕ1,ϕ2 − ϕ3,ϕ3 − ϕ1 = 0,π,2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' That is, x1 = 0,π,2π, α1x1 − x2 = 0,π,2π, α2x1 + x2 = 0,π,2π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=', the vertices, boundary and the three mid-segments of the triangular region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' These singularities divide the configuration space into four sub- triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Let us denote the four sub-triangular regions as I, II, III, and IV, as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The three vertices correspond to the total collisions, the three sides correspond to the double collisions, the three mid-segments correspond to antipodal singularities, and the intersections of the three mid-segments correspond to three collision-antipodal singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In Figure 2, we sketch the real configurations corresponding to typical points of the configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' m3 m2 0 m1C2 = α1 III 22 I II 1 IV 2 = α2X1Schubart orbits on S1 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The configurations corresponding to vertices, sides and interior of the four sub-triangular regions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Hill’s region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Consider the motion on the energy surface H = h, then the projection of the energy surface to the configuration space is called the Hill’s region corresponding to energy h, H(h) = {x ∶ U(x) + h ≥ 0} Recall that the energy surface lies over its projection H(h) as a kind of degenerate circle bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The boundary ∂H(h) = {x ∶ U(x) = −h} is the zero-velocity curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We sketched some zero-velocity curves in region I in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The zero-velocity curves in region III and IV are similar to that in region I, but they are more complex in region II, as we shall see soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that U is undefined at the three intersections of the mid-segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Other than the three points, U(x) → ∞ as x approaches the three sides, and U(x) → −∞ as x approaches the three mid-segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Thus we have the following Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The antipodal singularities are repelling for the three- body problem on S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 6 Shuqiang Zhu 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Eulerian homothetic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For three-body problem, it is enough to consider the motion in region I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Let us first consider a simple case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Consider the isosceles problem on S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The masses are m1 = m2 = 1−m 2 ,m3 = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The initial data is ϕ2 = ϕ ∈ (0,π),ϕ1 = −ϕ,ϕ3 = 0, ˙ϕ1 = − ˙ϕ2, ˙ϕ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' By the symmetry, the configuration would stay isosceles with ϕ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' That is, the system has just one degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' More precisely, since ϕ2 − ϕ3 = ϕ3 − ϕ1, we see x2 = 0, and x1 = 2ϕ ∈ (0,2π), µ1 = m1/2 = 1−m 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Let p1 = µ1u1, then H = 1 2 p2 1 µ1 − U(x1), U = (1 − m)2 4 (cotd12 + 4m 1 − m cotd23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The motion depends on the function, {cotx1 + 4m 1−m cot x1 2 , 0 < x1 < π −cotx1 + 4m 1−m cot x1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' π < x1 < 2π, On [0,π], the function is decreasing from ∞ to −∞ for any value of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' While on [π,2π], the graph depends on the value of m, since − cot2ϕ + 4m 1 − m cotϕ = sin2 ϕ − cos2 ϕ 2sinϕcosϕ + 4m 1 − m cotϕ = sin2 ϕ 2sinϕcosϕ + ( 4m 1 − m − 1 2)cotϕ = 1 2 tan x1 2 + 9m − 1 2(1 − m) cot x1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Graphs of U on [π,2π] for m < 1/9 (orange), m = 1/9 (blue), m > 1/9 (green) The motions are obtained by the conservation of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Recall that x1 = 0,π,2π are singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' More precisely, x1 = 0 is the total collision, x1 = π is the antipodal singularity, and x1 = 2π is the collision- antipodal singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' On (0,π), all motions would eventually go to x1 = 0, or, triple collision must happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' At x1 = π, −U is ∞, so it 4 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='0 2 6 8Schubart orbits on S1 7 is repelling, or, antipodal singularity is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The qualitative feature of motions on (π,2π) depends on the sign of 9m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' If m < 1/9, all motions would eventually go to x1 = 2π, or, collision-antipodal singularity must happen, and at that mo- ment, the velocity is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' If m = 1/9, −U is decreasing on (π,2π) and −U(2π) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then for H = h ≥ 0, collision-antipodal singularity must happen, and at that moment, the velocity is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' If m > 1/9, there is one critical value h0 of −U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' If H = h > h0, the motions are periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' If H = h = h0, it is a stable equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For later use, let us refer to those configurations as Eulerian central configurations and the orbits as Eulerian homothetic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The above example was first considered by Florin et al, see [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Regularization of the collision singularities We now focus on motions in region I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Intuitively, all motions in this region seems to end in a total collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' However, we will construct symmetric periodic orbits in region I, called Schubart orbits, in next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In this section, we regularize the double and triple collision singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Assume that m3 = m ∈ (0,1), m1 = m2 = n, then n = (1 − m)/2, and α1 = α2 = 1/2, µ1 = 1 − m 4 = n/2, µ2 = (1 − m)m = 2nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In region I, the distances are d12 = x1,d13 = 1 2x1 + x2,d23 = 1 2x1 − x2, so L = K + U, K = 1 2(µ1u2 1 + µ2u2 2), U(x1,x2) = n2 cotx1 + mn(cot(1 2x1 + x2) + cot(1 2x1 − x2)) Recall that region I is a triangular region bounded by 1 2x1±x2 = 0 and x1 = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The three vertices and sides are singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We perform first Mcgehee’s coordinates then another change of variables to eliminate the singularities corresponding to the collisions, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Let x1 = 1 √µ1 r cosθ, x2 = 1 √µ2 r sinθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then 2K = ˙r2 + r2 ˙θ2, and U(x1,x2) = n2 cot(rA1 cosθ)+mn(cot[rA2 sin(θ + θ∗)] + cot[rA2 sin(θ∗ − θ)]) 8 Shuqiang Zhu where A1 = √ 2 n,A2 = √ m+1 2nm, and θ∗ = arctan√m, θ∗ < π 4 The configuration space has been blew up to θ ∈ (−θ∗,θ∗), 0 ≤ r < √µ1π cosθ = √nπ √ 2cosθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The corresponding second-order Euler-Lagrange equations are: ¨r = r ˙θ2 + Ur, ˙ r2 ˙θ = Uθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' (2) Next, one can blow-up the triple collision singularity at r = 0 by intro- ducing the the time rescaling ′ = r 3 2 ˙ and the variable ν = r′/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Setting τ = θ′ gives the following first-order system of differential equations: r′ = νr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' ν′ = 1 2ν2 + τ 2 + r2Ur θ′ = τ τ ′ = −1 2τν + rUθ (3) with energy equation: (4) ν2 + τ 2 2 − rU = rh Explicitly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' the functions are rU = n2r cot(rA1 cosθ) + mn(r cot[rA2 sin(θ + θ∗)] + r cot[rA2 sin(θ∗ − θ)]) rUθ = n2 r2A1 sinθ sin2(rA1 cosθ) + mn(− r2A2 cos(θ + θ∗) sin2[rA2 sin(θ + θ∗)] + r2A2 cos(θ∗ − θ) sin2[rA2 sin(θ∗ − θ)]) r2Ur = −n2 r2A1 cosθ sin2(rA1 cosθ) − mn( r2A2 sin(θ + θ∗) sin2[rA2 sin(θ + θ∗)] + r2A2 sin(θ∗ − θ) sin2[rA2 sin(θ∗ − θ)]) They are well-defined at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Hence {r = 0} is now an invariant set for the flow, called the triple collision manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The differential equations are still singular due to the double col- lisions at θ = ±θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The final coordinate change will eliminate these singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Define new variables u,γ such that θ = θ∗ sinu, γ = τ cos2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that −θ∗ ≤ θ ≤ θ∗ corresponds to −π 2 ≤ u ≤ π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' After calculating the differential equations for u,γ, introduce a further rescaling of time by Schubart orbits on S1 9 multiplying the vector field by θ∗ cos2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Retaining the prime to denote differentiation with respect to the new time variable one finds r′ = θ∗νr cos2 u ν′ = θ∗ cos2 u(1 2ν2 + τ 2 + r2Ur) = θ∗(1 2ν2 cos2 u + γ2 cos2 u + r2Ur cos2 u) = θ∗ cos2 u(−1 2ν2 + 2rh + 2rU + r2Ur) u′ = γ cosu γ′ = −1 2θ∗νγ cos2 u + θ∗rUθ cos4 u − 2τ 2 sinucos2 u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' = −1 2θ∗νγ cos2 u + θ∗rUθ cos4 u − 2sinu γ2 cos2 u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' (5) with energy equation: (6) ν2 cos2 u + γ2/cos2 u 2 − rU cos2 u = rhcos2 u The configuration space is now u ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 0 ≤ r < √nπ √ 2cos(θ∗ sinu) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that there is still one singularity on the boundary of the con- figuration space, r = √nπ √ 2 cos(θ∗),u = π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Recall that it is one of the inter- sections of the mid-segments and that the potential is undefined there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Denote it by Q, see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Except this singularity, The vector field is smooth and continuous on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The differential equations (5) represent the three-body problem on S1 with the prescribed energy for configurations being an obtuse triangle and with m3 in the middle, with triple collision blown-up and double collisions regularized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The shape variable u need not be restricted to the interval [−π 2, π 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' As u ranges over the real axis, the configuration oscillates between the double collisions at ±θ∗ and the mass m3 collides with m1 and m2 successively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The equations has some symmetries and a Schubart orbit can be obtained from an orbit from u = 0,ν = 0 to u = π 2,ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Suppose that we have a quarter of the trajectory Γ(t),t ∈ [0,t1] with Γ(0) = (r0,0,0,γ0), Γ(1) = (r1,0, π 2 ,γ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 10 Shuqiang Zhu we can construct the second quarter of the trajectory from Γ(1) to Γ(2) = (r0,0,π,γ0) by [t1,2t1] → (r(2t1 − t),−ν(2t1 − t),π − u(2t1 − t),γ(2t1 − t)), (it satisfies the boundary condition and is a solution since Uθ(π − u) = Uθ(u),Ur(π − u) = Ur(u)) and so the third and the fourth quarters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then, using the symmetry of the vector field, it follows that one can piece together the four of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Hence, the existence of the required orbit is reduced to find the first quarter, which will be proved by a topological shooting argument in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Schubart orbits by the shooting method Consider the system (5) on the manifold of fixed energy h = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We construct the first quarter of the claimed Schubart orbit in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We will follow the shooting method in [11], where Moeckel used it to show the existence of Schubart orbit for the Newtonian collinear three- body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The idea is to construct a continuous map in the phase space and then apply a shooting argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The construction of the continuous map in the phase space is based on the result of Wazewski [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Roughly speaking, a subset, called a Wazewski set, of the phase space is carefully chosen such that the amount of time required to leave depends continuously on initial con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then the exit point also depends continuously on initial con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' This idea were developed by Conley and Easton [2, 4, 9] to isolating blocks, topological index for invariant sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' There are several technical computations in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' To not interrupt the flow of the argument, we will just claim them in this section and give the detail in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The Wazewski set W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Consider a flow φt(x) on a topological space X and a subset W ⊂ X of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Let W0 be the set of points in W which eventually leave W in forward time, and let E the set of points which exit immediately: W0 = {x ∈ W ∶ ∃t > 0,φt(x) ∉ W}, E = {x ∈ W ∶ ∀t > 0,φ[0,t)(x) ⊈ W}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Clearly, E ⊂ W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Given x ∈ W0 define the exit time τ(x) = sup{t ≥ 0 ∶ φ[0,t)(x) ⊆ W}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that τ(x) = 0 if and only if x ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then τ is continuous if If x ∈ W0 and φ[0,t](x) ⊆ W, then φ[0,t](x) ⊆ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' E is a relatively closed subset of W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Schubart orbits on S1 11 In this case, the set W is called a Wazewski set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Since we are considering the motion on the energy manifold h = −1, the configuration is in the region, {(r,u) ∶ U ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that U = 1 defines an implicit function r(u) since Ur < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Define r∗ = r(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Let W = {(r,ν,u,γ) ∶ (6) holds, 0 ≤ r ≤ r∗, ν ≤ 0,0 ≤ u ≤ π 2 ,γ ≥ 0}, The choice of the set W is motivated by that of Moeckel in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The major difference is that the value of r is confined to [0,r∗] in our case, and it is not in that of Moeckel’s proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Thus, the configuration space is restricted to a rectangle, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' As it turns out, the restriction 0 ≤ r ≤ r∗ is essential for our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' On one hand, this restriction avoids the singularity Q so that the system leads to a well-defined flow on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' On the other hand, the restriction leads to the estimate(8), which is essential for our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that the restriction makes no harm since r is non-increasing in W, so the exit points must have r ≤ r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' To visualize W, we use coordinates (r,ν,u) on the energy manifold, and the value of γ is determined by energy equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The energy manifold projects to the three-dimensional region ν2 cos2 u 2 − rU cos2 u + r cos2 u ≤ 0, 0 ≤ r ≤ r∗, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The south part of the upper surface in the figure, where equality holds in (6) corresponds to γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The figure also shows a sketch of the shooting argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The invariant manifold H = {u = 0,γ = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' It is easy to verify that it is invariant under the flow since u′ = γ cos u = 0 and γ′ = θ∗rUθ∣θ=0 where we have used that fact that τ = 0,γ = 0 and the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Claim 1: U,r2Ur are even in θ, rUθ is even in r, and limr→0 −r2Ur = rU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In words, the last identity implies that the function U is homoge- neous of degree −1 on r where r is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The dynamics on H is thus r′ = θ∗νr, ν′ = θ∗(1 2ν2 + r2Ur) Since u = 0, it is just the homothetic orbits considered in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='3, but is regularized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' There is one equilibrium point, the intersection of the collision manifold and H, denote it by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The exact coordinates is P = (0,−ν0,0,0), 1 2ν2 0 = rU∣r=0,θ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 12 Shuqiang Zhu Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The Wazewski set W and a sketch of the existence proof 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The equilibrium point P is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' As in the Newto- nian collinear three-body problem, the equilibrium P is found to be hyperbolic [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We use the coordinates r,u,γ, and the variable ν is treated as a function of r,u,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The energy relation gives ∂ν ∂r = (rU)r − 1 ν = 1 ν0 , ∂ν ∂u = θ∗ rUθ cosu − 2γ2 sinu/cos5 u ν = 0, ∂ν ∂γ = −γ ν cos4 u = 0, at the point P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then one finds that the linearized differential equations at P have matrix ⎡⎢⎢⎢⎢⎢⎣ θ∗ν cos2 u + θ∗r cos2 u ∂ν ∂r θ∗r cos2 u ∂ν ∂u θ∗r cos2 u ∂ν ∂γ 0 γ( 1 cos u)′ 1 cos u −θ∗ 2 γ cos2 u ∂ν ∂r + θ∗(rUθ)r cos4 u ☀ −2 sin u cos2 u ⎤⎥⎥⎥⎥⎥⎦ = ⎡⎢⎢⎢⎢⎢⎣ −θ∗ν0 0 0 0 0 1 0 θ2 ∗rUθθ 1 2θ∗ν0 ⎤⎥⎥⎥⎥⎥⎦ where ☀ = −θ∗ 2 γ(ν cos2 u)u + rUθθθ2 ∗ cos5 u + rUθθ∗(cos4 u)u − 2γ2( sinu cos2 u)u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' B s A F(s) T Wp) u P 1Schubart orbits on S1 13 and we use the fact in Claim 1 and the following Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Claim 2: At the point r = 0,u = 0, we have rUθθ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Thus, the equilibrium P is hyperbolic, with eigenvalues λ1 = −θ∗ν0 < 0 and λ2 < 0,λ3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then it has two-dimensional stable manifold and one-dimensional unstable manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The eigenvectors are (1,0,0), (0,1,λ2), and (0,1,λ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The first stable eigenvector is tangent to the homothetic orbit H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that the other stable eigenvector(0,1,λ2) points out of W since γ ≥ 0 in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' It follows that H ⋂W = W s(P)⋂W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The unstable manifold of P is on the collision manifold, with one branch in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The branch of W u(P) in W exits W at a point of the form (0,ν, π 2,γ) with ν < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The following fact will be used in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Claim 3: Restricted on r = 0, the maximum of 2rU cos2 u is at u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Consider the system for u,ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' By the energy relation, the equa- tions read ν′ = θ∗ cos2 u(−1 2ν2 + 2rh + 2rU + r2Ur) = θ∗ cos2 u(rU − 1 2ν2) u′ = γ cosu = cosu √ 2rU − ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then dν du = θ∗ 2 √ 2rU cos2 u − ν2 cos2 u ≤ θ∗ 2 √ 2rU cos2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' So dν du ≤ θ∗ 2 ν0 which implies hat the increment in ν for 0 ≤ u ≤ π 2 satisfies: ∆ν ≤ π 2 θ∗ 2 ν0 ≤ ν0 since θ∗ ≤ π 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Since the branch of W u(P) begins near P, and P has coordinates u = 0 and ν = −ν0, then it arrives at u = π 2 without crossing the line ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' W is a Wazewski set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In this subsection, we identify the subsets W0,E and show Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' W is a Wazewski set for the flow on the energy manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The first property obviously holds since the set W is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For the second property, we first identify the subsets W0,E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 14 Shuqiang Zhu Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' W0 = {x ∈ W ∶ ∃t > 0,φt(x) ∉ W} = W ∖ H The following facts will be used in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Claim 4: 2rU cos2 u∣u= π 2 has a positive lower bound c2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Claim 5: The function θ∗rUθ cos4 u sin u has a positive lower bound c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Let x0 = (r0,ν0,u0,γ0) ∈ W, It is easy to that the solution begin from x0 exist as long as it remains in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Now suppose x0 ∈ W ∖ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Our goal is to show that φt(x0) eventually leaves W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' If u0 = 0 then u′(0) = γ0 > 0 since x0 ∉ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' It follows that for every t0 > 0,u(t0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Thus it is enough to assume u0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Let u0 be a positive constant and Wu0 = {x ∈ W ∶ u ≥ u0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Since u(t) is non-decreasing in W, Wu0 is positively invariant relative to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We show below that there are two constants c0 > 0 and c1 > 0 such that for every x ∈ Wu0 either γ cosu ≥ c0, or ( γ cosu)′ ≥ c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then it is easy to see that φt(x0) must eventually leave W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that Wu0 = W+ u0 ⋃W− u0, where W + u0 = {x ∈ Wu0, γ cosu ≥ c0}, W− u0 = {x ∈ Wu0,0 ≤ γ cosu < c0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Since ( γ cos u)′ ≥ c1 > 0 in W− u0 it implies that an orbit segment can stay in W− u0 for time at most c0/c1, and then would enter W+ u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that W+ u0 is positively invariant relative to Wu0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Finally, an orbit can remain in W+ u0 for time not longer than π 2c0 since u′ = γ cos u ≥ c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Hence, every orbit starting in Wu0 must leaves W eventually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We now construct c0 > 0,c1 > 0 such that either γ cos u ≥ c0 or ( γ cos u)′ ≥ c1 for all x ∈ Wu0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For u = π 2, the equation (6) implies γ cos u = √ 2rU cos2 u∣u= π 2 ≥ c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We can choose c0 to be less than c2 then γ cos u ≥ c0 holds for u = π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For u0 ≤ u < π 2, we have ( γ cosu)′ = γ′ cosu + tanu( γ cosu)2 = −1 2θ∗νγ cosu + θ∗rUθ cos3 u − 2sinu γ2 cos3 u + tanu( γ cosu)2 ≥ tanu(θ∗rUθ cos4 u sinu − ( γ cosu)2) ≥ tanu0 (c3 − ( γ cosu)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Schubart orbits on S1 15 Then we take c0 such that c0 ≤ c2,c2 0 ≤ c3 2 , and take c1 = c3 tan u0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then on u = π 2, we have γ cos u ≥ c2 ≥ c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For u0 ≤ u < π 2, if γ cos u ≤ c0, then ( γ cos u)′ ≥ tanu0 (c3 − ( γ cos u)2) ≥ c1, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' □ It remains to identify the immediate exit set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' It is useful to dis- tinguish two subsets of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Let x = (r,ν,u,γ) and let B1 = {x ∈ W ∶ u = π 2 }, B2 = {x ∈ W ∶ ν = 0,0 ≤ u < π 2 ,2rU + r2Ur − 2r ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Obviously, the two subsets B1 and B2 are relatively closed in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Hence Theorem 1 is proved once we show Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The immediate exit set of W is E = B1 ⋃B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The configuration space and the rectangle [0, π 2] × [0,r∗] The following fact will be used in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Claim 6: Let F(r,u) = 2rU + r2Ur − 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Fu > 0 for 0 ≤ r ≤ r∗,0 < u ≤ π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' At u = 0,F = 0, we have Fu = 0,Fuu > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' T U= B 0=1 A n 0 216 Shuqiang Zhu Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' As claimed, √ 2rU cos2 u∣u= π 2 has a positive lower bound, so B1 ⊂ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Consider a point x ∈ B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that h = −1,ν = 0, then ν′ = θ∗( γ2 cos2 u + r2Ur cos2 u) = θ∗ cos2 u(2rU + r2Ur − 2r) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Let F(r,u) = 2rU+r2Ur−2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Since Fu > 0 in the rectangle (0, π 2]×[0,r∗], the set F = 0 in this rectangle is a curve bounded by the two points A,B (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The curve divides the rectangle into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The bottom r = 0 is in the set F > 0 since on which F = rU, while the vertex u = 0,r = r∗ is in the set F < 0 since U = 1 and that r2Ur < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' If F > 0, then ν′ > 0 and x is an immediate exit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' If F = 0 and u ≠ 0, one has ν = ν′ = 0 and one finds that the second derivative reduces to ν′′ = θ∗(−sin2uF + cos2 uFu)u′ + cos2 uFrr′ = cosuFuγ > 0, and x is an immediate exit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Finally, if u = ν = 0 and F = 0, one has ν = ν′ = ν′′ = 0, The third derivative at the point A is found to be ν′′′ = θ∗γ2Fuu > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Again, x is an immediate exit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' It remains to check that there are no other immediate exit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Suppose that x0 ∈ W is an immediate exit point and it is not in B1 ⋃B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Following the argument in [11], it is enough the check the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' First, it may happen that r0 = 0 but r(t) < 0 for small positive times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' This is impossible because the manifold {r = 0} is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Secondly, it may happen that u0 = 0 but u(t) < 0 for small positive times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' It requires u′(0) = γ0 ≤ 0 and since x0 ∈ W this means γ0 = 0, so x0 ∈ H and points of H are certainly not leaving W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Thirdly, it may happen that ν0 = 0 but ν(t) increases for small positive times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' This forces ν′(0) ≥ 0 and then x0 ∈ B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Fourthly, it may happen that r0 = r∗ but r(t) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' This forces r′(0) = 0 and then ν0 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=', the coordinates of the point is r = r∗,0 ≤ u < α, where (r∗,α) is the coordinates of the point B, see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' So ν′ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then one finds r′′ = θ∗(ν(r cos2 u)′ + r cosu ν′) = θ∗r∗ν′ cos2 u < 0, This mode of existing is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' At last, it may happen that γ0 = 0 but γ(t) decreases for small positive times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' If u0 = 0, then x0 ∈ H, and points of H are certainly not leaving W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' If u0 = π 2, then x0 ∈ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' One may assume 0 < u0 < π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In this case, it follows from the proof of Lemma 2 that there are positive Schubart orbits on S1 17 constants c0,c1 such that ( γ cos u)′ ≥ c1 > 0 whenever γ cos u < c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' So this mode of exiting is also impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The shooting argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Finally, we can complete the con- struction of the symmetric periodic orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Recall that it suffices to con- struct the first quarter, which is required to be an orbit from u = 0,ν = 0 to u = π 2,ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Since W is a Wazewski set, the time required to reach E depends continuously on initial conditions and so there is a continuous flow- defined map from W0 to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The map is also continuous if we restrict the domain to S = {(r,ν,u,γ) ∈ W0,u = ν = 0,0 ≤ r < r∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' That is, the flow-defined map F ∶ S → E is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Let T = {(r,ν,u,γ) ∈ W,u = π 2 ,ν = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that T ⊂ E and that S and T are two of the edges in the boundary of the three-dimensional Wazewski set W (shown as bold vertical lines in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then the construction of the first quarter of the orbit reduces to show that F(S)⋂T ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' First, note that part of S near r = 0 is contained in B2 ⊂ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' These points exit W immediately, so the map F is the identity there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Sec- ondly, points of S with r close to r∗ will enter the interior of W and exit elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' By continuous dependence of the initial conditions, these points will follow the homothetic orbit H to a neighborhood of the equi- librium point P = (0,−ν0,0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then the lambda lemma [16] implies that they will follow a branch of the unstable manifold W u(P), which is one-dimensional and is contained entirely in the invariant manifold r = 0, as shown in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Furthermore, by Lemma 1, one of the two branches lies in W0 and it goes to some point on r = 0,ν < 0,u = π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then the lambda lemma implies that the image of points near the upper endpoint of S under the continuous mapping F are on B1 ∖ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We can now complete the shooting argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Recall that there is continuous map F ∶ S → E, E = B1 ⋃B2 and that B1and B2 are two- dimensional continuum meeting along the edge T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' As we have shown, the image of points near r = 0 under F are in B2 ∖ T , while the image of points near r = r∗ under F are in B1 ∖ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' It follows that there must exist at least one intersection point U ∈ F(S)⋂T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' This shows that F(S)⋂T ≠ ∅ and completes the existence proof for the symmetric periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 18 Shuqiang Zhu Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The orbit constructed lies in the energy manifold h = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' By restricting the configuration to the rectangle 0 ≤ r ≤ r∗,0 ≤ u ≤ π 2, where u = 0,r = r∗ is the intersection of U = 1 and u = 0, the six Claims hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Consider an energy manifold h < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Since U is decreasing on u = 0, the intersection of U = −h and u = 0 is lower than r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then the six Claims made in this section still hold and all arguments can be applied as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Thus, we have the following Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Given three positive masses m1 = m2 and m3 and an energy h ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then there exists a symmetric periodic solution of the collinear three-body problem on S1 with energy h and regularized double collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The orbit has the following features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The configuration lies in region I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In the first quarter of the orbit, the masses move from the Euler- ian central configuration with m3 in the middle of m1,m2 to a double collision between m2 and m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' At the moment of the dou- ble collision the velocity of m1 is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The second quarter of the orbit is the time-reverse of the first, and the second half is the reflection of the first half with the roles of m1 and m2 reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Appendix: Proofs of the six Claims Recall that rU = n2r cot(rA1 cosθ) + mn(r cot[rA2 sin(θ + θ∗)] + r cot[rA2 sin(θ∗ − θ)]) rUθ = n2 r2A1 sinθ sin2(rA1 cosθ) + mn(− r2A2 cos(θ + θ∗) sin2[rA2 sin(θ + θ∗)] + r2A2 cos(θ∗ − θ) sin2[rA2 sin(θ∗ − θ)]) r2Ur = −n2 r2A1 cosθ sin2(rA1 cosθ) − mn( r2A2 sin(θ + θ∗) sin2[rA2 sin(θ + θ∗)] + r2A2 sin(θ∗ − θ) sin2[rA2 sin(θ∗ − θ)]) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Claim 1: U,r2Ur are even in θ, rUθ is even in r, and limr→0 −r2Ur = rU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' In words, the last identity implies that the function U is almost homogeneous of degree −1 on r where r is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' It is easy to see by the explicit form of the functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Claim 2: At the point r = 0,u = 0, we have rUθθ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We show that rUθ = n2 r2A1 sinθ sin2(rA1 cosθ)+mn(− r2A2 cos(θ + θ∗) sin2[rA2 sin(θ + θ∗)] + r2A2 cos(θ∗ − θ) sin2[rA2 sin(θ∗ − θ)]) is strictly increasing on θ at r = 0,θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Schubart orbits on S1 19 The first term is strictly increasing in θ, at r = θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Direct compu- tation gives ( r2A1 sinθ sin2(rA1 cosθ))θ = r2A1 sin2(rA1 cosθ)cosθ − rA1 sin2 θ sin(2rA1 cosθ) sin4(rA1 cosθ) → 1 A1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The second term, denoted by g(r,θ), is an increasing function on θ in a neighborhood of r = θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Indeed, g(r,0) = 0, and g(r,θ) > 0 if 0 < θ ≤ θ∗ and r is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that θ∗ ≤ π 4, then cos(θ∗ − θ) > cos(θ∗ + θ), rA2 sin(θ∗ − θ) < rA2 sin(θ∗ + θ) < π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Thus, g(r,θ) = mnr2A2 ( cos(θ∗ − θ) sin2[rA2 sin(θ∗ − θ)] − cos(θ + θ∗) sin2[rA2 sin(θ + θ∗)]) > 0, and the derivative gθ(0,0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Claim 3: Restricted on r = 0, the maximum of 2rU cos2 u is at u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Recall that rU = n2r cot(rA1 cosθ)+mn(r cot[rA2 sin(θ + θ∗)] + r cot[rA2 sin(θ∗ − θ)]) Let r → 0, we have rU cos2 u = n2 cos2 u A1 cos(θ∗ sinu) + mncos2 u( 1 A2 sin(θ∗ + θ) + 1 A2 sin(θ∗ − θ)) = n2 cos2 u A1 cos(θ∗ sinu) + 2mnsinθ∗ A2 cos2 ucosθ cos2 θ − cos2 θ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The first term is a decreasing function of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Since θ∗ ≤ π 4 < 1, we have ( cos2 u cos(θ∗ sinu))′ = cosu cos2(θ∗ sinu)[θ∗ cos2 usin(θ∗ sinu) − 2sinucos(θ∗ sinu)] ≤ cosu cos2(θ∗ sinu)2[cosusin(θ∗ sinu) − sinucos(θ∗ sinu)] = cosu cos2(θ∗ sinu)2[sin(θ∗ sinu − u)] ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' It remains to show (7) 1 1 − cos2 θ∗ ≥ cos2 ucosθ cos2 θ − cos2 θ∗ , u ∈ [0, π 2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 20 Shuqiang Zhu it is equivalent to J = cos2 θ − cos2 θ∗ − (1 − cos2 θ∗)cos2 ucosθ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' View J as a function of the two variables (θ∗,θ), on the triangular region 0 < θ∗ ≤ π 4,0 ≤ θ ≤ θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that J (θ∗,θ∗) = 0 and ∂J ∂θ∗ = 2sinθ∗ cosθ∗(1 − cos2 ucosθ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We conclude that the function J is non-negative on the triangular region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Claim 4: 2rU cos2 u∣u= π 2 has a positive lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Recall that rU = n2r cotd12 + mn(cotd13 + cotd23), and the fact that we are not at the singularity Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Hence, when u = π 2, we have d23 = 0, and the two distance d12,d13 are different from 0,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Hence rU cos2 u = mncotd23 cos2 u = mnr cot[rA2 sin(θ∗ − θ)]cos2 u Since cos2 u = sin2(π 2 − u), and θ∗ − θ = 2θ∗ sin2(π 2 − u), so we obtain rU cos2 u = mn 2θ∗A2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Claim 5: The function rUθ cos4 u sin u has a positive lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Recall that rUθ = n2 r2A1 sinθ sin2(rA1 cosθ)+mn(− r2A2 cos(θ + θ∗) sin2[rA2 sin(θ + θ∗)] + r2A2 cos(θ∗ − θ) sin2[rA2 sin(θ∗ − θ)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We first claim that the function rUθ is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The first term is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For the second term, which has been denoted by g(r,θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We have showed that g(r,0) = 0, and g(r,θ) > 0 if 0 < θ ≤ θ∗ and r is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Now we show that 0 ≤ r ≤ r∗, 0 < θ ≤ θ∗,⇒ g(r,θ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For this, it suffices to show that r∗A2 sin(2θ∗) ≤ π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Recall that at θ = 0,r = r∗, we have U = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that rA2 sinθ∗ = r √ m+1 2nm √ m m+1 = rA1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then n2 cot(r∗A1) + 2mncot(r∗A1/2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Let a = cot(r∗A1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Note that a > 0 since rA2 sinθ∗ = d23 < π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then 2mna + n2a2 − 1 2a = 1,⇒ (4mn + n2)a2 − 2a − n2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Schubart orbits on S1 21 So a = cot(r∗A1/2) = 4 −7m2 + 6m + 1 + √ −7m4 + 20m3 − 18m2 + 4m + 17 −7m2 + 6m + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Let g(m) = −7m4+20m3−18m2+4m+17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' We have g′ = −4(m−1)2(7m− 1), so g(m) ≥ min{g(0),g(1)} = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Since −7m2+6m+1 ≤ 16 7 , we obtain the desired estimate cot(r∗A1/2) ≥ 7 2,⇒ r∗A1/2 ≤ π 10 (8) r∗A2 sin(2θ∗) = r∗A1 cosθ∗ = r∗A1 √ m + 1 < π 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Now we show that θ∗rUθ cos4 u sin u ,0 < u0 < u < π 2,0 ≤ r ≤ r∗, has a positive lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Obviously, at u = π 2, the second term equals to lim u→ π 2 r2A2 cos(θ∗ − θ) sin2[rA2 sin(θ∗ − θ)] cos4 u sinu = lim u→ π 2 mnsin4(π/2 − u) A2 sin2[2θ∗ sin2(π/2−u 2 )] = 4mn A2θ2∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then there is some u1 < π 2 such that θ∗rUθ cos4 u sinu ≥ 2mn A2θ2∗ , u1 ≤ u ≤ π 2 ,0 ≤ r ≤ r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For the first term, let θ0 = u0 sinu, then r2A1 sinθ sin2(rA1 cosθ) cos4 u sinu ≥ r2A2 1 sinθ cos4 u A1 sin2(rA1) ≥ sinθ0 cos4 u1 A1 , u0 ≤ u ≤ u1,0 ≤ r ≤ r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Thus, we conclude that rUθ cos4 u sin u ,0 < u0 < u < π 2,0 ≤ r ≤ r∗, has a positive lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Claim 6: Let F(r,u) = 2rU + r2Ur − 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Fu > 0 for 0 ≤ r ≤ r∗,0 < u ≤ π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' At u = 0,F = 0, we have Fu = 0,Fuu > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Recall that 2rU + r2Ur = n2r[2cot(rA1 cosθ) − rA1 cosθ sin2(rA1 cosθ)] + mnr{2cot[rA2 sin(θ∗ − θ)] − rA2 sin(θ∗ − θ) sin2[rA2 sin(θ∗ − θ)] + 2cot[rA2 sin(θ + θ∗)] − rA2 sin(θ + θ∗) sin2[rA2 sin(θ + θ∗)]} Introduce new variables ρ = rA1 cosθ, ξ = rA2 sin(θ∗ − θ), η = rA2 sin(θ∗ + θ), and define f(x) = 2cotx − x sin2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Then 2rU + r2Ur = n2rf(ρ) + mnr[f(ξ) + f(η)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' 22 Shuqiang Zhu Let us first study the function f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' On [0, π 5] f ′(x) = −3 − 2xcotx sin2 x < 0, f ′′(x) = − 2 sin4 x(2x − 2sin2x + xcos2x) ≥ 2 sin4 x(2sin2x − 3x) ≥ 0, since k(x) = 2sin2x−3x is a concave function on [0, π 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Thus its value on [0, π 5] is at least min{k(0),k(π 5)} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' ρ′ = −rA1 sinθ, ξ′ = −rA2 cos(θ∗ − θ), η′ = rA2 cos(θ∗ + θ), ρ′′ = −ρ, ξ′′ = −ξ, η′′ = −η, ρ ≤ π 5 , ξ ≤ η ≤ r∗A2 sin2θ∗ < π 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For the first derivative, one finds Fu = Fθθ∗ cosu, then it suffices to show that Fθ = (2rU + r2Ur)θ > 0 for 0 ≤ r ≤ r∗,0 < θ < θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Fθ = n2rf ′(ρ)ρ′ + mnr(f ′(ξ)ξ′ + f ′(η)η′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The first term is positive if θ ∈ (0, π 5], and it is zero if θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The second term is zero if θ = 0, and it is positive if θ ∈ (0, π 5) since both −f ′(x) and cos(x) are decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Hence, we have proved that Fu > 0 for 0 ≤ r ≤ r∗,0 < u < π 2 and Fu = 0 for 0 ≤ r ≤ r∗,u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' For the second derivative at the point A, one finds Fuu = Fθθ(θ∗ cosu)2 − Fθθ∗ sinu = Fθθθ2 ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' and Fθθ = n2r[f ′′(ρ)(ρ′)2 − f ′(ρ)ρ] + mnr[f ′′(ξ)(ξ′)2 − f ′(ξ)ξ + f ′′(η)(η′)2 − f ′(η)η] = n2r[−f ′(ρ)ρ] + mnr[2f ′′(ξ)(ξ′)2 − 2f ′(ξ)ξ] > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Hence, we have proved that Fuu > 0 at the point A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The author would like to thank Cristina Stoica and Jean-Marie Becker for enlightening discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' References [1] AV Borisov, LC Garc´ıa-Naranjo, IS Mamaev, and James Mon- taldi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Reduction and relative equilibria for the two-body problem on spaces of constant curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Celestial Mechanics and Dynam- ical Astronomy, 130(6):1–36, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' [2] C.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=', 16:620–625, 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Schubart orbits on S1 23 [4] Charles Conley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Isolated invariant sets and the Morse index, vol- ume 38 of CBMS Regional Conference Series in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' American Mathematical Society, Providence, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=', 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' [5] Florin Diacu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' On the singularities of the curved n-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Transactions of the American Mathematical Society, 363(4):2249– 2264, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' [6] Florin Diacu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Relative equilibria in the 3-dimensional curved n- body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=', 228(1071):vi+80, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' [7] Florin Diacu, Ernesto P´erez-Chavela, and Manuele Santoprete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The n-body problem in spaces of constant curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' part ii: Sin- gularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Journal of nonlinear science, 22(2):267–275, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Princeton University Press, Princeton, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' University of Tokyo Press, Tokyo, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' With special emphasis on celestial mechanics, Hermann Weyl Lec- tures, the Institute for Advanced Study, Princeton, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Schubart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Numerische Aufsuchung periodischer L¨osungen im Dreik¨orperproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Astronom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Nachr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=', 283:17–22, 1956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' [14] Mitsuru Shibayama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Minimizing periodic orbits with regulariz- able collisions in the n-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=', 199(3):821–841, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' [15] Tadeusz Wa˙zewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Sur un principe topologique de l’examen de l’allure asymptotique des int´egrales des ´equations diff´erentielles ordinaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Polon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=', 20:279–313 (1948), 1947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' [16] Stephen Wiggins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Introduction to applied nonlinear dynamical systems and chaos, volume 2 of Texts in Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Springer-Verlag, New York, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' [17] Zhihong Xia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' The existence of noncollision singularities in Newto- nian systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} +page_content=' (2), 135(3):411–468, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9AyT4oBgHgl3EQfX_dB/content/2301.00193v1.pdf'} diff --git a/idAyT4oBgHgl3EQfxvnE/content/tmp_files/2301.00673v1.pdf.txt b/idAyT4oBgHgl3EQfxvnE/content/tmp_files/2301.00673v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..373506dc5c218243ee95e2eab5c1ad5449efb90b --- /dev/null +++ b/idAyT4oBgHgl3EQfxvnE/content/tmp_files/2301.00673v1.pdf.txt @@ -0,0 +1,1321 @@ +arXiv:2301.00673v1 [q-bio.PE] 29 Dec 2022 +Predator-Prey Linear Coupling with Hybrid Species +Jean-Luc Boulnois +Babson College, Babson Park, Wellesley, Massachusetts 02457, jlboulnois@msn.com +Abstract +The classical two-species non-linear Predator-Prey system, often used in pop- +ulation dynamics modeling, is expressed in terms of a single positive coupling +parameter λ. Based on standard logarithmic transformations, we derive a novel +λ-invariant Hamiltonian resulting in two coupled first-order ODEs for “hybrid- +species”, albeit with one being linear; we thus derive a new exact, closed-form, +single quadrature solution valid for any value of λ and the system’s energy. In +the particular case λ = 1 the ODE system completely uncouples and a new, +exact, energy-only dependent simple quadrature solution is derived. In the case +λ ̸= 1 an accurate practical approximation uncoupling the non-linear system is +proposed and solutions are provided in terms of explicit quadratures together +with high energy asymptotic solutions. A novel, exact, closed-form expression +of the system’s oscillation period valid for any value of λ and orbital energy is +also derived; two fundamental properties of the period are established; for λ = 1 +the period is expressed in terms of a universal energy function and shown to be +the shortest. +Keywords: +Single coupling parameter, Uncoupling, Quadrature solutions, +Hamiltonian, Asymptotic solutions, Period +Mathematics Subject Classification: 34A34, 34E05, 41A55, 92D25 +1. Introduction +The historic Predator-Prey problem, also known as the Lotka-Volterra (“LV”) +system of two coupled first-order nonlinear differential equations, has first been +investigated in ecological and chemical systems [23],[12]. This classical problem +models the competition of two isolated coexisting species: a ‘prey population’ +evolves while feeding from an infinitely large resource supply, whereas ‘preda- +tors’ interact by exclusively feeding on preys, either through direct predation +or as parasites. This idealized two-species model has further been generalized +to interactions between multiple coexisting species in biological mathematics +[4], ecology [1], virus propagation [2], and also in molecular vibration-vibration +energy transfers [21]. +Preprint submitted to Elsevier +January 3, 2023 + +Let u′ ⩾ 0 and v′ ⩾ 0 be the respective instantaneous populations of preys and +predators assumed to be continuous functions of time t′ : the net growth rates +of each species is modeled as a system of two coupled first-order autonomous +nonlinear ordinary differential equations (ODEs) according to +du′ +u′dt′ = α − βv′ +for preys +(1a) +dv′ +v′dt′ = γu′ − δ +for predators +(1b) +In the classical LV model, α, β, γ, δ are assumed to be time-independent, posi- +tive, and constant: the rates α and δ represent self-interaction while the rates +associated with β and δ characterize inter-species interaction. In absence of +predators, the natural exponential growth rate of the prey population is α ; +when interacting with predators this population decreases at a rate modeled as +−βv′. Similarly, when preys are scarce, the predator population decays at a +rate −δ , and when feeding on preys its growth rate is modeled as γu′. +Numerous solutions of the non-linear system (1) using a variety of techniques +have been proposed including trigonometric series [6], Lambert W-functions [18], +[19], mathematical transformations [5], Taylor series expansions [13], perturba- +tion techniques [16], [14], and numeric-analytic techniques [3]. Also, an exact +solution has been derived by Varma [22] in the special case when the rates α +and δ are identical in magnitude, but with α = −δ, a condition which pre- +cludes population oscillation. The basic system (1) is non-trivial and analytical +closed-form solutions are unknown. +2. Normalized Equations and Single Coupling Parameter +Without any loss of generality, the system (1) can further be simplified by simul- +taneously rescaling the predator and prey populations according to v = (β/α) v′ +and u = (γ/δ) u′ respectively, while also rescaling time through a “stretched” +time without unit t = +√ +αδt′. Upon introducing the positive coupling parameter +λ, ratio of the respective growth and decay rates of each species taken separately, +defined as +λ = +�α +δ +(2) +a normalized form of the LV system is obtained as a set of two coupled nonlinear +first-order ODEs exclusively depending on this single coupling ratio λ according +to +˙u = λu (1 − v) +for preys +(3a) +˙v = 1 +λv (u − 1) +for predators +(3b) +Here the “dot” on ˙u and ˙v indicates a derivative with respect to the time t: in the +2 + +sudden absence of coupling between species (β = γ = 0), the prey population +would grow at an exponential rate λ while predators would similarly decay at an +inverse rate −1/λ from their respective positive initial values. Remarkably, the +normalized ODE system (3) is invariant in the transformation u → v together +with λ → −1/λ: this fundamental property, subsequently referred to as “λ- +invariance”, is extensively used throughout to considerably simplify the LV +problem analysis. +Since the original publications [23], [12], the system (3) has been known to +possess a dynamical invariant or “constant of motion K” expressed here in λ- +invariant form +1 +λu + λv − ln +� +u +1 +λ vλ� += K +(4) +In the following sections, through a particular Hamiltonian transformation com- +bined with a suitable linear change of variables we introduce a novel λ-invariant +Hamiltonian based on new “hybrid-species” that reduces the system (3) to a new +set of two coupled first-order ODEs with one being linear. Upon exploiting this +linearity, a new, exact analytical solution is derived for one hybrid-species in +terms of a simple quadrature: we then proceed with an original method to un- +couple the system and derive complete, closed-form quadrature solutions of the +LV problem. The population oscillation period is further derived in terms of a +unique energy function and two fundamental properties are established. +3. Solutions with Hybrid Predator-Prey Species +The logarithmic functional transformation originally introduced by Kerner [10] +reduces the normalized LV system (3) to a Hamiltonian form: the coupling be- +tween the respective species is modified through a change of variables according +to +y = ln(u) and x = ln(v) with y ∈ (−∞, +∞), x ∈ (−∞, +∞) +(5) +The LV system (3) for the respective “logarithmic” prey-like and predator-like +species y(t) and x(t) becomes +˙y = λ(1 − ex) +˙x = 1 +λ(ey − 1) +(6) +Similarly to Eq. (4) this λ- invariant system (6) admits a primary conservation +integral H expressed as the linear combination of two positive convex functions +H(x, y) = λ(ex − x − 1) + 1 +λ(ey − y − 1) +(7) +As already established [15], [11], H(x, y) is the Hamiltonian of the conservative +LV system since Eqs. (6) satisfy Hamilton’s equations with x as the coordinate +conjugate to the canonical momentum y. Equation (7) expresses the conserva- +tive coupling between species x(t) and y(t): it is further rendered λ- invariant +by introducing a scaled Hamiltonian h(x, y) with total constant positive energy +3 + +simply labeled h, according to +H (x, y) = +� +λ + 1 +λ +� +h (x, y) +(8) +We introduce a λ- invariant linear first-order ODE between the species x(t) and +y(t) by further combining the system (6) with (7) and (8) +˙x − ˙y − +� +λx + y +λ +� += +� +λ + 1 +λ +� +h +(9) +Equation (9) suggests introducing a λ-invariant linear transformation of the +set {x(t), y(t)} to a new set {ξ(t), η(t)} representing the symbiotic coupling +between ”hybrid predator-prey species” +ξ = λx + 1 +λy +λ + 1 +λ +(10a) +η = x − y +λ + 1 +λ +(10b) +The original Hamiltonian (7) together with (8) and the linear transformation +(10) then becomes +h(η, ξ) = λe +η +λ + 1 +λe−λη +λ + 1 +λ +eξ − ξ − 1 +(11) +Here h (η, ξ) is a new Hamiltonian for the coordinate η and conjugate momentum +ξ. Notice that for small amplitudes, h (η, ξ) is the Hamiltonian of a harmonic +oscillator. Upon further introducing the following λ-invariant G-function +Gλ(η) = λe +η +λ + 1 +λe−λη +λ + 1 +λ +with Gλ(η) = G1/λ(−η) +(λ-invariance) +(12) +the conservation relationship (11) between the conjugate functions η(t) and ξ(t) +is recast into a compact form which provides a natural separation of variables +Gλ(η) = (h + 1 + ξ)e−ξ +(13) +In the following we define the function U(ξ) that appears throughout as +U(ξ) = (h + 1 + ξ)e−ξ +(14) +Even though still nonlinear, the fundamental conservation relationship (13) par- +tially uncouples the ξ(t)-function from the η(t)-function, resulting in three es- +sential G-function properties: +1. the system’s energy h ⩾ 0 is explicitly associated with the function U(ξ) +only; +4 + +2. the positive function Gλ(η) is a generalized hyperbolic cosine function +that reaches its minimum Gλ = 1 at η = 0 for any value of λ : hence +its inverse function G−1 +λ +exists, and, for any value of λ , Eq. (13) admits +two respective positive and negative roots η±(ξ, λ) functions of ξ only +satisfying +η±(ξ, λ) = G−1 +λ +� +U(ξ) +� +(15) +3. since the η-function is associated with the coupling ratio λ only, λ-invariance +of the G-function (12) implies that, for a given λ, any positive solution +η+(ξ, λ) is directly derived from the negative solution associated with the +ratio 1/λ, and reciprocally +η±(ξ, λ) = −η∓(ξ, 1/λ) +(16) +From Eq. (14) the hybrid-species population ξ(t) thus oscillates between the +λ-independent respective negative and positive roots ξ−(h) and ξ+(h), solutions +of the equation U(ξ) = 1, solely dependent on the system’s energy h as displayed +in Table 1 for several increasing values of h +eξ − ξ − 1 = h +with h ⩾ 0 +(17) +h +0.3 +0.5 +1 +2 +3 +5 +7 +10 +ξ−(h) +-0.889 +-1.198 +-1.841 +-2.948 +-3.981 +-5.998 +-8.000 +-11.00 +ξ+(h) +0.686 +0.858 +1.146 +1.505 +1.749 +2.091 +2.336 +2.611 +Table 1: Roots of eξ − ξ − 1 = h as a function of the energy h from Eq. (17) +In the ξ − η plane, Eq. (13) represents a closed-orbit mapping around the fixed +point (0, 0). On the η = 0 horizontal axis this orbit is bounded by the limits +ξ−(h) and ξ+(h), and since U(ξ) admits a maximum eh located at ξ = −h , it +is also bounded by the two respective positive and negative roots solutions of +the equation η±(−h, λ) = G−1 +λ (eh). For any given energy h this orbit consists +of two respective branches η+(ξ, λ) and η−(ξ, λ) as displayed on Fig. 1 where +the respective values chosen are h = 2 and coupling ratios λ = 2 and λ = 1/2. +Per Eq. (16), the respective branches associated with the λ and 1/λ-mappings +are readily observed to be symmetric with respect to the η = 0 axis. +Except when λ = 1, algebraic solutions of Eq.(15) may generally not be obtained +directly. However, for any value ξ ∈ {ξ−(h), ξ+(h)} the two roots η±(ξ, λ) of Eq. +(15) may numerically be obtained through a standard ”Newton-Raphson” algo- +rithm. Appendix 1 establishes that each root admits lower and upper bounds +for any value of U(ξ), thereby ensuring algorithm convergence. +5 + +−3 +−2 +−1 +0 +1 +2 +ξ +−4 +−2 +0 +2 +4 +η +λ =2 +λ =1/2 +Figure 1: +ξ − η Mapping for λ = 2 and λ = 1/2, and energy h = 2 +Lastly, upon inserting the linear transformation (10) into the modified LV sys- +tem (6), or equivalently using the standard Hamilton equations with Eq. (11), +a new semi-linear system of coupled 1st order ODEs is obtained +˙η = ξ + h +(18a) +˙ξ = −G′ +λ(η)eξ +(18b) +The solution of the system (18), in which G′ +λ(η) is the derivative G′ +λ(η) = +dGλ/dη, represents the time-evolution of the hybrid-species η(t) and ξ(t), albeit +due to the linear transformation (10), the first coupled equation (18a) becomes +linear since it directly expresses ODE (9). +Remarkably, as a result of this +hybrid-species transformation, up to the constant energy h, the time derivative +of the function η(t) is directly equal to the instantaneous value of the species +population ξ(t), considerably simplifying the solution of (18). The exact solution +of the LV problem is then derived by integration of the linear ODE (18a) as a +simple closed-form quadrature for t(ξ), time as a function of ξ: upon using the +initial conditions η0 = 0 and ξ0 = ξ±(h) when t = 0, the exact LV solution +corresponding to the respective negative and positive branches η−(ξ, λ) and +η+(ξ, λ) simply becomes +t(ξ) = +� ξ +ξ± +dη±(x, λ) +h + x +(19) +This quadrature is not divergent at x = −h , since the differential dη in Eq. +6 + +(15) contains the derivative U ′(ξ) = −(h + ξ)e−ξ in the numerator. +Upon +using the same initial conditions for η0 and ξ0, the solution (19) is expressed in +terms of the function η±(ξ, λ) itself through a standard integration by parts in +which the singularity at ξ = −h is further eliminated by adding and subtracting +the expression η±(−h,λ) +h+ξ +in the integral. The final, exact, closed-form, regular +solution of the entire LV problem for any value of the coupling ratio λ and any +value of the orbital energy h is thus explicitly expressed as a simple quadrature +over each of the two branches η±(ξ, λ) solutions of (15) +t(ξ) = η±(ξ, λ) − η±(−h, λ) +h + ξ ++ η±(−h, λ) +h + ξ± ++ +� ξ +ξ± +η±(x, λ) − η±(−h, λ) +(h + x)2 +dx (20) +This exact solution is further analyzed in the following section. Numerical solu- +tions for ξ(t) and η(t) are also obtained by integrating Eqs. (18) using a standard +fourth-order Runge-Kutta (RK4) method as presented in Fig. 2 for values of +h and λ exactly identical to those of Fig. 1, together with initial conditions η0 +and ξ0 defined above. The function ξ(t) is observed to principally depend on +two time constants: a quasi-exponential increase at a rate of order λ followed +by an exponential decrease at a rate −1/λ. As expected from λ-invariance (16) +the two functions ξ(t) respectively corresponding to the coupling ratio λ = 2 +and its inverse λ = 1/2 are mirrors of each other; so are the functions η(t), but +with the change η → −η. +It may generally not be possible to algebraically solve (15) for η(ξ, λ) for in- +sertion into the exact solution (20). +A strategy consists in eliminating the +η-dependence in (18b) and seeking an ODE for ξ(t) only: upon explicitly re- +lating Gλ(η) to its derivative G′λ(η) and expressing the latter as an analytical +function of ξ only through (13), a critical relationship is derived below. +7 + +0 +5 +10 +15 +20 +25 +t +−4 +−2 +0 +2 +4 +ξ(t),η(t) +ξ(t) +η(t) +Figure 2: Solutions for ξ(t) and η(t) as a function of time t with λ = 2 and energy h = 2: +numerical integration of Eq. (18) by RK4 +Case λ = 1 +The particular λ = 1 case is exactly solved since an explicit relationship exists +between Gλ and G′λ : it enables to entirely uncouple the ODE system (18) +and provides exact closed-form solutions for ξ(t) and η(t) in terms of simple +quadratures. +In this case, the G-function (12) (omitting the index for simplicity) reduces to +the hyperbolic cosine function ; the conservation equation (13) becomes +G(η) = cosh(η) = (h + 1 + ξ)e−ξ +(21) +The resulting ξ – η closed-orbit mapping is symmetric: on the ξ-axis, for any +value of the orbital energy h, the mapping is bounded by ξ−(h) and ξ+(h) +defined in (17); the two symmetric branches η±(ξ) are explicitly expressed in +terms of the inverse hyperbolic cosine function +η±(ξ) = ± cosh−1� +(h + 1 + ξ)e−ξ� +(22) +Equation (22) again establishes the symbiotic coupling between the hybrid +species η and ξ. +In this λ = 1 case, the explicit relationship sought earlier +in the discussion of (18b) between G(η) and its derivative G′(η) = sinh(η) is +G′(η) = ±(G2 − 1)1/2 +(23) +8 + +Upon inserting (23) together with (21) into (18b), the nonlinear LV problem +completely uncouples, consisting in the 1st order linear ODE (18a) together +with a 1st order nonlinear autonomous ODE for the species ξ population +˙η = ξ + h +(24a) +˙ξ = ±eξ� +(U(ξ))2 − 1 +�1/2 = ± +� +(h + 1 + ξ)2 − e2ξ�1/2 +(24b) +The linear equation (24a) is directly solved by inserting η(ξ) from (22) into +the solution (20). Together with U(ξ) defined in (14), the exact, closed-form +analytic solution on the interval ξ− ⩽ ξ ⩽ ξ+ is thus expressed as a simple +quadrature in terms of elementary functions +t(ξ) = cosh−1(eh) − cosh−1� +U(ξ) +� +h + ξ +− cosh−1(eh) +h + ξ− ++ +� ξ +ξ− +cosh−1(eh) − cosh−1� +U(x) +� +(h + x)2 +dx +(25) +By applying l’Hˆopital’s rule, it is readily verified that the integrand in (25) is +regular at ξ = −h. Figure 3 presents the ξ(t)-solution obtained by numerical +integration of (25) for an energy h = 2. The complete solution of the LV problem +for λ = 1 is finalized for η(t) by inserting ξ(t) derived above into Eq. (22). +0 +5 +10 +15 +20 +t +−3 +−2 +−1 +0 +1 +2 +3 +ξ(t),η(t) +ξ(t) +η(t) +Figure 3: Solutions for ξ(t) and η(t) as a function of time t obtained by numerical integration +of the quadrature solution Eq. (25) with λ = 1 and energy h = 2 +9 + +Another expression for t(ξ) may be obtained by integrating ξ(t) over the positive +root in (24b), yielding a simple alternative quadrature solution +t(ξ) = +� ξ +ξ− +dx +� +(h + 1 + x)2 − e2x +(26) +It is readily verified that upon inserting U(x) into the integrand of (26) and +integrating by parts the resulting expression is identical to that of solution (25). +The integrand of (26) has a weak singularity of the square root type at the +respective limits ξ−(h) and ξ+(h), but is strictly continuous and the integral +is absolutely convergent. +Finally, even though the oscillation of the hybrid- +species population ξ(t) is not expressed as an explicit function of time t, the +function t(ξ) being monotonic and continuous on each integration interval for +ξ, its inverse function ξ(t), which uniquely depends on the energy level h, exists +and is monotonic and continuous on each interval. The exact solution (26) is +similar in form to a solution derived by Evans and Findley (Eq. (17) in [5]); +however, this integral expression lends itself to simpler analytical or numerical +integration by standard methods. An exact expression for (26) is further derived +in Appendix 2 in terms of a series of exponential integrals. +Case λ ̸= 1 +In the general case when λ ̸= 1 the relationship between Gλ(η) and its derivative +G′ +λ(η) is obtained by observing that +G′ +λ(η) = e +η +λ − e−λη +λ + 1 +λ +with G′ +λ(η) = −G′ +1/λ(−η) +(λ-invariance) +(27) +Upon eliminating η between Eqs. (12) and (27), an implicit non-linear 1st order +ODE relating G to its derivative G′ is derived (for clarity the index λ is omitted +in the remainder of this section) +� +G + 1 +λG′ +�λ +(G − λG′)1/λ = 1 +(28) +Equation (28) is completely invariant in the change λ → −1/λ, or equivalently +changing λ → 1/λ together with G′ → −G′. As a result, similar to Eq. (23), in +the G − G′ phase space, Eq. (28) represents the positive and negative branches +of a “skewed” hyperbola with orthogonal asymptotes, respectively G′ = G/λ +and G′ = −λG , together with a vertex G’ = 0 located at G = 1. For any value +taken by the coupling ratio λ, the function G′(η) reaches its extremes at the two +roots of G(η) = eh. Also, as expected, in the case λ = 1 Eq. (28) identically +reduces to (23). Being implicit, (28) can generally not be solved for G′ as a +function of G by standard algebraic techniques. +A practical yet accurate approximation for the function G′(G) predicated on Eq. +(23), which removes the dependence on η in (18b) and uncouples the system, is +10 + +proposed below. +For the positive branch G′ ⩾ 0 , for large G the function G′ is asymptotic to +G′ = G/λ: Eq. (28) is thus reformulated as +λG′ +G = 1 − +1 +Gλ2+1 � +1 + 1 +λ +G′ +G +�λ2 +(29) +Furthermore, the factor in parenthesis in the denominator always satisfies the +following inequality +� +1 + 1 +λ +G′ +G +�λ2 +< eλ G′ +G +(30) +Upon approximating this factor by its exponential limit, Eq. (29) becomes +eλ G′ +G +� +1 − λG′ +G +� +∼= +1 +Gλ2+1 +(31) +Since the G-function is bounded by eh, the right hand side of (31) satisfies the +following inequalities +e−h(λ2+1) ⩽ +1 +Gλ2+1 ⩽ 1 +(32) +In order for (31) to be consistent with (32), the left hand side of (31) must at +most be of order O(1). Consequently, a Taylor expansion of the exponential +function to first order yields an explicit approximation for G′(G). For the pos- +itive branch G’ ⩾ 0 it is formulated as (33a); for the negative branch G′ ⩽ 0, +λ-invariance applied to (33a) directly yields (33b). +G′(G) ∼= G +λ +� +1 − +1 +Gλ2+1 +�1/2 +(positive branch G′ ⩾ 0) +(33a) +G′(G) ∼= −λG +� +1 − +1 +G1/λ2+1 +�1/2 +(negative branch G′ ⩽ 0) +(33b) +Remarkably, the above approximate function G′(G) satisfies the following three +basic properties identical to those of an exact numerical solution of Eq. (28): +1. at its vertex, when G = 1, the function G′(G) reaches G′ = 0, +2. for G ≫ 1, as expected, the positive branch of the function G′(G) is +asymptotic to G′ = G/λ whereas the negative branch is asymptotic to +G′ = −λG, +3. for λ = 1, the function G′(G) reduces to the exact predicate expression +(23). +Thus, in the G− G′ phase space, the explicit expressions (33) represent approx- +imate positive and negative branches of the “skewed” hyperbola defined by Eq. +(28) with the same orthogonal asymptotes. Upon comparing graphic represen- +tations of the explicit expressions (33) to the exact numerical solution of (28) +11 + +for the implicit function G′(G) it is found that the agreement is quite reason- +able particularly for the positive G′(G)-branch when λ ⩾ 1, and conversely for +the negative branch when λ ⩽ 1. This is understandable in light of the above +first two properties of (33). As λ → 1 the approximation (33) approaches the +exact solution (23); for λ ≫ 1 the graph of (33) exhibits two branches tightly +bounded by their respective orthogonal asymptotes with the accuracy of this +approximation increasing with increasing λ. +As intended, approximation (33) effectively uncouples the system (18) by explic- +itly removing the dependence on η in the original ODE (18b): upon inserting +the conservation Eq. (13) into (33), Eq. (18b) is replaced by a pair of two +λ-invariant 1st order nonlinear ODEs for the hybrid species population ξ(t) +˙ξ = −h + 1 + ξ +λ +� +1 − +eξ(λ2+1) +(h + 1 + ξ)(λ2+1) +�1/2 +(positive η-branch: η ⩾ 0) +(34a) +˙ξ = λ(h + 1 + ξ) +� +1 − +eξ(1/λ2+1) +(h + 1 + ξ)(1/λ2+1) +�1/2 +(negative η-branch: η ⩽ 0) +(34b) +Evidently, for λ = 1 the two branches of (24b) are recovered. Even though ξ(t) +is not explicitly expressed as a function of time t, the arbitrary λ ̸= 1 problem +has thus been reduced to a pair of simple quadratures for the function t(ξ). As +already stated, the function ξ(t) oscillates between the λ-independent respective +roots ξ−(h) and ξ+(h) solutions of Eq. (17). The process for solving Eq. (34) is +identical to that of Eq. (24b): upon again choosing the time origin t = 0 when +ξ0 = ξ−(h), a complete period is obtained by integration over the corresponding +negative η-branch in (34b) until ξ(t) reaches ξ+(h), followed by an integration +over the positive η-branch (34a) until ξ−(h) is reached +t(ξ) = +� ξ +ξ− +1 +λ(h + 1 + x) +� +1 − +ex(1/λ2+1) +(h + 1 + x)(1/λ2+1) +�−1/2 +dx +(negative η-branch) +(35a) +t(ξ) = − +� ξ +ξ+ +λ +h + 1 + x +� +1 − +ex(λ2+1) +(h + 1 + x)(λ2+1) +�−1/2 +dx +(positive η-branch ) +(35b) +The function t(ξ) being monotonic and continuous on the respective integra- +tion intervals ξ− ⩽ ξ ⩽ ξ+ and ξ+ ⩾ ξ ⩾ ξ− its inverse function ξ(t) exists +and is unique, monotonic, and continuous on each interval. The LV problem is +then completed for the function η(t) by directly integrating the linear Eq. (18a) +through standard numerical techniques. +12 + +To assess the accuracy of the uncoupled approximate solutions (34), a compari- +son is made with the exact numerical solutions of the original coupled LV system +(18). Upon using the respective values λ = 2 and h = 2 identical to those of +Fig. 2 for the coupling ratio and system energy, Fig. 4 presents the comparison +between the functions ξ(t) and η(t) respectively obtained by numerically inte- +grating (34) and (18) simultaneously through a standard 4th-order RK4 method. +From the figure it is observed that the ODEs (34) provide a reasonably accurate +solution for both functions ξ(t) and η(t) over an entire period, yet, when λ > 1, +with an underestimation of the time taken to reach ξ+(h) compensated by an +overestimation of the time to reach ξ−(h). As expected, the accuracy of the +solutions obtained with approximations (34) increases with increasing λ. +0 +2 +4 +6 +8 +10 +12 +t +−2 +0 +2 +4 +6 +ξ(t),η(t) +ξ(t) +η(t) +ξApprox(t) +ηApprox(t) +Figure 4: Solutions for ξ(t) and η(t) as a function of time t with λ = 2 and energy h = 2; +comparison between RK4 numerical integration of Eq. (18) and Eq. (34) +From Fig. 4, regardless of the value of λ, the hybrid species population ξ(t) +is observed to oscillate with exponential-like growth and decay phases with its +energy-dependent amplitude determined by the difference ξ+(h) - ξ−(h). +Remarkably, in the high energy limit (h ≫ 1), upon keeping the leading asymp- +totic term in (34), the asymptotic behavior of the LV system becomes modeled +as a system of two coupled linear 1st order ODEs for each hybrid species. In +this asymptotic limit, together with the linear ODE (18a) for η(t), the system +admits trivial exponential solutions remarkably representative of the exact so- +13 + +lutions of (18). For example, the asymptotic solutions (h ≫ 1) for the growth +phase (ξ− ⩽ ξ ⩽ ξ+) simply are +ξ(t) = eξ−+λt − (h + 1) +(36a) +η(t) = 1 +λ +� +ξ(t) − ξ−(h) +� +− t +(36b) +The decay phase asymptotic solutions for ξ(t) are obtained by λ-invariance, +namely λ → −1/λ together with ξ−(h) → ξ+(h). +Lastly, upon inserting the hybrid-species populations ξ(t) and η(t) derived from +Eqs. (35) together with the transformation (10) into the definition (5) of the +prey and predator species, the respective standard solutions for the original +populations u(t) and v(t) are fully recovered +u(t) = eξ(t)−λη(t) +for preys +(37a) +v(t) = eξ(t)+η(t)/λ +for predators +(37b) +4. Oscillation Period of the LV System +The unique λ-invariance property of η±(ξ, λ) in (16) directly enables to estab- +lish two fundamental properties of the LV system period. Consider the double +mapping of Fig. 1 and follow in a counterclockwise direction the two branches +AB− and BA+ corresponding to the respective branches η−(ξ, λ) and η+(ξ, λ): +the negative branch AB− starts at ξ−(h) and ends at ξ+(h) and conversely for +the positive BA+ branch. Upon integrating (19) over the ξ-variable and re- +calling the earlier definition t = +√ +αδt′ , the oscillation period Tλ(h) associated +with the λ-mapping is directly obtained as a quadrature over these two branches +(38a); here the negative sign for the second integral reflects integration from ξ+ +to ξ−. Similarly for the 1/λ-mapping the oscillation period is expressed as (38b) +Tλ(h) = +1 +√ +αδ +�� +AB− +dη−(ξ, λ) +h + ξ +− +� +BA+ +dη+(ξ, λ) +h + ξ +� +(38a) +T1/λ(h) = +1 +√ +αδ +�� +AB− +dη−(ξ, 1/λ) +h + ξ +− +� +BA+ +dη+(ξ, 1/λ) +h + ξ +� +(38b) +Upon recalling the λ-invariance property of Eq. (16), substitution into (38b) +establishes that: +Tλ(h) = T1/λ(h) +(39) +Theorem 1. For any value of the positive orbital energy h , the LV system os- +cillation periods respectively corresponding to the coupling ratio λ and its inverse +1/λ are equal. +Consequently, an exact, closed-form, regular expression for the nonlinear LV +system oscillation period, valid for any value of the coupling ratio λ and any +14 + +value of the orbital energy h, is directly derived from (38a) as an integral over +the two branches of the ξ - η mapping +Tλ(h) = +1 +√ +αδ +� +η−(−h, λ) − η+(−h, λ) +� +(ξ+ − ξ−) +(h + ξ+)(h + ξ−) ++ +1 +√ +αδ +� ξ+ +ξ− +η−(x, λ) − η−(−h, λ) + η+(−h, λ) − η+(x, λ) +(h + x)2 +dx +(40) +In Appendix 1, for any ξ ∈ {ξ−(h), ξ+(h)}, the interval η+ (ξ, λ) − η−(ξ, λ) +is shown to be a positive increasing function of λ when λ ⩾ 1 (and decreasing +when 0 < λ ⩽ 1) admitting respective lower and upper bounds, both of which +are minimal when λ = 1. Together with Eq. (40) this establishes: +Theorem 2. For any value of the positive orbital energy h , the LV system +oscillation period Tλ(h) is an increasing function of λ for λ ⩾ 1 (decreasing for +0 < λ ⩽ 1) and the period is shortest for λ = 1. +In the particular case when λ = 1, the exact LV system period T1(h) is uniquely +expressed in terms of a universal energy function Θ1(h) as +T1(h) = +2π +√ +αδ +Θ1(h) +(41) +The LV energy function Θ1(h) introduced here is readily defined from (26) as +Θ1(h) = 1 +π +� ξ+ +ξ− +dx +� +(h + 1 + x)2 − e2x +(42) +At small orbital energy (h ≪ 1), Θ1(h) is directly expressed in terms of the +complete elliptic integral of the first kind K(k) with its modulus k +Θ1(h) = +1 +� +1 + +√ +2h +2 +π K(k) +with +k = +� +2 +√ +2h +1 + +√ +2h +(43) +A standard series expansion for K(k) yields +Θ1(h) = 1 + 1 +6h + 35 +432h2 + O(h3) +(44) +As expected, for small oscillation amplitudes, the integral (42) is independent +of the energy h and exactly equates π: hence Θ1(h) approaches unity in (44) +and the LV system period T1(h) is that of a harmonic oscillator with time factor +1/ +√ +αδ, as already established [23], [24]. +At high orbital energy (h ≫ 1), the contribution from the exponential term in +(42) is negligible over the integration interval except when ξ approaches ξ+(h): +since by definition ξ ⩾ ξ−(h), approximating the exponential term by its lowest +15 + +value e2ξ−(h) and performing the integration yields an asymptotic expression for +Θ1(h) +Θasymp(h) ∼= 1 +π cosh−1 � +eξ+(h)−ξ−(h)� +with h ≫ 1 +(45) +When λ ̸= 1 the exact LV oscillation period Tλ(h) is obtained by numerically +solving the ODE system (18) as done for Fig. 2. Similar to Eq. (41), for each +value of the coupling ratio λ, the period Tλ(h) is then uniquely expressed in +terms of universal LV energy functions Θλ(h) +Tλ(h) = +2π +√ +αδ +Θλ(h) +(46) +As shown on Fig. 5 and consistent with Theorem 2, for any value of the cou- +pling ratio λ, each function Θλ(h) is a monotonically increasing function of the +system’s energy h; so is the LV system period Tλ(h), [24]. Also displayed is +the asymptotic approximation (45) of the exact function Θ1(h) ; for h ⩾ 3 the +difference between the exact solution and its asymptotic approximation is ⩽ 3%. +0 +1 +2 +3 +4 +5 +Energy h +0 +1 +2 +3 +4 +5 +6 +7 +8 +Energy Function Θλ(h) +λ =1 +λ =2 +λ =3 +λ =4 +λ =5 +λ =1 Approximation +Figure 5: Energy function Θλ(h) for λ = 1, 2, 3, 4, 5 and asymptotic approximation for λ = 1 +In this general λ ̸= 1 case, an asymptotic formula for the LV system oscillation +period Tλ(h) valid at high energy (h ≫ 1) is obtained from the asymptotic +solutions (36). The contribution T + +λ (h) of the exponential growth phase of ξ(t) +to the period is readily obtained from Eq. (36b) since η(t) = 0 when ξ(t) reaches +its maximum ξ+(h); the contribution T − +λ (h) of the decay phase is obtained by +λ-invariance. As a result the high energy (h ≫ 1) asymptotic expression for +16 + +the LV system period Tλ(h) simply becomes proportional to the sum of the +ξ(t)-function growth and decay rates, λ and 1/λ, respectively +Tλ(h) ∼= +π +√ +αδ +� +λ + 1 +λ +� � +ξ+(h) − ξ−(h) +� +(47) +This asymptotic formula which separately factorizes the LV system coupling +from the λ-independent energy contribution satisfies both Theorem 1 and The- +orem 2 since it is minimal when λ = 1. +Shih performed an exhaustive review of integral representations of the period +of the two-species LV system: he compared the methods of Volterra [23], Hsu +[9], Waldvogel [24], and Rothe [17] and demonstrated that all of these repre- +sentations are equivalent to his own solution in terms of a sum of convolution +integrals [18]. Subsequent approximations of the LV system period in terms of +power series [20] or perturbation expansions [8] have also been published. In +Appendix 3, following the derivation of Rothe [17], we show that, even though +not ”planar” in Rothe’s sense (Eq. (7)), the Hamiltonian (11) based on hybrid- +species populations provides a ”state sum” Z(β) identical to that of Rothe +thereby establishing direct equivalence with Rothe’s convolution integral for +the LV oscillator period. +5. Conclusion +The coupled 1st order non-linear ODE system for the LV problem of two in- +teracting species has been re-formulated in terms of a single positive coupling +parameter λ, ratio of the relative growth/decay rates of each species taken inde- +pendently. Based on a Hamiltonian formulation combined with a linear trans- +formation introducing ”hybrid-species populations”, a novel λ-invariant set of +two 1st order ODEs is obtained with one being linear. As a result, an exact, +closed-form quadrature solution of the LV problem is derived for any value of +the coupling ratio λ and any value of the system’s energy (Eq. (20)). +In the λ = 1 case, the LV problem completely uncouples and an exact explicit +closed-form solution is expressed in terms of the orbital energy h as a simple +quadrature for the population of one hybrid-species whereas the other hybrid +species’ solution is explicitly expressed in terms of the former. +In the λ ̸= 1 case, a λ-invariant accurate practical approximation is derived that +explicitly uncouples the LV system and provides a closed-form solution in terms +of a single quadrature for one of the hybrid-species populations. Remarkably, +at high orbital energies (h ≫ 1), the original coupled non-linear LV ODE sys- +tem totally uncouples and becomes entirely linear admitting trivial asymptotic +exponential solutions. +Further, as a consequence of λ-invariance, for any value of the orbital energy +h, the LV system oscillation period is shown to be identical when the coupling +parameter λ is inverted to 1/λ and is smallest when λ = 1. In this particular +case, an exact, closed-form expression for the non-linear LV system oscillation +17 + +period is derived in terms of a universal LV energy function. In the λ ̸= 1 case, +a simple asymptotic expression for the LV system oscillation period is derived +for high energies (h ≫ 1). +Appendix 1 +This Appendix presents a proof of Theorem 2 introduced after Eq. (40). For +the positive root η+(ξ, λ) , Eq. (13) is written +λ2e +η +λ + e−ηλ = (λ2 + 1)U(ξ) +(A1.1) +For any given value of ξ ∈ {ξ−(h), ξ+(h)}, since we seek a positive root and +since by definition 0 ⩽ e−ηλ ⩽ 1, this root admits a lower and an upper bound +λ ln +�� +1 + 1 +λ2 +� +U (ξ) − 1 +λ2 +� +⩽ η+(ξ, λ) ⩽ λ ln +� � +1 + 1 +λ2 +� +U (ξ) +� +(A1.2a) +Similarly, by λ-invariance, the negative root satisfies +− 1 +λ ln +� � +1 + λ2� +U (ξ) +� +⩽ η− (ξ, λ) ⩽ − 1 +λ ln +� � +1 + λ2� +U (ξ) − λ2� +(A1.2b) +From Eqs. (A1.2) the lower and upper bounding of the roots η±(ξ, λ) of Eq. +(15) enables to prove Theorem 2. From Eq. (40), the period depends on the +magnitude of the positive interval η+ (ξ, λ) − η−(ξ, λ). Upon introducing the +“outer limit” ∆out (ξ, λ) as +∆out (ξ, λ) = λ ln +� � +1 + 1 +λ2 +� +U (ξ) +� ++ 1 +λ ln +� � +1 + λ2� +U (ξ) +� +(A1.3a) +it is readily seen that ∆out (ξ, λ) is a positive, increasing function of λ when +λ ⩾ 1 (and decreasing when λ ⩽ 1) whose partial derivative ∂∆out(ξ, λ)/∂λ +vanishes when λ = 1. Similarly, upon introducing the “inner limit” ∆in (ξ, λ) as +∆in (ξ, λ) = λ ln +� � +1 + 1 +λ2 +� +U (ξ)− 1 +λ2 +� ++ 1 +λ ln +� � +1 + λ2� +U (ξ)−λ2� +(A1.3b) +it is also seen that ∆in (ξ, λ) is a positive, increasing function of λ when λ ⩾ +1 (and decreasing when λ ⩽ 1) whose partial derivative ∂∆in(ξ, λ)/∂λ also +vanishes when λ = 1. Since the positive interval η+ (ξ, λ) − η−(ξ, λ) obviously +satisfies +∆in (ξ, λ) ⩽ η+ (ξ, λ) − η−(ξ, λ) ⩽ ∆out(ξ, λ) +(A1.4) +This proves Theorem 2. +18 + +Appendix 2 +Upon recalling the definition (14) of U(ξ), a series expansion for the quadrature +solution (26) is derived by first writing the integral as +t(ξ) = cosh−1� +U(ξ) +� ++ +� ξ +ξ− +1 +� +1 − U(x)−2 dx +(A2.1) +Since 1 ⩽ U(ξ) ⩽ eh, a binomial expansion of the integrand with binomial coef- +ficients expressed in terms of the Gamma function Γ(p) defined by its standard +Euler integral of the second kind yields the solution in terms of a converging +series +t(ξ) = cosh−1� +U(ξ) +� ++ +∞ +� +p=0 +Γ +� 1 +2 +� +Γ +� 1 +2 − p +� +Γ (p + 1) +� ξ +ξ− U(x)−2pdx +(A2.2) +Each integral I2p(ξ) in the expansion (A2.2) is of the form +I2p(ξ) = +� ξ +ξ− +e2pxdx +(h + 1 + x)2p +(A2.3) +Successive integrations by parts and substitution into (A2.2) result in a slowly +convergent series of exponential integral functions with positive argument of the +form Ei +� +2p(h + 1 + ξ) +� +where the integer p is 1, 2, 3, . . .. +Appendix 3 +Based on thermodynamics, Rothe [17] established that the Laplace transform +of the period function T (h), in which h is the system’s energy, is the canonical +state sum Z(β) of the Hamiltonian (7), with β ∈ (0, ∞) as the inverse of the +absolute temperature, namely +Z(β) = +� +∞ +−∞ +� +∞ +−∞ +e−βH(x,y)dxdy = +� ∞ +0 +e−βhT (h)dh +(A3.1) +From Eqs. (8) and (11) together with the definition (12) of the G-function, the +LV system’s Hamiltonian is +H(η, ξ) = +� +λ + 1 +λ +� � +Gλ(η)eξ − ξ − 1 +� +(A3.2) +For notation purposes, we introduce the reduced g-function gλ(η) defined as +gλ(η) = λe +η +λ + 1 +λe−ηλ +(A3.3) +19 + +Consequently, upon inserting the Jacobian | J |= +� +λ + 1 +λ +� +of the linear trans- +formation (10) +Z(β) = +� +λ + 1 +λ +� � +∞ +−∞ +� +∞ +−∞ +e−βgλ(η)eξ+(λ+ 1 +λ)β(ξ+1)dξdη +(A3.4) +Upon substituting s = eξ with s ∈ (0, ∞), (A3.4) becomes +Z(β) = +� +λ + 1 +λ +� +eβ(λ+ 1 +λ) +� +∞ +−∞ +� ∞ +0 +sβ(λ+ 1 +λ)−1e−βsgλ(η)dsdη +(A3.5) +The integration over s is expressed in terms of the Gamma function Γ(s): +Z(β) = +� +λ + 1 +λ +� � e +β +�β(λ+ 1 +λ) +Γ +� +β +� +λ + 1 +λ +�� � +∞ +−∞ +(gλ(η))−β(λ+ 1 +λ)dη (A3.6) +Together with the above definition of gλ(η) this definite integral has been eval- +uated (see 3.314 in [7]); the λ-invariant state sum Z(β) thus becomes +Z(β) = +� e +βλ +�βλ +Γ(βλ) +�eλ +β +�( β +λ) +Γ +�β +λ +� +(A3.7) +Although the Hamiltonian (A3.2) is defined in the ξ −η space, the result (A3.7) +for the state sum Z(β) is identical to that of Rothe (Eqs. (8) and (9) in [17]) +who used the ”planar” Hamiltonian (7) in the x − y space. The derivation of +the period then directly follows Rothe who defines a function τ(h) (Eqs. (14), +(15), and (16) in [17]) whose Laplace transform is +� ∞ +0 +e−βhτ(h)dh = +� e +β +�β +Γ(β) +(A3.8) +Since our state sum (A3.7) is expressed as the product of two Laplace transforms +similar to (A3.8), use of the Hamiltonian (A3.2) establishes that the period +Tλ(h) of the LV system (18) is directly equivalent to that of Rothe. +Upon +recalling the earlier definition of time t = +√ +αδt′ , the period is formulated as a +λ-invariant convolution integral satisfying Theorem 1 with τ(h) defined above +Tλ(h) = +1 +√ +αδ +� h +0 +τ +� s +λ +� +τ +� +λ(h − s) +� +ds +(A3.9) +References +[1] E. Chauvet, J. E. Paullet, J. P. Previte, and Z. Walls. A lotka-volterra +three-species food chain. In Mathematics Magazine, volume 75, pages 243– +255, 2002. +20 + +[2] B. M. Chen-Charpentier and D. Stanescu. 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Approximating the periodic solutions +of the Lotka-Volterra system. ANZIAM J., 49((C)):C243–C257, 2007/08. +[9] S. B. Hsu. A remark on the period of the periodic solution in the Lotka- +Volterra system. J. Math. Anal. Appl., 95(2):428–436, 1983. +[10] E. H. Kerner. Dynamical aspects of kinetics. Bull. Math. Biophys., 26:333– +349, 1964. +[11] E. H. Kerner. Comment on Hamiltonian structures for the n-dimensional +Lotka-Volterra equations. J. Math. Phys., 38(2):1218–1223, 1997. +[12] A. J. Lotka. Undamped oscillations derived from the law of mass action. +Journal of the American Chemical Society, 42(8):1595–1599, 1920. +[13] G. Mingari Scarpello and D. Ritelli. A new method for the explicit inte- +gration of Lotka-Volterra equations. 11:1–17, 01 2003. +[14] K. N. Murty and D. V. G. Rao. Approximate analytical solutions of general +Lotka-Volterra equations. J. Math. Anal. Appl., 122(2):582–588, 1987. +[15] M. Plank. Hamiltonian structures for the n-dimensional Lotka-Volterra +equations. J. Math. Phys., 36(7):3520–3534, 1995. +[16] D. V. G. Rao and Y. L. P. Thorani. A study of the solutions of the Lotka- +Volterra prey-predator system using perturbation technique. Int. Math. +Forum, 5(53-56):2667–2673, 2010. +[17] F. Rothe. +The periods of the Volterra-Lotka system. +J. Reine Angew. +Math., 355:129–138, 1985. +21 + +[18] S.-D. Shih. The period of a Lotka-Volterra system. Taiwanese J. Math., +1(4):451–470, 12 1997. +[19] S.-D. Shih. +Comments on “a new method for the explicit integration +of Lotka-Volterra equations”. Divulgaciones Matem´aticas, 13(2):99–106, +2005. +[20] S.-D. Shih and S.-S. Chow. A power series in small energy for the period +of the Lotka-Volterra system. Taiwanese J. Math., 8(4):569–591, 12 2004. +[21] C. E. Treanor, J. W. Rich, and R. Rehm. Vibrational relaxation of an- +harmonic oscillators with exchange-dominated collisions. J. Chem. Phys., +48:1798–1807, 02 1968. +[22] V. S. Varma. +Exact solutions for a special prey-predator or competing +species system. Bull. Math. Biology, 39(5):619–622, 1977. +[23] V. Volterra. Variation and fluctuations of the number of individuals of +animal species living together. In R. N. Chapman, editor, Animal Ecology, +pages 31–113. McGraw-Hill, 1926. +[24] J. Waldvogel. The period in the Lotka-Volterra system is monotonic. J. +Math. Anal. Appl., 114(1):178–184, 1986. +22 + diff --git a/idFAT4oBgHgl3EQfZx1V/content/2301.08547v1.pdf b/idFAT4oBgHgl3EQfZx1V/content/2301.08547v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..524318b8b1b976b7075c6bdd8be788ce5baf265e --- /dev/null +++ b/idFAT4oBgHgl3EQfZx1V/content/2301.08547v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b689d696f59cc8dda5fc19364f6e5559b0d8f5c1b75f0074e0d267f10ea7e5d9 +size 206851 diff --git a/idFAT4oBgHgl3EQfZx1V/vector_store/index.faiss b/idFAT4oBgHgl3EQfZx1V/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..7bce96739db840a7476d6cbb27ea2f473588b0ff --- /dev/null +++ b/idFAT4oBgHgl3EQfZx1V/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:996a36de79160427845ea623cbd094d496b73d8c51df16c82ea9f37fc4a31393 +size 1441837 diff --git a/idFAT4oBgHgl3EQfZx1V/vector_store/index.pkl b/idFAT4oBgHgl3EQfZx1V/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..2ef8b3e7d7cb07b146b7c72b4a91ce1dafa1ec82 --- /dev/null +++ b/idFAT4oBgHgl3EQfZx1V/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6d30f6a31d4149a99ba79c32424a7ee394f67ec4e7e1a4e85e179d4d2c8d1c66 +size 59236 diff --git a/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf b/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..961a1f030a10beba7ab859244357e229889f8c47 --- /dev/null +++ b/jNA0T4oBgHgl3EQfIv8h/content/2301.02078v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:eebcc600972c29db5f53d446573317da01f6bf68fd6f4ef8233765eaa504cedc +size 3073772 diff --git a/jdFKT4oBgHgl3EQfBS3T/content/tmp_files/2301.11703v1.pdf.txt b/jdFKT4oBgHgl3EQfBS3T/content/tmp_files/2301.11703v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d98a4c8ed9bf4d36fb0fd1d0e8d18bd309d18b5 --- /dev/null +++ b/jdFKT4oBgHgl3EQfBS3T/content/tmp_files/2301.11703v1.pdf.txt @@ -0,0 +1,3763 @@ +Competitive exclusion and Hebbian couplings in random generalised +Lotka–Volterra systems +Enrique Rozas Garcia,1, 2, ∗ Mark J. Crumpton,3, 4, † and Tobias Galla4, 2, ‡ +1Department of Physics, Gothenburg University, 41296 Gothenburg, Sweden +2Instituto de F´ısica Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), +Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain +3Department of Mathematics, King’s College London, London WC2R 2LS, United Kingdom +4Department of Physics and Astronomy, School of Natural Sciences, +The University of Manchester, Manchester M13 9PL, UK +(Dated: January 30, 2023) +We study communities emerging from generalised random Lotka–Volterra dynamics with a large +number of species and with competitive exclusion. Each species is endowed with a number of traits, +and competition between pairs of species increases with their similarity in trait space. This leads +to a model with random Hebbian interactions. We use tools from the theory of disordered systems, +notably dynamic mean field theory, to characterise the statistics of the resulting communities at +stable fixed points and determine analytically when stability breaks down. Two distinct types of +transition are identified in this way, both marked by diverging abundances, but differing in the +behaviour of the integrated response function. At fixed points only a fraction of the initial pool +of species survives. We numerically study the eigenvalue spectra of the interaction matrix between +extant species. +We find evidence that the two types of dynamical transition are, respectively, +associated with the bulk spectrum or an outlier eigenvalue crossing into the right half of the complex +plane. +I. +INTRODUCTION +The foundations of the theory of disordered sys- +tems date back close to 50 years [1]. +Initially the +aim was to understand certain magnetic states in con- +densed matter physics (‘spin glasses’) [2]. However it +became clear that applications of the tools developed +for disordered systems had a reach far beyond the +boundaries of physics. Methods such as replica the- +ory or dynamic generating functionals were quickly +adapted and used to answer questions in neural net- +works [3–5], to study the Minority Game [6] (some- +times presented as a simple model of a financial mar- +ket), or indeed evolutionary bi-matrix games and so- +called Nash equilibria [7]. +The defining feature of disordered systems is the +presence of quenched disorder. That is, the system +is made up of many constituents, and the interac- +tions between these is determined by coefficients that +are drawn at random at the beginning, but then re- +main fixed as the dynamics of the system unfolds. +The disorder leads to complicated energy landscapes. +The number of local minima can grow exponentially +in the size of the system, and is often organised in +a hierarchical manner. Dynamic phenomena in dis- +ordered systems include ergodicity breaking and so- +called ageing [2, 8]. +Ideas and methods from the physics of disordered +systems have also been used to study complex ecosys- +∗ enrique.rozas.garcia@physics.gu.se +† mark.j.crumpton@kcl.ac.uk +‡ tobias.galla@ifisc.uib-csic.es +tems [9–16]. The word ‘complex’ in this context in- +dicates that the ecosystem is composed of a large +number of species, and that these species are sub- +ject to randomly drawn interaction coefficients. +In +this paper we continue this line of work, and focus +on a Lotka–Volterra system with ‘Hebbian’ interac- +tions [3–5]. More specifically, we are interested in a +set of N species (N ≫ 1), whose abundances develop +in time following a generalised Lotka–Volterra equa- +tion (details will follow in Sec. II). This involves an +N ×N matrix aij of interaction coefficients. Existing +work on the statistical physics of complex ecosystems +has mostly focused on the case in which the interac- +tion matrix is drawn from a Gaussian distribution +[9, 11, 14–18]. +This includes work on random ma- +trix ensembles with relatively little structure, but also +cases in which the matrix is composed of blocks, and +where the elements in different blocks have different +statistics [19]. One common element shared by Gaus- +sian Lotka–Volterra models is that the finest level of +modelling is set by the interaction coefficients. No +further assumptions are made about the properties +of the species, and how the species interactions come +about from these properties. +The Hebbian model is inspired by structures first +used in neural networks [5, 20]. +Translated to the +language of ecology, the starting point is now a set +of species and a set of traits. Each species can ei- +ther possess or not possess a given trait. +This as- +signment of traits to species, in turn determines how +species will interact. Broadly speaking, the interac- +tion between two species will be more competitive +the more traits they share (i.e. the more similar the +two species are). This type of interaction structure +arXiv:2301.11703v1 [q-bio.PE] 27 Jan 2023 + +2 +has also been studied in models combining resources +and consumers, both in economics and in ecology +[12, 13, 21–23]. Analyses of random replicator sys- +tems with Hebbian interactions [24] have shown inter- +esting statistical mechanics, and in particular types +of phase transition that are different from those seen +in replicator systems with Gaussian couplings. +In this paper, we set out to characterise the be- +haviour of a Lotka–Volterra system with Hebbian in- +teractions, where we allow for a degree of ‘mild’ di- +lution (the system is not fully connected, but each +species still interacts with an extensive number of +other species). A system of replicator equations with +such interactions was studied in [24]. Our aim is to +calculate the statistics of fixed points in the phase +where such fixed points are attained and identify the +onset of instability. As in the system with Gaussian +interactions we find that only a proportion of the ini- +tial species survive at stable fixed points. +Recent +work [25] on Gaussian systems has shown that the +reduced interaction matrix (the matrix of interaction +coefficients among the surviving species) has intri- +cate statistics. Specifically, that its bulk and outlier +eigenvalues can be related to different types of dy- +namic phase transitions. As we will show, the types +of phase transition seen in the Hebbian model dif- +fer from those in the Gaussian model. One aim of +the current paper, is therefore to establish (in simu- +lations) how these transitions relate to the spectra of +the interaction matrix of the extant species. +The remainder of the paper is organised as follows. +In Sec. II we define the model and introduce the nec- +essary notation. +Sec. III then contains the mathe- +matical analysis. This is based on so-called ‘generat- +ing functionals’ and dynamic mean field theory. The +phase diagram and further behaviour of the model is +then discussed in Sec. IV. In Sec. V we finally turn +to a study of the spectra of the reduced interaction +matrix and their relation to the phase diagram. We +conclude the paper with a discussion and an outlook +in Sec. VI. +II. +MODEL DEFINITIONS +We will study the following generalised Lotka– +Volterra equation (gLVE) +˙xi(t) = xi(t) +� +�Ki − uixi + +� +j̸=i +cijJijxj +� +� , +(1) +where the xi ≥ 0 represent the abundances (or popu- +lation densities) of different species, i = 1, . . . N. We +always assume initial conditions for which all xi are +strictly positive. +The quantities ui > 0 denote the strength of in- +traspecific competition, and the aij = cijJij represent +the interspecific interactions. The Ki (together with +the ui) set the carrying capacities of the species in the +absence of interactions between different species (xi +then tends to Ki/u in the long run). We focus on the +case ui ≡ u for all i, noting that u controls the time +scale on which the non-interacting system approaches +the fixed point xi ≡ Ki/u. We allow for general pos- +itive values of u throughout our analysis, but in an +effort keep the number of parameters manageable we +set Ki ≡ 1. +The dilution variables cij ∈ {0, 1} (i ̸= j) deter- +mine which species interact with one another, i.e. +they set the topology of the interaction network. For +each pair i < j, the coefficients cij and cji are chosen +from a Bernoulli distribution with +⟨cij⟩ = c, +⟨cijcji⟩ − c2 = Γc(1 − c). +(2) +We thus have P(cij = 1) = c for all i ̸= j, i.e. c is +the analog of what May called ‘connectance’ [9]. The +parameter Γ is restricted to the range from −1 to 1 +by construction, but we note that not all choices of +pairs (c, Γ) are possible (see Supplemental Material +[26] for details). We note that the choice of the di- +agonal coefficients cii is irrelevant as we set Jii = 0 +below +Throughout our paper, c is chosen not to scale with +N [c = O(N 0)]. This means that each species inter- +acts with an O(N) number of other species. We are +therefore not studying a ‘dilute’ system in the sense +of random matrix theory. The extensive connectivity +allows us to use established methods from dynamic +mean-field theory. An analysis of a truly dilute sys- +tem with c = O(1/N) (and where consequently each +species only interacts with a finite number of other +species) would be much more intricate. +Interaction links in our system are directed, that is, +an effect of the presence of species j on the dynamics +of i does not necessarily imply the reverse. The pa- +rameter Γ measures the correlations between cij and +cji. A choice of Γ = −1 implies cij = 1 − cji with +probability one, and Γ = 1 means that cij = cji with +probability one. +The matrix Jij determines the strength of the ef- +fects of the presence of species j on the dynamics +of the abundance of species i. Positive values of Jij +imply that the population of species j is beneficial +to the growth of species i, while negative values im- +ply a detrimental interaction. In ecological terms the +signs of the pair of interactions (sgn Jij, sgn Jji) de- +termine whether two species are in a mutualistic rela- +tion (+, +), whether they compete with one another +(−, −), or whether there is an antagonistic predator- +prey relation between i and j (±, ∓) [10]. +In this work, the interspecific interaction is chosen +according to a competitive exclusion heuristic (see +e.g. [27]). We assume each species is described by a +set of binary traits, labelled µ = 1, . . . , P, and that +a pair of species will compete in proportion to the +similarity between the two species (i.e, the number + +3 +of traits which both or neither species possess). We +write ξµ +i = +1, if species i has trait µ, and ξµ +i = −1 +if the species does not possess the trait. Interactions +are then assumed to be of the form +Jij = +� +� +� +� +� +� +� +− 1 +cN +αcN +� +µ=1 +ξµ +i ξµ +j +i ̸= j +0 +i = j +. +(3) +We have here set P = αcN, with α > 0 a model pa- +rameter (in simulations P is restricted to integer val- +ues.) That is to say, we assume that the number of +traits is proportional to the number of species in the +system. The interaction in Eq. (3) is known as ‘Heb- +bian’ in the context of neural networks [3, 20], and +interesting phase behaviour occurs when α = O(N 0). +This is the regime we focus on. We have normalised +the interaction strength by cN, the mean number of +species that any one species will interact with. We +will refer to the random variables cij and ξµ +i as the +disorder of the system. +The traits ξµ +i are chosen to be ξµ +i = ±1 with equal +probability, and there is no correlation between the +different ξµ +i . This implies that the distribution of the +Jij approaches a Gaussian as N → ∞, reminiscent of +the model studied for example in [14, 15]. We note +however, that the Hebbian structure introduces cor- +relations between the different Jij, which are different +from the correlations studied in the earlier literature. +III. +GENERATING FUNCTIONAL +ANALYSIS AND STABILITY +A. +Generating functional and effective process +We analyse the system in Eq. (1) using dynamic +generating functionals, an established method in the +theory of disordered systems [4, 28]. +This is also +known as ‘dynamic mean field theory’, and has been +used to study Lotka–Volterra systems with Gaussian +random couplings [11, 15, 29]. We note that an alter- +native approach is based on the the so-called cavity +method [14, 17, 30]. +The outcome of the application of these tech- +niques is an effective stochastic process for a ‘rep- +resentative species’. The ensemble of realisations of +stochastic processes is statistically equivalent to the +set of single-species trajectories xi(t) of the disor- +dered dynamical system in Eq. (1). +The dynamic +mean-field description becomes exact in the thermo- +dynamic limit (N → ∞). Overall, in this limit, the +infinite-dimensional deterministic dynamical system +in Eq. (1) is traded for an effective single-species pro- +cess which is non-local in time (it involves retarded +self-interaction) and contains coloured noise. +The generating functional analysis begins from +˙xi(t) = xi(t) +� +�1 − uxi + +� +j̸=i +cijJijxj − hi(t) +� +� , +(4) +where we have introduced the perturbation fields +hi(t) in order to calculate linear response functions. +These fields are not actually part of the model and are +set to zero at the end of the calculation, as well as in +all simulations shown in the paper. For more details +see the Supplementary Material (SM). The generat- +ing functional of this dynamical system is given by +Z[ψi(t), hi(t)] = +� +exp +� +i +� +i +� +dt xi(t)ψi(t) +�� +paths +, +(5) +where the average is over paths [x1(t), . . . , xN(t)] of +the dynamics in Eq. (4). The ψi(t) constitute a source +field. +The generating functional in Eq. (5) is the +Fourier transform of the probability measure in the +space of paths generated by Eq. (4). +The final outcome of the generating-functional +analysis is a set of equations for the dynamic macro- +scopic order parameters of the problem. +For the +Lotka–Volterra model these are +M(t) = lim +N→∞ +1 +N +N +� +i=1 +⟨xi(t)⟩0, +C(t, t′) = lim +N→∞ +1 +N +N +� +i=1 +⟨xi(t)xi(t′)⟩0, +G(t, t′) = lim +N→∞ +1 +N +N +� +i=1 +δ⟨xi(t)⟩0 +δhi(t′) , +(6) +where δ denotes a functional derivative and ⟨· · · ⟩0 +stands for an average over random initial conditions. +The overbar · · · represents the average over the dis- +order, i.e. over the cij and ξµ +i . The order parame- +ters can be obtained from the disorder-averaged gen- +erating functional as derivatives with respect to the +fields ψi(t) and/or hi(t), evaluated at ψi(t) ≡ 0 and +hi(t) ≡ 0. +The order parameters in Eqs. (6) are determined +self-consistently from an effective process for a single +representative (‘mean field’) species. The procedure +to derive the effective equations is well-documented +[3, 4, 11, 31], therefore, we only report the final result +(a more detailed derivation can be found in the SM). +The effective single-species process for the model is +given by +˙x(t) = x(t) +� +1 − ux(t) − α +� t +0 +dt′[cG(I − G)−1 ++Γ(1 − c)G](t, t′)x(t′) − η(t) +� +, +(7) + +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +u +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +α +a) c = 1 +Divergence +Fixed point +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +u +b) c = 0.5, Γ = 1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +u +c) c = 0.5, Γ = −0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +FIG. 1. Fixed points and divergence in the model with Hebbian disorder. Each panel illustrates the behaviour of the +model for different choices of c and Γ. The heatmap indicates the fraction of samples that converge to a fixed point +in numerical integration of the gLVE. The criteria for the identification of convergence or divergence are described in +Appendix A. The dashed lines are theoretical predictions for the onset of divergence (see Sec. III C). +where I is the identity operator and η(t) is coloured +Gaussian noise with zero mean and correlations in +time, given by +⟨η(t)η(t′)⟩ = α[c(I − G)−1C(I − GT )−1 ++(1 − c)C](t, t′). +(8) +The order parameters in Eqs. (6) are to be obtained +self-consistently from the following expressions, +M(t) = ⟨x(t)⟩∗, +C(t, t′) = ⟨x(t)x(t′)⟩∗, +G(t, t′) = +δ +δη(t′)⟨x(t)⟩∗, +(9) +where the average ⟨· · · ⟩∗ is performed over realisa- +tions of the process in Eq. (7). Eqs. (7)-(9) form a +closed system and have to be solved self-consistently. +B. +Fixed point analysis +There is no realistic prospect for a general analyti- +cal solution of the effective dynamics in Eq. (7). One +alternative is to use Monte-Carlo methods to con- +struct sample paths for the effective process and so- +lutions for the dynamic order parameters. +For ex- +ample via the Eissfeller-Opper procedure [32], or us- +ing the more recent approach in [17]. +The latter +reference explicitly discusses applications to random +Lotka-Volterra systems with Gaussian disorder. +Here we will instead follow [11, 15, 29] and focus on +analytical solutions in the parameter regime in which +the dynamics approach stable fixed points. This is +motivated by observations from the numerical inte- +gration of Eq. (1). We find that, for certain parame- +ters, the system tends to a unique fixed point, which +is independent of initial conditions. Fig. 1 shows ex- +amples of parameter regions in which this is the case. +Broadly speaking, we observe two different types of +behaviour: (i) the population densities converge to a +fixed point, or (ii) they diverge. These types of be- +haviour occur in different regions of parameter space +(Fig. 1). There is a thin boundary between the two re- +gions where other behaviour (e.g. periodic behaviour +or persistent irregular motion) can appear, as evi- +denced by the occasional green or light blue pixel in +Fig. 1. We attribute this to the fact that the system +size N is necessarily finite in numerical experiments, +and we expect that this behaviour will become in- +creasingly more rare as N → ∞. +We will thus assume that each path in the ensemble +of trajectories of the effective process eventually ar- +rives at a unique fixed point, x = limt→∞ x(t). Each +realisation of the noise variable η(t) in Eq. (9) also ap- +proaches a stationary value η. We note that x and η +will be random variables, differing across realisations +of the effective dynamics. We can then write +M = ⟨x⟩∗, +G(τ) = lim +t→∞ G(t + τ, t), +q = lim +t→∞ C(t + τ, t) = ⟨x2⟩∗. +(10) +These relations can be understood as follows: if all +realisations of the effective dynamics approach sta- +tionary values then M(t) will approach a constant, +given by ⟨x⟩∗. Furthermore, we assume that the re- +sponse function G(t, t′) becomes time-translation in- +variant for large t, i.e. G(t, t′) = G(t − t′). Causal- +ity implies that G(t − t′) = 0 for t < t′. +Finally, +given that all trajectories of the effective dynam- +ics approach fixed points, the correlation function +C loses all time dependence and so we have writ- +ten C(t, t′) = q. This is consistent with Eq. (8), the +noise variables η(t) also approach a random but time- +independent value for all realisations. The mean of +the random variable η is zero and using Eq. (8), its + +5 +variance is given by +⟨η2⟩ = αq +� +c +(1 − χ)2 + 1 − c +� +, +(11) +where +χ = +� ∞ +0 +dτ G(τ). +(12) +From now on, we will write η = √q Σz, where z is a +standard Gaussian random variable, and +Σ2 = α +� +c +(1 − χ)2 + 1 − c +� +. +(13) +Setting the time derivative on the left-hand side of +Eq. (7) to zero, and using Eqs. (10)-(12) we find +x +� +1 − ux − √qΣz − αx +� +c +χ +1 − χ + Γ(1 − c)χ +�� += 0. +(14) +For a given value of z this is to be solved for x, +subject to the constraint that abundances are non- +negative, i.e, x(z) ≥ 0. Irrespective of the value of +z, Eq. (15) always has the solution x(z) = 0. Addi- +tionally, a second non-negative solution is possible for +some values of z. As we will confirm in simulations, +the physically meaningful solution is given by +x = max +� +� +�0, +1 − √qΣz +u + α +� +c +χ +1−χ + Γ(1 − c)χ +� +� +� +� . +(15) +The denominator in Eq. (15) always comes out posi- +tive. Therefore, we have the solution x(z) > 0 when +z < ∆, and x(z) = 0 when z ≥ ∆, with +∆ = +1 +Σ√q . +(16) +Given that z is a Gaussian random variable, the abun- +dances of extant species at the fixed point therefore +follow a clipped Gaussian distribution. +Using this fixed point ansatz, the relations for the +order parameters in Eq. (10) can be written in the +following form [14, 15] +M = +� ∆ +−∞ +Dz x(z), +χ = +1 +√qΣ +� ∆ +−∞ +Dz ∂x(z) +∂z +, +q = +� ∆ +−∞ +Dz x(z)2, +(17) +where Dz = +dz +√ +2πe− z2 +2 . It is now convenient to intro- +duce the following functions +fn(∆) := +� ∆ +−∞ +Dz (∆ − z)n, +(18) +for n = 0, 1, 2. We then find from Eqs. (17) +−χ +� +u + α +� +c +χ +1 − χ + Γ(1 − c)χ +�� += f0(∆), +M +u + α +� +c +χ +1−χ + Γ(1 − c)χ +� +� +αq +� +c +(1−χ)2 + (1 − c) +� += f1(∆), +� +u + α +� +c +χ +1−χ + Γ(1 − c)χ +��2 +α +� +c +(1−χ)2 + (1 − c) +� += f2(∆). +(19) +Eqs. (19) together with Eq. (16) form a closed set for +the set of unknowns q, χ, M and ∆, which is to be +solved as a function of the model parameters u, c, Γ +and α. +Recalling that x(z) > 0 if, and only if, z < ∆ we +identify f0(∆) as the fraction of surviving species, +φ ≡ f0(∆) = +� ∆ +−∞ +Dz. +(20) +Eqs. +(19) can be solved parametrically. +We fix +u, c, Γ and ∆ and then solve for the set of χ, q, M and +α. +In detail, we find the following cubic equation for +χ, valid for c < 1, +0 =f0(c(Γ − 1)f0 − Γf0 − f2) ++ χ +� +f 2 +0 [c + 2Γ − 2cΓ] + 2f0f2[1 − c] − f2u +� ++ χ2(c − 1) +� +Γf 2 +0 + f0f2 − 2f2u +� ++ χ3uf2(c − 1). +(21) +Further, we have from Eqs. (19), +M = χ +f 2 +1 +f0(f0 − f2), +q = χ2 +� +f1 +f2 − f0 +�2 f2 +f 2 +0 +, +α = f 2 +0 +f2 +1 +χ2(1 − c + +c +(1−χ)2 ), +(22) +where the fn are to be evaluated at ∆. +The relations in Eqs. (21) and (22) are also valid +for c = 1, and can then be simplified as outlined in +Appendix B. +The validity of the predictions from Eqs. (21) and +(22) is confirmed by direct numerical integration of +the gLVE in Fig. 2. +C. +Stability analysis +1. +Diverging abundance +Model with c < 1. The first and second relations +in Eqs. (22) indicate that the order parameters M + +6 +0 +1 +2 +3 +4 +α +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +φ +a) +u = 0.3 +u = 1 +u = 2 +0 +1 +2 +3 +4 +α +0 +1 +2 +3 +4 +5 +6 +M +b) +0 +1 +2 +3 +4 +α +0 +2 +4 +6 +8 +10 +Var[x] +c) +FIG. 2. Test of analytical predictions for the order parameters against numerical simulations. The figure shows the +fraction of surviving species φ, the mean abundance M, and the variance of abundances (q − M 2) as a function of +the model parameter α (where P = αcN is number of traits each species in the original system possesses). Lines are +from the theory, derived in Eqs. (21) and (22), markers from numerical integration of the gLVE (N = 1000, tmax = 30, +averaged over 10 realisations of the disorder). Remaining model parameters are c = 0.5, and Γ = 0.3. Vertical dashed +lines indicate the onset of divergence as determined from the theory in Sec. III C. +and q both diverge in the system with c < 1 when +f0(∆) = f2(∆). The latter implies ∆ = 0. The value +of α for which this occurs can (for a given choice of +c, u and Γ) be obtained from the third relation in +Eqs. (22), with χ being the relevant root of Eq. (21). +Using Eq. (21) the susceptibility χ is found to remain +finite at the transition. We note that f0(∆) > 0 for +all relevant values of ∆. +Model with c = 1. +The fully connected system +also shows two types of divergences: (i) The quan- +tities M and q both diverge when f0(∆) = f2(∆), see +Eqs. (B1). The susceptibility then remains finite; (ii) +Eqs. (B1) further indicate, that M and q also diverge +in the model with c = 1 when f 2 +0 (∆) = uf2(∆). This +latter condition results in α = u. From Eqs. (B1) the +susceptibility χ is then seen to diverge as well (the di- +vergences of M, q, and χ take place simultaneously). +We note that the divergencies resulting from +f0 = f2 and f 2 +0 = uf2 can take place at different lo- +cations in parameter space for the model with c = 1. +If this is the case, and starting in the stable phase, +the divergence that occurs first will determine the +loss of stability in the fully connected system. For +u < 1/2 the transition of type (ii) takes place first as +α is increased (M, q, and χ diverge), and for u > 1/2 +the transition of type (i) is instead observed (q, M +diverge, χ remains finite). +2. +Linear instability +The system also shows a linear instability which +can be identified using the procedure established in +[11, 33]. +We write x(t) = x + y(t) and η(t) = +√q Σz +ζ(t), where y(t) and ζ(t) are small perturba- +tions about the fixed point of the trajectories of the +effective process in Eq. (7). Expanding to first order +in these perturbations we find that +˙y(t) =x +� +−uy(t) − α +� t +−∞ +dt′ K (t, t′) y (t′) − ζ(t) +� ++ y(t) +� +1 − ux − αx +� t +−∞ +dt′ K (t, t′) − √q Σz +� +, +(23) +with K(t, t′) = [cG(I − G)−1 + Γ(1 − c)G](t, t′). We +also have the self-consistency relation +⟨ζ(t)ζ (t′)⟩ = α +� +c(I − G)−1D(I − GT )−1 ++ (1 − c)D +� +(t, t′) , +(24) +where D (t, t′) = ⟨y(t)y (t′)⟩∗. +When x = 0, Eq. (23) becomes +˙y(t) = y(t) (1 − √q Σz) . +(25) +Eq. (15), together with the observation that the de- +nominator in this equation is strictly positive, implies +that 1 − √q Σz < 0 when x = 0. This allows us to +conclude that perturbations on extinct species decay, +and do not contribute to any linear instability. +For fixed points x > 0 we find from Eqs. (15) and +(23) that +˙y(t) = −x +� +uy(t) + α +� t +−∞ +dt′ K (t − t′) y (t′) + ζ(t) +� +. +(26) +To identify the onset of linear instability we follow +[11, 33]. +We move to Fourier space, writing ω for +the variable conjugate to time t, and using tildes to +indicate Fourier transforms. +Focusing on the mode with ω = 0 and following +steps similar to those in [11, 15, 33] we then find from + +7 +Eq. (26) +� +|˜y(0)|2� += +� +φ−1 +� +u + αc +χ +1 − χ + αΓ(1 − c)χ +�2 +− α +� +c +(1 − χ)2 + 1 − c +��−1 +. (27) +The left-hand side is manifestly non-negative, so that +a change of sign of the expression inside the square +bracket on the right-hand side indicates an inconsis- +tency (and divergence of +� +|˜y(0)|2� +). Using Eqs. (19) +this is shown to occur when +α +� +c +(1 − χ)2 + 1 − c +� +(f0 − f2) = 0. +(28) +For c < 1 the expression in the square brackets is +never zero. This leaves us with the condition f0 = f2, +which is the same we obtained for the divergence of +M and q. If c = 1, the term in the square bracket is +zero if χ → ∞, which using Eq. (B1) we can write as +f 2 +0 − uf2 = 0. +From this, we conclude that in our model the +linear instability is always accompanied by the in- +stability with diverging mean abundance. +This is +markedly different from the behaviour of the gLVE +model with Gaussian random interactions. +In this +Gaussian model there are instances where the linear +instability sets in as the variance of interactions is +increased, but where abundances remain finite and +the divergence only occurs at a later point at even +higher variance of the interactions. This leads to a +phase with multiple attractors between the two tran- +sitions [14, 16, 30]. Our analysis indicates that the +model with Hebbian couplings does not have such a +multiple-attractor regime. +IV. +PHASE DIAGRAM AND FURTHER +BEHAVIOUR OF THE MODEL +A. +Phase diagram for the fully connected +system (c = 1) +The phase diagram of the fully connected model +is shown in Fig. 3(a). +We recall that, for c = 1, +the only model parameters are the self-interaction co- +efficient u and the ratio of the number of traits to +the number of interspecies interactions in the origi- +nal pool (α = P/cN). For a fixed value of α, the sys- +tem shows a unique stable fixed point for u > uc(α), +where uc(α) marks the onset of instability. The line in +Fig. 3(a), obtained from Eq. (B1), shows the phase +boundary between the stable and unstable regions. +At this boundary M and q diverge, and if uc < 1/2 +we also observe a divergence of χ. +The two types of trajectory in the stable and di- +vergent phases are illustrated in the right panel of +Fig. 3(b). In the stable phase the system reaches a +fixed point, for any one realisation of the interaction +matrix (two examples are illustrated in green and red +respectively). +Fig. 3(b) also shows two examples in which the +species abundances diverge (blue and orange). The +divergence occurs at a finite time. We will discuss +this further in Sec. IV C. +B. +Phase diagram for connectivity c < 1 +Fig. 4 shows how the phase diagram for the system +with c < 1 depends on the connectivity c and the +symmetry parameter Γ. In all cases there is a sin- +gle phase boundary, where the divergence of M and +q and the onset of linear instabilities coincide. This +phase boundary separates a region where trajectories +converge to a single globally stable fixed point (phase +to the right of the line), from a region where trajecto- +ries are unbounded and diverge in finite time (phase +on the left). +The phase diagrams in Figs. 3 and 4 show that +the system is in the stable phase for small values +of α (i.e. +a small number of traits relative to the +number of species in the initial pool), or large val- +ues of u (i.e. large negative self-interaction). This +is the consequence of two competing effects, the self- +interaction (parametrised by u) which stabilises the +system, and the interaction between species (induced +by competition of similar species) which promotes in- +stability. When u is large and/or α is small, the stabi- +lizing effect of the intra-species interaction dominates +over the interactions across species. In the extreme +limit α = 0 (no interaction between different species), +each abundance follows a separate logistic equation, +˙xi = xi(1−uxi), and converges to xi = 1/u. When α +is small but non-zero, the system consists of weakly +interacting species. The effect of the interactions be- +tween species is then a small perturbation to the lo- +gistic behaviour of individual species, and does not +change the convergence to a fixed point. This can be +confirmed from Eqs. (21) and (22) by taking the limit +α → 0, which results in all species surviving with +fixed point abundance x∗ +i = 1/u (φ → 1, M → +1 +u, +and Var[x] → 0). +A similar result is obtained for +u ≫ 1 at fixed value of α. +Conversely, for low values of u or large values of +α the system is unstable. In this situation the sta- +bilising self-interaction is not sufficient to overcome +the destabilising effect of the random interactions be- +tween species. +The most interesting behaviour takes place at the +phase boundary, where the effect of the intraspe- +cific and interspecific nonlinearities are of comparable +magnitude. From Eqs. (21) and (22) we can conclude +that φ → 1/2, and M ∼ (αc − α)−1 as the system +approaches the instability (from the stable phase). + +8 +0.00 +0.25 +0.50 +0.75 +1.00 +u +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +α +a) +0 +2 +4 +6 +8 +10 +t +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +x(t) +b) +FIG. 3. Phase behaviour of the fully connected model (c = 1). Panel (a): Phase diagram for the model with c = 1, +the only model parameters are then u and α. The system is stable to the right of the lines. At the dot-dashed line +(u < 1/2) q, M and χ all diverge, and at the dashed line M and q diverge, but χ remains finite. Panel (b): Illustration +of the behaviour of the abundances of individual species in the two different phases (convergence to fixed point shown +in green and red, diverging abundances in orange and blue). +0.0 +0.5 +1.0 +1.5 +2.0 +u +0.0 +0.5 +1.0 +1.5 +2.0 +α +a) +Γ = −0.5 +c = 1 +c = 0.33 +c = 0.5 +c = 0.67 +0.0 +0.5 +1.0 +1.5 +2.0 +u +0.0 +0.5 +1.0 +1.5 +2.0 +b) +Γ = 0 +c = 1 +c = 0.25 +c = 0.5 +c = 0.75 +c = 0.9 +0.0 +0.5 +1.0 +1.5 +2.0 +u +0.0 +0.5 +1.0 +1.5 +2.0 +c) +Γ = 1 +c = 1 +c = 0.25 +c = 0.5 +c = 0.75 +c = 0.9 +FIG. 4. Phase diagram for different choices of the connectivity c, and the symmetry parameter Γ. The coloured lines +in each panel indicate where the linear instability occurs. The instability coincides with the divergence of M and q. +The system is stable to the right of the line, abundances diverge on the left. +Further details can be found in Appendix C. +We further note that decreasing the value of the +symmetry parameter Γ, increases the range of the +stable region in the phase diagrams in Fig. 4. This +is similar to the effect of increasing the fraction of +predator-prey interactions in Lotka–Volterra models +with Gaussian interactions [10, 14, 15]. Indeed, the +effect of a reduction of Γ is to increase the fraction of +species pairs i, j with cij = 1 and cji = 0, that is the +proportion of uni-directional interactions. +Interestingly, the effect of varying the ‘connectance’ +c is not straightforward. As can be seen in Fig. 4 an +increased connectivity can, depending on the other +model parameters, turn a previously stable system +into an unstable one, or vice versa, stabilise a previ- +ously unstable system. +C. +Finite-time divergence of the mean +abundance +As mentioned earlier, the divergence of the abun- +dances in the divergent phase occurs at finite time. +This has previously been reported in the model +with Gaussian interactions [17], and can be justified +heuristically from the Lotka-Volterra equations. In- +deed, Eq. (1) has a second order non-linearity in the +abundances xi. This can lead to dynamics of the form +˙x ∼ x2, which in turn implies a solution of the form +x(t) = (c − t)−1, where c is an integration constant. +This results in a divergence at finite time. +Fig. 5 shows the time, tdiv, at which the divergence +occurs for different choices of the model parameters. +This time grows as one approaches the stability line +(from inside the unstable phase). When the stability +line is crossed (into the stable phase), the time-to- +divergence diverges itself (tdiv → ∞), i.e. the diver- +gence no longer occurs. Results from the numerical +integration of the gLVE suggest that the divergence of + +9 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +u +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +α +a) c = 1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +u +b) c = 0.5, Γ = 1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +u +c) c = 0.5, Γ = −0.5 +10−2 +10−1 +100 +FIG. 5. Finite-time divergence of abundances. The heatmaps indicate the time, tdiv, at which abundances diverge, for +initial conditions xi(0) = 1. Data is obtained from numerical integration of the gLVE. The dotted line is the phase +boundary predicted by the theory. To the right of the phase boundary the system is in the stable phase, so that no +divergence occurs. +the abundances is of the form M ∼ (tdiv − t)ν, where +ν ≈ 1, as shown in Fig. 6. This behaviour appears to +be independent of initial conditions, the values of the +parameters u and α, and the initial number of species +N. +10−13 +10−11 +10−9 +10−7 +10−5 +10−3 +10−1 +101 +t∞ − t +10−2 +101 +104 +107 +1010 +1013 +M +α = 2 +α = 5 +α = 10 +FIG. 6. Divergence of M for initial conditions uniformly +distributed in (0, 10−3) and parameters N = 1000, u = 1, +and different values of α. +The dotted black line corre- +sponds to (t∞ −t)−1. The deviation from M ∼ (t∞ −t)−1 +close to the divergence is attributed to numerical error. +V. +REDUCED INTERACTION MATRIX +AND ITS EIGENVALUE SPECTRUM +Ref. [25] recently established a close connection be- +tween different instabilities in the Gaussian random +Lotka–Volterra model and the eigenvalue spectrum of +the interaction matrix of the surviving species. More +specifically, the spectrum of this reduced interaction +matrix is composed of a bulk region and a potential +outlier eigenvalue. As parameters are changed (start- +ing from within the stable phase) either the bulk spec- +trum or the outlier eigenvalue can cross into the right +half of the complex plane. In the Gaussian model, the +crossing of the outlier is associated with a transition +marked by the divergence of abundances, and a cross- +ing of the bulk with a linear instability. +In this section we explore in numerical simulations +how the different transitions in the gLVE model with +Hebbian interactions relate to the eigenvalue spec- +trum of the matrix of interactions between surviving +species. +A. +Spectrum of the original interaction matrix +Before we discuss spectra of the reduced interac- +tion matrix for the Hebbian model, we make a few +remarks on the initial interaction matrix αij = cijJij +among all species. +Throughout this section we set +the diagonal elements of this matrix to zero, the only +effect of self-interaction (the term −uxi) is a simple +shift of this spectrum. In the large-N limit the cen- +tral limit theorem applies to Jij = − 1 +cN +�αcN +µ=1 ξµ +i ξµ +j , +so each off-diagonal entry αij of the interaction ma- +trix is either a Gaussian random variable (if cij = 1) +or equals zero (if cij = 0). The variance of αij is +Var(αij) = +1 +cN 2 +αcN +� +µ,µ′ +� +ξµ +i ξµ′ +i ξµ +j ξµ′ +j +� += α +N . +(29) +Calculating the correlations between pairs of ele- +ments we obtain +Corr[αij, αnm] = ⟨αijαnm⟩ − ⟨αij⟩⟨αnm⟩ +Var(αij) += +� +� +� +� +� +Γ(1 − c) + c +(i, j) = (m, n) +1 +(i, j) = (n, m) +0 +else +, +(30) + +10 +−6 +−4 +−2 +0 +2 +4 +6 +ℜ [λ] +−6 +−4 +−2 +0 +2 +4 +6 +ℑ [λ] +α = 35 +α = 3.5 +α = 0.35 +α = 0.03 +FIG. 7. +Examples of the eigenvalue spectrum of the +original interaction matrix. +The dashed lines are the +na¨ıve predictions of Eq. (31). +Model parameters are +c = 0.4, Γ = c/(c − 1) and N = 5000. +where we have used Eq. +(2) and the fact that Jij +is symmetric. This means that only diagonally op- +posed pairs of elements are correlated, and that their +correlation is determined by both, Γ and c. +Based on a theory that only takes into account cor- +relations between diagonally opposed matrix entries, +one might then expect an elliptic spectrum [34], with +support given by the ellipse +� +x +√α(1 + τ) +�2 ++ +� +y +√α(1 − τ) +�2 += 1, +(31) +with τ = Γ(1 − c) + c. +However, as illustrated in +Fig. 7, this is an approximation to the true spectrum +at best for large values of α. For intermediate values +of α (an example is shown in orange in the figure) +the eigenvalue spectrum appears to have a triangu- +lar shape, and for small values of α (shown in green) +the spectrum becomes even more skewed, and even- +tually appears to consist of two separate components +(example shown in red). While we cannot fully ex- +clude finite-size effects (the spectra in Fig. 7 are for +N = 5000), we believe that the deviations from an el- +liptical spectrum in Eq. (31) are due to higher-order +correlations between entries of the interaction ma- +trix. +For example, it has been shown in Ref. [35] +that cyclic correlations can result in eigenvalue spec- +tra with shapes similar to the ones in Fig. 7. +−10 +−8 +−6 +−4 +−2 +0 +ℜ [λ] +−4 +−2 +0 +2 +4 +ℑ [λ] +α/αc = 0.90 +α/αc = 0.75 +α/αc = 0.50 +α/αc = 0.20 +FIG. 8. Eigenvalue spectrum of the reduced interaction +matrix for u = 5, c = 0.5, Γ = c/(c − 1), for different +choices of the model parameter α. The vertical dashed +lines indicate the real part of the right-most eigenvalue. +B. +Eigenvalues of the reduced interaction +matrix +We now conclude the analysis of the model with a +numerical study of the spectra of the reduced interac- +tion matrix, that is, the interaction matrix between +species that survive at the fixed points of the gLVE. +Fig. 8 shows the spectra of this matrix for the case +c < 1, and for a choice of Γ less than one. This means +that the initial interaction matrix is not symmetric. +The reduced matrix is not symmetric either, and as a +consequence its eigenvalues will generally be complex. +As seen in Fig. 8, the spectrum is not elliptic, and we +have found no evidence of an outlier eigenvalue in this +scenario (c < 1). In the figure we have fixed u, and +varied α. The data suggests that the phase transition +at α = αc(u) coincides with the point at which the +right-most bulk eigenvalue crosses the imaginary axis +into the right half-plane. +In Fig. 9 we study the fully connected system for +two different values of the self-interaction strength +u. The original interaction matrix in the fully con- +nected model is symmetric by construction, and so is +the reduced interaction matrix. As a consequence all +eigenvalues are real. +Panel (a) focuses on the case u > 1/2. We find no +signs of outlier eigenvalues, and again the data indi- +cates that the transition to instability occurs when +the leading bulk eigenvalue crosses into the positive +half of the real axis. +Panel (b) shows a scenario in which u < 1/2. In +contrast with the situation in (a), outlier an eigen- + +11 +−2.5 +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +λ +10−3 +10−2 +10−1 +100 +P(λ) +a) +α/αc=0.95 +α/αc=0.90 +α/αc=0.80 +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +λ +10−3 +10−2 +10−1 +100 +101 +b) +−0.10 +−0.08 +−0.06 +−0.04 +−0.02 +0.00 +FIG. 9. +Eigenvalue spectrum of the reduced interaction matrix in the fully connected system. +The matrices are +symmetric, and their eigenvalues are therefore real valued. Panel (a) is for u = 0.7, panel (b) for u = 0.3. +value now becomes apparent, and the transition to +instability in the gLVE at α = αc(u) now appears to +coincide with the point at which the outlier becomes +positive. +The connection between the different types of tran- +sition and the behaviour of the spectrum of the re- +duced interaction matrix is shown in Table I. We re- +call that the mean abundance and the second moment +of the abundances diverge at all transitions, and that +the onset of the linear instability always coincides +with the point of diverging abundances. There are +thus only two types of transition, one in which the +susceptibility remains finite (χ < ∞), and another +for which it diverges (χ → ∞). The table indicates +that the former transition (χ finite) appears to co- +incide with the bulk spectrum of the reduced matrix +crossing into the right half of the complex plane. The +transition at which χ → ∞ (along with the diver- +gences of M and q) on the other hand seems to be +seen when the outlier eigenvalue of the reduced ma- +trix in the fully connected system reaches the origin. +We stress that these are numerical observations, +and that these findings should therefore be seen +mostly as conjectures at this point. +In principle, +the spectrum of the reduced interaction matrix can +likely be calculated in the Hebbian model, adapting +c < 1 +q, M diverge +bulk spectrum +χ remains finite +crosses axis +c = 1 +u > 1/2 +q, M diverge +bulk spectrum +χ remains finite +crosses axis +u < 1/2 +q, M, χ +outlier eigenvalue +all diverge +crosses axis +(at u = α) +TABLE I. Types of phase transition in the gLVE model +with Hebbian interactions. The table summarises the dif- +ferent transitions, giving details about the nature of the +divergence at the transition, and the associated behaviour +of the spectrum of the reduced interaction matrix. +the method used in Ref. [25]. However this involves +a substantial calculation and is beyond the scope of +the current paper. +VI. +DISCUSSION +To summarise, we have carried out a generating +functional analysis of a random generalised Lotka– +Volterra system with competitive exclusion. Species +interactions in the model are governed by Hebbian +couplings subject to mild dilution (the remaining con- +nectivity is still extensive). We have computed the +statistics of surviving species in the stable fixed point +phase, and we have analytically determined the on- +set of instability. Similar to the gLVE with Gaussian +interactions, asymmetry in the connectivity matrix +promotes stability. +That is to say, the system be- +comes more stable when there is a larger fraction of +unidirectional interactions (cij = 1, but cji = 0). In +contrast with the Gaussian model, the linear instabil- +ity against small perturbations cannot be separated +from an instability at which species abundances di- +verge. As a consequence, there is no phase with mul- +tiple stable fixed points in the Hebbian model. De- +spite some common features, the statistical mechan- +ics of the Gaussian and Hebbian models are therefore +rather distinct. +Our analysis shows that there are two types of +transitions to divergent abundances in the Hebbian +model, one in which the integrated response χ re- +mains finite, and another in which χ diverges. This +raises interesting questions about the exact nature +of memory onset in the system (a diverging inte- +grated response indicates persistent memory of per- +turbations). Future work could focus on the precise +shape of the response function, where the numeri- +cal methods in [17] might prove particularly useful. +Given that the fully connected system has symmet- +ric couplings it would also be interesting to see how +crossing each of the different types of transition af- +fects the energy landscape. A natural approach here + +12 +might be the replica method and suitable levels of +replica symmetry breaking [16, 18]. +Numerical simulations provide evidence that the +transition at which the integrate response remains fi- +nite (χ < ∞) is associated with the bulk spectrum of +the reduced interaction matrix (the matrix of interac- +tions between extant species) crossing the axis. The +transition at which χ diverges on the other hand ap- +pears to be signalled by an outlier eigenvalue crossing +the imaginary axis. +These findings in simulations reinforce the in- +triguing analytical result obtained recently in [25]. +Namely, the eigenvalues of the interaction matrix in +the community of surviving species can be used to de- +cide the stability of feasible equilibria, that is to say +fixed points with non-negative species abundances. +In the traditional approach to ecosystem stability by +Robert May [9], based on the eigenvalue spectra of +random matrices, no actual dynamics is specified, and +the feasibility of the assumed equilibria therefore re- +mains unclear. +Any fixed point of the generalised +Lotka–Volterra model on the contrary is feasible by +construction. The study of the spectra of reduced in- +teraction matrices resulting from Lotka–Volterra dy- +namics can therefore contribute to establishing how +May’s approach can be adapted to include feasible +equilibria. +On a broader level, our study highlights two com- +mon facets of work on the statistical physics of com- +plex systems, which were also seen for example 30-40 +years ago when physicists studied neural networks, or +15-20 years ago when a number of physicists worked +on the Minority Game. On the one hand, tools from +physics can make a difference for problems in other +disciplines. In our system (and other models of com- +plex ecosystems more generally) this is the study +of feasible equilibria with methods from spin glass +physics. At the same time, studying problems aris- +ing in other areas can reveal new types of physics and +complexity, which one would perhaps not find within +the strict boundaries of traditional physics. In our +case these are the different types of phase transition +in the generalised Lotka–Volterra model. We think +that this mutually beneficial relation of physics and +adjacent disciplines is what makes the field of com- +plex systems particularly attractive. +ACKNOWLEDGEMENTS +Partial financial support has been received from the +Agencia Estatal de Investigaci´on and Fondo Europeo +de Desarrollo Regional (FEDER, UE) under project +APASOS +(PID2021-122256NB-C21/PID2021- +122256NB-C22), and the Maria de Maeztu program +for Units of Excellence, CEX2021-001164-M funded +by MCIN/AEI/10.13039/501100011033. +[1] S. F. Edwards and P. W. Anderson, Journal of +Physics F: Metal Physics 5, 965 (1975). +[2] M. M´ezard, G. Parisi, and M. A. Virasoro, Spin glass +theory and beyond: An Introduction to the Replica +Method and Its Applications, Vol. 9 (World Scientific +Publishing Company, 1987). +[3] A. C. C. 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Stein, Physical review letters 60, 1895 (1988). +[35] P. V. Aceituno, T. Rogers, and H. Schomerus, Phys- +ical Review E 100, 010302 (2019). +Appendix A: Details of numerical procedures +For the numerical integration of the gLVE (1) we +used scypi’s solve ivp function, which makes use of +a RK45 integration scheme. +To determine the fraction of survivors we counted +the number of species above a threshold abundance of +10−4. There are two sources of systematic error asso- +ciated to this method. The most relevant is the over- +estimation of the fraction of survivors if the system +is not close enough to the equilibrium configuration. +This can be addressed by extending the simulation +time. +The second source of error comes from the fact +there is no ‘gap’ between zero and the lowest non- +zero abundance [see the clipped Gaussian Eq. (15)]. +This implies that for any value of the threshold there +is a nonzero probability of finding equilibrium abun- +dances below it. A possible solution, making use of +the facts that in simulations N is finite and that we +know the abundance distribution analytically, is to +choose the threshold value so that the expected num- +ber of surviving species with an abundance below the +threshold is small (e.g. smaller than one). The chosen +value of 10−4 provides good results in the parameter +ranges we have explored. +As part of our measurements it is necessary to de- +tect divergences in the species abundances. To de- +tect this divergence we have used the failure of the +integration method as an indicator. Indeed, as the +abundances grow with each iteration, so does the es- +timated error used to adapt the step size. This causes +the solver to lower the time-step until it eventually +drops below machine precision, at which point inte- +gration is stopped. The agreement of the theoretical +and numerical phase boundaries in Fig. 5 confirm the +validity of this method. +Appendix B: Order parameters at the fixed point +of the fully connected system +The parametric solution for the order parameters of +the fully connected system (c = 1) in the fixed point +phase can be obtained from the following relations, +α = (u + f0)2f2 +(f0 + f2)2 , +M = +f 2 +1 (f0 + f2) +(f0 − f2)(f 2 +0 − uf2), +q = +f 2 +1 f2(f0 + f2)2 +(f0 − f2)2(f 2 +0 − uf2)2 , +χ = f0(f0 + f2) +f 2 +0 − uf2 +. +(B1) +The functions fn(∆) on the right provide α, M, q, χ +as implicit functions of ∆. +Keeping Eq. (18) in mind one sees that M and q +can only diverge if f0 = f2 or f 2 +0 = uf2, as indicated +in the main text. +Appendix C: Limiting behaviour of the order +parameters +1. +Limit α → 0 +The weak interaction limit α → 0 corresponds to +∆ → ∞. [This can be seen from Eq. (22), keeping in +mind that f0 > 0]. From the definition in Eqs. (18) +we have, in this limit, +f0(∆) = φ = 1 + e− ∆2 +2 +√ +2π +� +− 1 +∆ + O +� +∆−3�� +(C1) +f1(∆) = ∆ + e− ∆2 +2 +√ +2π +� 1 +∆2 + O +� +∆−4�� +(C2) +f2(∆) = 1 + ∆2 + e− ∆2 +2 +√ +2π +� +− 2 +∆3 + O +� +∆−5�� +. (C3) +Next, we compute the value of χ. Since only f0 and +f2 are present in Eq. (21), and only f2 is divergent, +we group in terms proportional to f2 to obtain +0 = 1−χ [2(1 − c) − u]−χ2(c−1)(1−2u)−χ3u(c−1), +which we can check always has χ = −1/u as its +negative solution. Finally, from Eq. (22) we obtain +M = 1/u and Var[x] = 0. +As expected, these values are independent of c and +Γ, since in the limit of absent interactions Eq. (4) +becomes a set of independent logistic maps. In this +case all species survive with abundance 1/u, which is +what we obtain. + +14 +2. +Limit α → αc +There are two different scenarios for the limit α → +αc (where αc is the location of the phase transition). +(1) If c = 1, u < 1/2 the divergence takes place as +uf2 → f 2 +0 . Using Eq. (B1) we see that αc = u, and +αc − α = +u − f2 +(f0 + f2)2 (f 2 +0 − uf2). +(C4) +This implies that both χ and M diverge as (αc−α)−1, +and q diverges as (αc − α)−2. +(2) The other type of transition occurs when f0 → +f2, which implies ∆ → 0. In this case χ remains finite +and we have near ∆ = 0, +f0(∆) = 1 +2 + +∆ +√ +2π − +∆3 +6 +√ +2π + O +� +∆4� +f1(∆) = +1 +√ +2π + ∆ +2 + +∆2 +2 +√ +2π + O +� +∆4� +f2(∆) = 1 +2 + +� +2 +π ∆ + ∆2 +2 + +∆3 +3 +√ +2π + O +� +∆4� +. +(C5) +We note from these expansions that f0 − f2 ∝ ∆ +as ∆ → 0. Using Eq. (22) we then find M ∼ ∆−1. +Similarly, we have αc − α ∝ ∆ [this can be seen from +expanding f 2 +0 /f2 in the third relation in Eq. (22)], so +we can conclude that M ∼ (α − αc)−1. These results +are consistent with simulations (see for example Fig. +2). + +S1 +SUPPLEMENTAL MATERIAL +Competitive exclusion and Hebbian couplings in random generalised +Lotka–Volterra systems +Enrique Rozas Garcia1,2, Mark J. Crumpton3,4, Tobias Galla1,4 +1 Instituto de F´ısica Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat Illes +Balears, E-07122 Palma de Mallorca, Spain +2 Department of Physics, Gothenburg University, 41296 Gothenburg, Sweden +3 Department of Mathematics, King’s College London, London WC2R 2LS, United Kingdom +4 Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, +Manchester M13 9PL, UK +S1. +GENERATION OF RANDOMLY DILUTED INTERACTIONS +The dynamics studied in this work, of the form in Eq. (1), requires the generation of pairs of +identically distributed correlated Bernoulli random variables (cij, cji) as part of the simulations. +To do this we have to find their joint probability distribution. Let +p(cij = 1, cji = 1) = x, +p(cij = 1, cji = 0) = y, +p(cij = 0, cji = 1) = w, +p(cij = 0, cji = 0) = z. +(S1) +We can solve for x, y, w, z using the desired moments in Eq. (2) and the normalization condi- +tion, i.e. +⟨cij⟩ = x + y = c , +⟨cji⟩ = x + w = c , +(S2) +⟨cijcji⟩ = x = Γc(1 − c) + c2 , +x + y + w + z = 1 , +to obtain +p(cij = 1, cji = 1) = Γc(1 − c) + c2, +p(cij = 1, cji = 0) = p(cij = 0, cji = 1) = c − Γc(1 − c) − c2, +p(cij = 0, cji = 0) = 1 − 2c + Γc(1 − c) + c2. +(S3) +Since c ∈ [0, 1] and all probabilities need to be non-negative, we find that the values that +Γ can take are restricted, as shown in Fig. S1. Intuitively, since our variables are identically +distributed and can only take two possible values, negative correlations imply exclusion. For +example, it is impossible for both variables to have mean 1 if they are perfectly negatively +correlated, in fact, in that case one will have mean c and the other 1 − c, making c = 1/2 the +only possible choice (see Fig. S1). + +S2 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +c +FIG. S1. Shaded area indicates the possible pairs of c and Γ. +S2. +DETAILS OF GENERATING FUNCTIONAL ANALYSIS +A. +The generating functional +To study Lotka-Volterra equations of the form in Eq. (1), we will follow [11, 15] and use the +generating functional +Z[ψ(t)] = +� +D[x(t)] p[x(0)]ei � +i +� +dtψi(t)xi(t) +(S4) +N +� +i=1 +δ +� +d +dt ln xi − 1 + uxi − +� +i̸=j +cijJijxj + σζi(t) + hi(t) +� +, +where ψ is a source field. +We have introduced both a Gaussian white noise ζi(t) and a +perturbation field hi(t). These auxiliary fields will allow us to calculate magnitudes of interest +later on, but they do not play any role in the upcoming calculation and can be ignored for +the most part. +Writing the δ-functions in their integral representation (and absorbing any resulting factors +in the integration measure) we find +Z[ψ(t)] = +� +D[x(t), ˆx(t)]p[x(0)] ei � +i +� +dt ψi(t)xi(t)ei � +i +� +dt ˆxi[yi−1+uxi−� +j̸=i cijJijxj+σζi(t)+hi(t)] +(S5) +where we use the short-hand yi = d +dt ln xi. +Our objective is to manipulate the generating functional (S5) until we can identify a sim- +plified effective dynamics in terms of a single mean-field population x(t). +To do this we calculate the disorder average of Z. This consists of an average over both +random variables {cij} and {ξµ +i }: +Z[ψ(t)] = +� +D[x(t), ˆx(t)]p[x(0)] ei � +i +� +dt ψi(t)xi(t)ei � +i +� +dt ˆxi[yi−1+uxi+σζi(t)+hi(t)] +× e−i � +i +� +dt ˆxi[ +� +j̸=i cijJijxj], +(S6) +where the overbar denotes the disorder average. + +S3 +Once the disorder-averaged generating functional is known, we can calculate magnitudes of +interest by taking derivatives. For example, the mean abundance, correlation function, and +response function can be written as +M(t) = 1 +N +� +i +⟨xi(t)⟩ = −i +N +� +i +δZ +δψi(t) +���� +ψ(t)=0 +(S7) +C(t, t′) = 1 +N +� +i +⟨xi(t)xi(t′)⟩ = −1 +N +� +i +δ2Z +δψi(t)δψi(t′) +���� +ψ(t)=0 +(S8) +G(t, t′) = 1 +N +� +i +δ⟨xi(t)⟩ +δhi(t′) = −i +N +� +i +δ2Z +δψi(t)δhi(t′) +���� +ψ(t)=0 +. +(S9) +B. +Introduction of the order parameters +Before we explicitly carry out the disorder average it is convenient to introduce the following +order parameters +a(t) = 1 +N +� +i +xi(t) +k(t) = i 1 +N +� +i +ˆxi(t) +Q(t, t′) = 1 +N +� +i +xi(t)xi(t′) +K(t, t′)) = 1 +N +� +i +xi(t)ˆxi(t′) +L(t, t′) = 1 +N +� +i +ˆxi(t)ˆxi(t′). +(S10) +To introduce the order parameters into the generating functional (S6) we use δ-functions in +their exponential representation, and insert the following expressions (all equal to one), +1 = +� +D[ˆa(t), a(t)] eiN +� +dt ˆa(t)[a(t)− 1 +N +� +i xi(t)], +1 = +� +D[ˆk(t), k(t)] eiN +� +dt ˆk(t)[k(t)− i +N +� +i ˆxi(t)], +1 = +� +D[ ˆQ(t, t′), Q(t, t′)] eiN +� +dt dt′ ˆQ(t,t′)[Q(t,t′)− 1 +N +� +i xi(t)xi(t′)], +1 = +� +D[ ˆK(t, t′), K(t, t′)] eiN +� +dt dt′ ˆ +K(t,t′)[K(t,t′)− 1 +N +� +i xi(t)ˆxi(t′)], +1 = +� +D[ˆL(t, t′), L(t, t′)] eiN +� +dt dt′ ˆL(t,t′)[L(t,t′)− 1 +N +� +i ˆxi(t)ˆxi(t′)]. +(S11) +Relevant factors of 2π have here been absorbed into the measure. By introducing these + +S4 +expressions into (S6) we obtain +Z[ψ(t)] = +� +D[a, ˆa, k, ˆk, Q, ˆQ, K, ˆK, L, ˆL] eNΨ +× +� +D[x(t), ˆx(t)]p[x(0)] ei � +i +� +dt ψi(t)xi(t)ei � +i +� +dt ˆxi[yi−1+uxi+σζi(t)+hi(t)] +× e−i � +i +� +dt [ˆa(t)xi(t)+iˆk(t)ˆxi(t)]e−i � +i +� +dt dt′ [ ˆQ(t,t′)xi(t)xi(t′)+ ˆ +K(t,t′)xi(t)ˆxi(t′)+ˆL(t,t′)ˆxi(t)ˆxi(t′)] +× e−i � +i +� +dt ˆxi[ +� +j̸=i cijJijxj], +(S12) +where +Ψ[a, ˆa, k, ˆk, Q, ˆQ, K, ˆK, L, ˆL] =i +� +dt +� +ˆa(t)a(t) + ˆk(t)k(t) +� +(S13) ++ i +� +dt dt′ � +ˆQ(t, t′)Q(t, t′) + ˆK(t, t′)K(t, t′) + ˆL(t, t′)L(t, t′) +� +. +For later purposes we define +Ω[ˆa, ˆk, ˆQ, ˆK, ˆL] = 1 +N +� +i +log +� � +D[xi(t), ˆxi(t)]p[xi(0)] exp +� +i +� +dt ψi(t)xi(t) +� +× exp +� +i +� +dt ˆxi [yi − 1 + uxi + σζi(t) + hi(t)] +� +× exp +� +−i +� +dt [ˆa(t)xi(t) + iˆk(t)ˆxi(t)] +� +× exp +� +−i +� +dt dt′ [ ˆQ(t, t′)xi(t)xi(t′) + ˆK(t, t′)xi(t)ˆxi(t′) + ˆL(t, t′)ˆxi(t)ˆxi(t′)] +�� +. +(S14) +C. +Disorder average +We follow mainly [4, 31]. The only term in Eq. (S5) containing the disorder variables {cij} +and {ξµ +i } is +exp +� +−i +� +i>j +� +dt (ˆxicijJijxj + ˆxjcjiJjixi) +� +. +(S15) +To simplify the calculations we will only keeps the terms that are leading order in N, since +we eventually intend to take the thermodynamic limit N → ∞. To estimate the order of each +term we use their variance, thus +cij ∼ +� +⟨c2 +ij⟩ = √c ∼ O(N 0), +(S16) +and +⟨J2 +ij⟩ = +�� +1 +cN +αcN +� +µ=1 +ξµ +i ξµ +j +�2� += +1 +(cN)2 +αcN +� +µ,µ′=1 +⟨ξµ +i ξµ +j ξµ′ +i ξµ′ +j ⟩ = +1 +(cN)2 +αcN +� +µ,µ′=1 +δµ,µ′ = α +cN , +(S17) + +S5 +implying that +Jij ∼ O(N −1/2) . +(S18) +We first perform the average of (S15) over the {cij}. We begin by Taylor expanding the +exponential +� +i t′. +The second and third assumptions relate to the response function G(t, t′). We assume that +the integrated response +χ = +� ∞ +0 +dτ G(τ) +(S56) +will be finite (FIR). Notice that since the perturbation field h(t) appears with a global minus +sign in Eq. (S52), we expect that an increase in h will decrease the value of x, implying χ < 0. +We then also have +� +dt′ (I + G)−1(t − t′) = +� +n +(−1)n +� +dt′ Gn(t − t′) = +� +n +(−1)nχn = +1 +1 + χ. +(S57) + +S12 +The final condition, weak long-term memory, is related to the question of whether the system +always reaches the same stationary state. WLTM is equivalent to the assumption that a fixed +point is globally attractive, so any initial condition or perturbation is eventually “forgotten” +as the system evolves. For our purposes it is enough to note that simulations confirm that +this is indeed the case in the fixed-point phase. +With these assumptions we can make the following replacements, +� +dt′ f[G(t, t′)] → f(χ), +C(t, t′) → q, +x(t) → x, +η(t) → √qΣz. +(S58) +From Eq. (7) we then have at the fixed point, +0 = x +� +1 − ux − √qΣz − αx +� +c +χ +1 − χ + Γ(1 − c)χ +�� +, +(S59) +and Eq. (8) for the correlation of the noise turns into the variance of the static random variable +z +⟨η2⟩ = α +� +c +q +(1 − χ)2 + (1 − c)q +� +≡ qΣ2 . +(S60) +In Eq. (9) averages over realizations of the dynamics ⟨·⟩∗ turn into averages over the static +random variable z, so we can express them using Gaussian integrals. We find +χ = +1 +√qΣ +�∂x +∂z +� +∗ += +1 +√qΣ +� ∞ +−∞ +dz +√ +2πe− z2 +2 ∂x +∂z = +1 +√qΣ +� ∆ +−∞ +Dz +−√qΣ +u + α +� +c +χ +1−χ + Γ(1 − c)χ +� += f0(∆) +−1 +u + α +� +c +χ +1−χ + Γ(1 − c)χ +�, +M = ⟨x⟩∗ = +1 +u + α +� +c +χ +1−χ + Γ(1 − c)χ +� +� ∆ +−∞ +Dz(1 − √qΣz) += +√qΣ +u + α +� +c +χ +1−χ + Γ(1 − c)χ +� +� ∆ +−∞ +Dz(∆ − z) = f1(∆) +� +αq +� +c +(1−χ)2 + (1 − c) +� +u + α +� +c +χ +1−χ + Γ(1 − c)χ +�, +q = +1 +� +u + α +� +c +χ +1−χ + Γ(1 − c)χ +��2 +� ∆ +−∞ +Dz(1 − √qΣz)2 += +qΣ2 +� +u + α +� +c +χ +1−χ + Γ(1 − c)χ +��2 +� ∆ +−∞ +Dz(∆ − z)2 = f2(∆) +αq +� +c +(1−χ)2 + (1 − c) +� +� +u + α +� +c +χ +1−χ + Γ(1 − c)χ +��2, +(S61) +where +fn(∆) = +� +Dz(∆ − z)n, +Dz = +dz +√ +2πe− z2 +2 . +(S62) + +S13 +The integrals fn can be written as +f0(∆) =1 +2 +� +1 + Erf +� ∆ +√ +2 +�� +, +(S63) +f1(∆) =1 +2 +� +e− ∆2 +2 +� +2 +π + ∆ +� +1 + Erf +� ∆ +√ +2 +��� +, +(S64) +f2(∆) =1 +2(1 + ∆2) +� +1 + Erf +� ∆ +√ +2 +�� ++ +∆ +√ +2πe− ∆2 +2 , +(S65) +and we have the useful identity +f2(∆) − f0(∆) = ∆f1(∆). +(S66) +The order parameters in the fixed-point regime are determined by Eqs. (19), which are +equivalent to Eqs. (S61). We now describe how to solve these equations. The fully connected +and dilute models are treated separately, because they result in qualitatively different solutions. +We next seek to obtain α, χ, M and q parametrically as a function of ∆, for given values of +u, c and Γ. +B. +Fully connected system +For c = 1 Eqs. (S61) turn into +f0(∆) = +− χ +� αχ +1 − χ + u +� +, +(S67) +f1(∆) =M(1 − χ) +√αq +� αχ +1 − χ + u +� +, +(S68) +f2(∆) = (1 − χ)2 +α +� αχ +1 − χ + u +�2 +. +(S69) +We notice that the first and last equations only contain χ and α. Dividing equation (S69) by +the square of equation (S67) we find +α = (1 − χ)2f 2 +0 +f2χ2 +. +(S70) +Substituting this back into Eq. (S67) leads to +f0 = −χ +�(1 − χ)2f 2 +0 +f2χ2 +χ +1 − χ + u +� += (χ − 1)f 2 +0 +f2 +− χu =⇒ χ = +f0 + f2 +0 +f2 +f2 +0 +f2 − u += f0(f2 + f0) +f 2 +0 − f2u . (S71) +Substituting the expression for χ into Eq. (S70) we then arrive at +α = +� 1 +χ − 1 +�2 f 2 +0 +f2 += +� f 2 +0 − uf2 +f0(f2 + f0) − 1 +�2 f 2 +0 +f2 += +� f2(u + f0) +f0(f2 + f0) +�2 f 2 +0 +f2 += f2(u + f0)2 +(f2 + f0)2 . +(S72) + +S14 +To obtain M and q we use Eq. (S66) and the fact that ∆ = (1 − χ)/√αq to rewrite Eq. (S68) +as +M = +f1√αq +(1 − χ + αχ) = +f1(1 − χ) +(1 − χ + αχ) +1 +∆ += +f1 +(1 + α +χ +1−χ) +f1 +f2 − f0 += +f 2 +1χ +f0(f0 − f2) = +f 2 +1(f2 + f0) +(f0 − f2)(f 2 +0 − uf2), +(S73) +where we have used Eq. (S67). +Dividing the square of Eq. (S68) by Eq. (S69) we further obtain +q = M 2 f2 +f 2 +1 += +f 2 +1f2(f0 + f2)2 +(f0 − f2)2(f 2 +0 − uf2)2. +(S74) +C. +General values of the connectivity (c < 1) +1. +Relations for order parameters in the fixed-point regime +Using the same reasoning we used in the fully connected case (dividing f2 by f 2 +0 using +Eqs. (S61) and simplifying), we can obtain a cubic expression for χ, +0 =f0(c(Γ − 1)f0 − Γf0 − f2) ++ χ +� +f 2 +0[c + 2Γ − 2cΓ] + 2f0f2[1 − c] − f2u +� ++ χ2(c − 1) +� +Γf 2 +0 + f0f2 − 2f2u +� ++ χ3uf2(c − 1). +(S75) +The remaining order parameters, and α, can be expressed as functions of χ +M = χ +f 2 +1 +f0(f0 − f2), +q = χ2 +� +f1 +f2 − f0 +�2 f2 +f 2 +0 +, +α = f 2 +0 +f2 +1 +χ2(1 − c + +c +(1−χ)2). +(S76) +2. +Selection of the physically meaningful solution for χ +For a given value of ∆ (and fixed model parameters c, Γ and u) we obtain χ as one root of +the above cubic equation (S75). +Fig. S2 shows how the solutions of Eq. (S75) look when three real solutions exist. Notice +that only one root is negative, as suggested by the fact that η appears in the equations with +a global negative sign, and that it is continuously connected with the case with a single real +solution. +The regions with three real solutions can be found by calculating the discriminant of the +cubic polynomial in Eq. (S75). When the discriminant is positive (and the coefficients are +real) there are three real roots, if it is negative there is only one real root and a further pair +of complex conjugate solutions. + +S15 +0 +2 +4 +6 +8 +10 +∆ +−25 +−20 +−15 +−10 +−5 +0 +5 +10 +15 +Re[χ] +0 +2 +4 +6 +8 +10 +∆ +−10.0 +−7.5 +−5.0 +−2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +Im[χ] +FIG. S2. Real and imaginary parts of the roots of Eq. (S75) for u = 0.1, c = 0.99, and Γ = 0.5. +However, while the discriminant is easily calculated, it results in a very long equation, not +suitable for direct analysis. We therefore investigate this numerically. Fig. S3 shows the regions +where the discriminant is positive for ∆ = 0 (this is where Fig. S2 suggests real solutions will +first appear). +-1.0 +-0.5 +0.0 +0.5 +1.0 +0.00 +0.05 +0.10 +0.15 +0.20 +Γ +u +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +-1.0 +-0.5 +0.0 +0.5 +1.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Γ +u +0.99999 +0.999 +0.99 +0.9 +0.8 +0.5 +FIG. S3. Parameter regions with positive values of the discriminant at ∆ = 0. The different colours represent different +values of c. Remember that, as discussed in Appendix S1, not all pairs of (c, Γ) are possible. +The right-hand panel of Fig. S3 indicates that, close to c = 1, the condition to have three +real solutions is independent of Γ. This can be further confirmed by Taylor expanding the +discriminant to first order in c about c = 1. This results in 4f2u(uf2 − f 2 +0)3(c − 1), which is +equal to zero whenever uf2 − f 2 +0 = 0, that is, at the phase boundary for c = 1 and u < 1/2. +For values of c that are not close to one we find that three solutions only exist when u is small +(left-hand panel of Fig. S3). +The takeaway of all this is the following: There exists a small region of parameter space, +close to the line where χ diverges for c = 1, where Eq. (21) has three real solutions. Of these +three solutions only one is always negative; this branch connects continuously with the regions +that have a single real solution. We will only use this value of χ. + +S16 +D. +Solution for remaining order parameters +Once χ is known we can substitute into +α = f 2 +0 +f2 +1 +χ2(1 − c + +c +(1−χ)2) +(S77) +to obtain α. +Once α and χ are known, we can proceed to find M and q. From Eq. (S66) we obtain +q = +� +f1 +f2 − f0 +�2 +1 +α +� +1 − c + +c +(1−χ)2 +� = χ2 +� +f1 +f2 − f0 +�2 f2 +f 2 +0 +, +(S78) +and thus, using +q = M 2 f2 +f 2 +1 +, +(S79) +we have +M = χ +f 2 +1 +f0(f0 − f2). +(S80) +E. +The limit c ↑ 1 +We briefly discuss the limit c ↑ 1. The observable order parameters M, q and χ are continuous +in this limit. However, the fraction of survivors can show a discontinuity. +Indeed, for c < 1 the divergence always occurs when f0(∆) = f2(∆), that is ∆ = 0 and +φ(∆ = 0) = f0(∆ = 0) = 1/2. However, we have seen that for the case c = 1, u < 1/2, the +divergence occurs when f 2 +0 − uf2 = 0 which implies ∆ > 0 and φ > 1/2. +As a result, we have φ(αc) = 1/2 for u > 1/2, but if u < 1/2 we have limc↑1 φ[αc(u, c, Γ)] ̸= +φ[αc(u, c = 1, Γ)]. Fig. S4 provides some more insight on how this limit takes place. The curve +for the fraction of survivors has pointwise, but not uniform, convergence to the case c = 1. +As a final note we consider the curve φ(α) in the intervals [0, αs(u, c, Γ)] in which there is +only a single real solution for χ for c < 1 (αs marks the onset of multiple solutions). We find +continuous convergence to the curve φ(α) for the fully connected model. This can be seen +from the Taylor expansion of the discriminant of (S75), and numerically in figure Fig. S4. +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +α +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +φ +c = 1 +c = 0.9 +c = 0.99 +c = 0.999 +c = 0.9999 +c = 0.999999 +FIG. S4. Convergence in c of the fraction of survivors for Γ = 0.5 and u = 0.1 to the case c = 1. The dots represent +the points where three real solutions start to appear (the discriminant becomes zero). + diff --git a/jdFKT4oBgHgl3EQfBS3T/content/tmp_files/load_file.txt b/jdFKT4oBgHgl3EQfBS3T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..29fb424c02172b015fbb97b78feee66cc69263f0 --- /dev/null +++ b/jdFKT4oBgHgl3EQfBS3T/content/tmp_files/load_file.txt @@ -0,0 +1,1325 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf,len=1324 +page_content='Competitive exclusion and Hebbian couplings in random generalised Lotka–Volterra systems Enrique Rozas Garcia,1, 2, ∗ Mark J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Crumpton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' † and Tobias Galla4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' ‡ 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Gothenburg University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 41296 Gothenburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Sweden 2Instituto de F´ısica Interdisciplinar y Sistemas Complejos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' IFISC (CSIC-UIB),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Campus Universitat Illes Balears,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' E-07122 Palma de Mallorca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Spain 3Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' King’s College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' London WC2R 2LS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' United Kingdom 4Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' School of Natural Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The University of Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Manchester M13 9PL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' UK (Dated: January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 2023) We study communities emerging from generalised random Lotka–Volterra dynamics with a large number of species and with competitive exclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Each species is endowed with a number of traits, and competition between pairs of species increases with their similarity in trait space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This leads to a model with random Hebbian interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We use tools from the theory of disordered systems, notably dynamic mean field theory, to characterise the statistics of the resulting communities at stable fixed points and determine analytically when stability breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Two distinct types of transition are identified in this way, both marked by diverging abundances, but differing in the behaviour of the integrated response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' At fixed points only a fraction of the initial pool of species survives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We numerically study the eigenvalue spectra of the interaction matrix between extant species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We find evidence that the two types of dynamical transition are, respectively, associated with the bulk spectrum or an outlier eigenvalue crossing into the right half of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' INTRODUCTION The foundations of the theory of disordered sys- tems date back close to 50 years [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Initially the aim was to understand certain magnetic states in con- densed matter physics (‘spin glasses’) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' However it became clear that applications of the tools developed for disordered systems had a reach far beyond the boundaries of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Methods such as replica the- ory or dynamic generating functionals were quickly adapted and used to answer questions in neural net- works [3–5], to study the Minority Game [6] (some- times presented as a simple model of a financial mar- ket), or indeed evolutionary bi-matrix games and so- called Nash equilibria [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The defining feature of disordered systems is the presence of quenched disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' That is, the system is made up of many constituents, and the interac- tions between these is determined by coefficients that are drawn at random at the beginning, but then re- main fixed as the dynamics of the system unfolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The disorder leads to complicated energy landscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The number of local minima can grow exponentially in the size of the system, and is often organised in a hierarchical manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Dynamic phenomena in dis- ordered systems include ergodicity breaking and so- called ageing [2, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Ideas and methods from the physics of disordered systems have also been used to study complex ecosys- ∗ enrique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='rozas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='garcia@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='se † mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='crumpton@kcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='uk ‡ tobias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='galla@ifisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='uib-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='es tems [9–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The word ‘complex’ in this context in- dicates that the ecosystem is composed of a large number of species, and that these species are sub- ject to randomly drawn interaction coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In this paper we continue this line of work, and focus on a Lotka–Volterra system with ‘Hebbian’ interac- tions [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' More specifically, we are interested in a set of N species (N ≫ 1), whose abundances develop in time following a generalised Lotka–Volterra equa- tion (details will follow in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This involves an N ×N matrix aij of interaction coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Existing work on the statistical physics of complex ecosystems has mostly focused on the case in which the interac- tion matrix is drawn from a Gaussian distribution [9, 11, 14–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This includes work on random ma- trix ensembles with relatively little structure, but also cases in which the matrix is composed of blocks, and where the elements in different blocks have different statistics [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' One common element shared by Gaus- sian Lotka–Volterra models is that the finest level of modelling is set by the interaction coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' No further assumptions are made about the properties of the species, and how the species interactions come about from these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The Hebbian model is inspired by structures first used in neural networks [5, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Translated to the language of ecology, the starting point is now a set of species and a set of traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Each species can ei- ther possess or not possess a given trait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This as- signment of traits to species, in turn determines how species will interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Broadly speaking, the interac- tion between two species will be more competitive the more traits they share (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' the more similar the two species are).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This type of interaction structure arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='11703v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='PE] 27 Jan 2023 2 has also been studied in models combining resources and consumers, both in economics and in ecology [12, 13, 21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Analyses of random replicator sys- tems with Hebbian interactions [24] have shown inter- esting statistical mechanics, and in particular types of phase transition that are different from those seen in replicator systems with Gaussian couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In this paper, we set out to characterise the be- haviour of a Lotka–Volterra system with Hebbian in- teractions, where we allow for a degree of ‘mild’ di- lution (the system is not fully connected, but each species still interacts with an extensive number of other species).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' A system of replicator equations with such interactions was studied in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Our aim is to calculate the statistics of fixed points in the phase where such fixed points are attained and identify the onset of instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' As in the system with Gaussian interactions we find that only a proportion of the ini- tial species survive at stable fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Recent work [25] on Gaussian systems has shown that the reduced interaction matrix (the matrix of interaction coefficients among the surviving species) has intri- cate statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Specifically, that its bulk and outlier eigenvalues can be related to different types of dy- namic phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' As we will show, the types of phase transition seen in the Hebbian model dif- fer from those in the Gaussian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' One aim of the current paper, is therefore to establish (in simu- lations) how these transitions relate to the spectra of the interaction matrix of the extant species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The remainder of the paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' II we define the model and introduce the nec- essary notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' III then contains the mathe- matical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This is based on so-called ‘generat- ing functionals’ and dynamic mean field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The phase diagram and further behaviour of the model is then discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' V we finally turn to a study of the spectra of the reduced interaction matrix and their relation to the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We conclude the paper with a discussion and an outlook in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' MODEL DEFINITIONS We will study the following generalised Lotka– Volterra equation (gLVE) ˙xi(t) = xi(t) � �Ki − uixi + � j̸=i cijJijxj � � , (1) where the xi ≥ 0 represent the abundances (or popu- lation densities) of different species, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We always assume initial conditions for which all xi are strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The quantities ui > 0 denote the strength of in- traspecific competition, and the aij = cijJij represent the interspecific interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The Ki (together with the ui) set the carrying capacities of the species in the absence of interactions between different species (xi then tends to Ki/u in the long run).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We focus on the case ui ≡ u for all i, noting that u controls the time scale on which the non-interacting system approaches the fixed point xi ≡ Ki/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We allow for general pos- itive values of u throughout our analysis, but in an effort keep the number of parameters manageable we set Ki ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The dilution variables cij ∈ {0, 1} (i ̸= j) deter- mine which species interact with one another, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' they set the topology of the interaction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' For each pair i < j, the coefficients cij and cji are chosen from a Bernoulli distribution with ⟨cij⟩ = c, ⟨cijcji⟩ − c2 = Γc(1 − c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (2) We thus have P(cij = 1) = c for all i ̸= j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' c is the analog of what May called ‘connectance’ [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The parameter Γ is restricted to the range from −1 to 1 by construction, but we note that not all choices of pairs (c, Γ) are possible (see Supplemental Material [26] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We note that the choice of the di- agonal coefficients cii is irrelevant as we set Jii = 0 below Throughout our paper, c is chosen not to scale with N [c = O(N 0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This means that each species inter- acts with an O(N) number of other species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We are therefore not studying a ‘dilute’ system in the sense of random matrix theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The extensive connectivity allows us to use established methods from dynamic mean-field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' An analysis of a truly dilute sys- tem with c = O(1/N) (and where consequently each species only interacts with a finite number of other species) would be much more intricate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Interaction links in our system are directed, that is, an effect of the presence of species j on the dynamics of i does not necessarily imply the reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The pa- rameter Γ measures the correlations between cij and cji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' A choice of Γ = −1 implies cij = 1 − cji with probability one, and Γ = 1 means that cij = cji with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The matrix Jij determines the strength of the ef- fects of the presence of species j on the dynamics of the abundance of species i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Positive values of Jij imply that the population of species j is beneficial to the growth of species i, while negative values im- ply a detrimental interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In ecological terms the signs of the pair of interactions (sgn Jij, sgn Jji) de- termine whether two species are in a mutualistic rela- tion (+, +), whether they compete with one another (−, −), or whether there is an antagonistic predator- prey relation between i and j (±, ∓) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In this work, the interspecific interaction is chosen according to a competitive exclusion heuristic (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We assume each species is described by a set of binary traits, labelled µ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' , P, and that a pair of species will compete in proportion to the similarity between the two species (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='e, the number 3 of traits which both or neither species possess).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We write ξµ i = +1, if species i has trait µ, and ξµ i = −1 if the species does not possess the trait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Interactions are then assumed to be of the form Jij = � � � � � � � − 1 cN αcN � µ=1 ξµ i ξµ j i ̸= j 0 i = j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (3) We have here set P = αcN, with α > 0 a model pa- rameter (in simulations P is restricted to integer val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=') That is to say, we assume that the number of traits is proportional to the number of species in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The interaction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (3) is known as ‘Heb- bian’ in the context of neural networks [3, 20], and interesting phase behaviour occurs when α = O(N 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This is the regime we focus on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We have normalised the interaction strength by cN, the mean number of species that any one species will interact with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We will refer to the random variables cij and ξµ i as the disorder of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The traits ξµ i are chosen to be ξµ i = ±1 with equal probability, and there is no correlation between the different ξµ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This implies that the distribution of the Jij approaches a Gaussian as N → ∞, reminiscent of the model studied for example in [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We note however, that the Hebbian structure introduces cor- relations between the different Jij, which are different from the correlations studied in the earlier literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' GENERATING FUNCTIONAL ANALYSIS AND STABILITY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Generating functional and effective process We analyse the system in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (1) using dynamic generating functionals, an established method in the theory of disordered systems [4, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This is also known as ‘dynamic mean field theory’, and has been used to study Lotka–Volterra systems with Gaussian random couplings [11, 15, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We note that an alter- native approach is based on the the so-called cavity method [14, 17, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The outcome of the application of these tech- niques is an effective stochastic process for a ‘rep- resentative species’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The ensemble of realisations of stochastic processes is statistically equivalent to the set of single-species trajectories xi(t) of the disor- dered dynamical system in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The dynamic mean-field description becomes exact in the thermo- dynamic limit (N → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Overall, in this limit, the infinite-dimensional deterministic dynamical system in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (1) is traded for an effective single-species pro- cess which is non-local in time (it involves retarded self-interaction) and contains coloured noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The generating functional analysis begins from ˙xi(t) = xi(t) � �1 − uxi + � j̸=i cijJijxj − hi(t) � � , (4) where we have introduced the perturbation fields hi(t) in order to calculate linear response functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' These fields are not actually part of the model and are set to zero at the end of the calculation, as well as in all simulations shown in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' For more details see the Supplementary Material (SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The generat- ing functional of this dynamical system is given by Z[ψi(t), hi(t)] = � exp � i � i � dt xi(t)ψi(t) �� paths , (5) where the average is over paths [x1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' , xN(t)] of the dynamics in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The ψi(t) constitute a source field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The generating functional in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (5) is the Fourier transform of the probability measure in the space of paths generated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The final outcome of the generating-functional analysis is a set of equations for the dynamic macro- scopic order parameters of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' For the Lotka–Volterra model these are M(t) = lim N→∞ 1 N N � i=1 ⟨xi(t)⟩0, C(t, t′) = lim N→∞ 1 N N � i=1 ⟨xi(t)xi(t′)⟩0, G(t, t′) = lim N→∞ 1 N N � i=1 δ⟨xi(t)⟩0 δhi(t′) , (6) where δ denotes a functional derivative and ⟨· · · ⟩0 stands for an average over random initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The overbar · · · represents the average over the dis- order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' over the cij and ξµ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The order parame- ters can be obtained from the disorder-averaged gen- erating functional as derivatives with respect to the fields ψi(t) and/or hi(t), evaluated at ψi(t) ≡ 0 and hi(t) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The order parameters in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (6) are determined self-consistently from an effective process for a single representative (‘mean field’) species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The procedure to derive the effective equations is well-documented [3, 4, 11, 31], therefore, we only report the final result (a more detailed derivation can be found in the SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The effective single-species process for the model is given by ˙x(t) = x(t) � 1 − ux(t) − α � t 0 dt′[cG(I − G)−1 +Γ(1 − c)G](t, t′)x(t′) − η(t) � , (7) 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 α a) c = 1 Divergence Fixed point 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 u b) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5, Γ = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 u c) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5, Γ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Fixed points and divergence in the model with Hebbian disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Each panel illustrates the behaviour of the model for different choices of c and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The heatmap indicates the fraction of samples that converge to a fixed point in numerical integration of the gLVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The criteria for the identification of convergence or divergence are described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The dashed lines are theoretical predictions for the onset of divergence (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' III C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' where I is the identity operator and η(t) is coloured Gaussian noise with zero mean and correlations in time, given by ⟨η(t)η(t′)⟩ = α[c(I − G)−1C(I − GT )−1 +(1 − c)C](t, t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (8) The order parameters in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (6) are to be obtained self-consistently from the following expressions, M(t) = ⟨x(t)⟩∗, C(t, t′) = ⟨x(t)x(t′)⟩∗, G(t, t′) = δ δη(t′)⟨x(t)⟩∗, (9) where the average ⟨· · · ⟩∗ is performed over realisa- tions of the process in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (7)-(9) form a closed system and have to be solved self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Fixed point analysis There is no realistic prospect for a general analyti- cal solution of the effective dynamics in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' One alternative is to use Monte-Carlo methods to con- struct sample paths for the effective process and so- lutions for the dynamic order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' For ex- ample via the Eissfeller-Opper procedure [32], or us- ing the more recent approach in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The latter reference explicitly discusses applications to random Lotka-Volterra systems with Gaussian disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Here we will instead follow [11, 15, 29] and focus on analytical solutions in the parameter regime in which the dynamics approach stable fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This is motivated by observations from the numerical inte- gration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We find that, for certain parame- ters, the system tends to a unique fixed point, which is independent of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 1 shows ex- amples of parameter regions in which this is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Broadly speaking, we observe two different types of behaviour: (i) the population densities converge to a fixed point, or (ii) they diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' These types of be- haviour occur in different regions of parameter space (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' There is a thin boundary between the two re- gions where other behaviour (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' periodic behaviour or persistent irregular motion) can appear, as evi- denced by the occasional green or light blue pixel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We attribute this to the fact that the system size N is necessarily finite in numerical experiments, and we expect that this behaviour will become in- creasingly more rare as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We will thus assume that each path in the ensemble of trajectories of the effective process eventually ar- rives at a unique fixed point, x = limt→∞ x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Each realisation of the noise variable η(t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (9) also ap- proaches a stationary value η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We note that x and η will be random variables, differing across realisations of the effective dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We can then write M = ⟨x⟩∗, G(τ) = lim t→∞ G(t + τ, t), q = lim t→∞ C(t + τ, t) = ⟨x2⟩∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (10) These relations can be understood as follows: if all realisations of the effective dynamics approach sta- tionary values then M(t) will approach a constant, given by ⟨x⟩∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Furthermore, we assume that the re- sponse function G(t, t′) becomes time-translation in- variant for large t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' G(t, t′) = G(t − t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Causal- ity implies that G(t − t′) = 0 for t < t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Finally, given that all trajectories of the effective dynam- ics approach fixed points, the correlation function C loses all time dependence and so we have writ- ten C(t, t′) = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This is consistent with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (8), the noise variables η(t) also approach a random but time- independent value for all realisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The mean of the random variable η is zero and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (8), its 5 variance is given by ⟨η2⟩ = αq � c (1 − χ)2 + 1 − c � , (11) where χ = � ∞ 0 dτ G(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (12) From now on, we will write η = √q Σz, where z is a standard Gaussian random variable, and Σ2 = α � c (1 − χ)2 + 1 − c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (13) Setting the time derivative on the left-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (7) to zero, and using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (10)-(12) we find x � 1 − ux − √qΣz − αx � c χ 1 − χ + Γ(1 − c)χ �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (14) For a given value of z this is to be solved for x, subject to the constraint that abundances are non- negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='e, x(z) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Irrespective of the value of z, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (15) always has the solution x(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Addi- tionally, a second non-negative solution is possible for some values of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' As we will confirm in simulations, the physically meaningful solution is given by x = max � � �0, 1 − √qΣz u + α � c χ 1−χ + Γ(1 − c)χ � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (15) The denominator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (15) always comes out posi- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Therefore, we have the solution x(z) > 0 when z < ∆, and x(z) = 0 when z ≥ ∆, with ∆ = 1 Σ√q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (16) Given that z is a Gaussian random variable, the abun- dances of extant species at the fixed point therefore follow a clipped Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Using this fixed point ansatz, the relations for the order parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (10) can be written in the following form [14, 15] M = � ∆ −∞ Dz x(z), χ = 1 √qΣ � ∆ −∞ Dz ∂x(z) ∂z , q = � ∆ −∞ Dz x(z)2, (17) where Dz = dz √ 2πe− z2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' It is now convenient to intro- duce the following functions fn(∆) := � ∆ −∞ Dz (∆ − z)n, (18) for n = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We then find from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (17) −χ � u + α � c χ 1 − χ + Γ(1 − c)χ �� = f0(∆), M u + α � c χ 1−χ + Γ(1 − c)χ � � αq � c (1−χ)2 + (1 − c) � = f1(∆), � u + α � c χ 1−χ + Γ(1 − c)χ ��2 α � c (1−χ)2 + (1 − c) � = f2(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (19) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (19) together with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (16) form a closed set for the set of unknowns q, χ, M and ∆, which is to be solved as a function of the model parameters u, c, Γ and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Recalling that x(z) > 0 if, and only if, z < ∆ we identify f0(∆) as the fraction of surviving species, φ ≡ f0(∆) = � ∆ −∞ Dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (20) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (19) can be solved parametrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We fix u, c, Γ and ∆ and then solve for the set of χ, q, M and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In detail, we find the following cubic equation for χ, valid for c < 1, 0 =f0(c(Γ − 1)f0 − Γf0 − f2) + χ � f 2 0 [c + 2Γ − 2cΓ] + 2f0f2[1 − c] − f2u � + χ2(c − 1) � Γf 2 0 + f0f2 − 2f2u � + χ3uf2(c − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (21) Further, we have from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (19), M = χ f 2 1 f0(f0 − f2), q = χ2 � f1 f2 − f0 �2 f2 f 2 0 , α = f 2 0 f2 1 χ2(1 − c + c (1−χ)2 ), (22) where the fn are to be evaluated at ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The relations in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (21) and (22) are also valid for c = 1, and can then be simplified as outlined in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The validity of the predictions from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (21) and (22) is confirmed by direct numerical integration of the gLVE in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Stability analysis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Diverging abundance Model with c < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The first and second relations in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (22) indicate that the order parameters M 6 0 1 2 3 4 α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 φ a) u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='3 u = 1 u = 2 0 1 2 3 4 α 0 1 2 3 4 5 6 M b) 0 1 2 3 4 α 0 2 4 6 8 10 Var[x] c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Test of analytical predictions for the order parameters against numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The figure shows the fraction of surviving species φ, the mean abundance M, and the variance of abundances (q − M 2) as a function of the model parameter α (where P = αcN is number of traits each species in the original system possesses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Lines are from the theory, derived in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (21) and (22), markers from numerical integration of the gLVE (N = 1000, tmax = 30, averaged over 10 realisations of the disorder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Remaining model parameters are c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5, and Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Vertical dashed lines indicate the onset of divergence as determined from the theory in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' and q both diverge in the system with c < 1 when f0(∆) = f2(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The latter implies ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The value of α for which this occurs can (for a given choice of c, u and Γ) be obtained from the third relation in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (22), with χ being the relevant root of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (21) the susceptibility χ is found to remain finite at the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We note that f0(∆) > 0 for all relevant values of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Model with c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The fully connected system also shows two types of divergences: (i) The quan- tities M and q both diverge when f0(∆) = f2(∆), see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The susceptibility then remains finite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (ii) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (B1) further indicate, that M and q also diverge in the model with c = 1 when f 2 0 (∆) = uf2(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This latter condition results in α = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (B1) the susceptibility χ is then seen to diverge as well (the di- vergences of M, q, and χ take place simultaneously).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We note that the divergencies resulting from f0 = f2 and f 2 0 = uf2 can take place at different lo- cations in parameter space for the model with c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' If this is the case, and starting in the stable phase, the divergence that occurs first will determine the loss of stability in the fully connected system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' For u < 1/2 the transition of type (ii) takes place first as α is increased (M, q, and χ diverge), and for u > 1/2 the transition of type (i) is instead observed (q, M diverge, χ remains finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Linear instability The system also shows a linear instability which can be identified using the procedure established in [11, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We write x(t) = x + y(t) and η(t) = √q Σz +ζ(t), where y(t) and ζ(t) are small perturba- tions about the fixed point of the trajectories of the effective process in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Expanding to first order in these perturbations we find that ˙y(t) =x � −uy(t) − α � t −∞ dt′ K (t, t′) y (t′) − ζ(t) � + y(t) � 1 − ux − αx � t −∞ dt′ K (t, t′) − √q Σz � , (23) with K(t, t′) = [cG(I − G)−1 + Γ(1 − c)G](t, t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We also have the self-consistency relation ⟨ζ(t)ζ (t′)⟩ = α � c(I − G)−1D(I − GT )−1 + (1 − c)D � (t, t′) , (24) where D (t, t′) = ⟨y(t)y (t′)⟩∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' When x = 0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (23) becomes ˙y(t) = y(t) (1 − √q Σz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (25) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (15), together with the observation that the de- nominator in this equation is strictly positive, implies that 1 − √q Σz < 0 when x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This allows us to conclude that perturbations on extinct species decay, and do not contribute to any linear instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' For fixed points x > 0 we find from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (15) and (23) that ˙y(t) = −x � uy(t) + α � t −∞ dt′ K (t − t′) y (t′) + ζ(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (26) To identify the onset of linear instability we follow [11, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We move to Fourier space, writing ω for the variable conjugate to time t, and using tildes to indicate Fourier transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Focusing on the mode with ω = 0 and following steps similar to those in [11, 15, 33] we then find from 7 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (26) � |˜y(0)|2� = � φ−1 � u + αc χ 1 − χ + αΓ(1 − c)χ �2 − α � c (1 − χ)2 + 1 − c ��−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (27) The left-hand side is manifestly non-negative, so that a change of sign of the expression inside the square bracket on the right-hand side indicates an inconsis- tency (and divergence of � |˜y(0)|2� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (19) this is shown to occur when α � c (1 − χ)2 + 1 − c � (f0 − f2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (28) For c < 1 the expression in the square brackets is never zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This leaves us with the condition f0 = f2, which is the same we obtained for the divergence of M and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' If c = 1, the term in the square bracket is zero if χ → ∞, which using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (B1) we can write as f 2 0 − uf2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' From this, we conclude that in our model the linear instability is always accompanied by the in- stability with diverging mean abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This is markedly different from the behaviour of the gLVE model with Gaussian random interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In this Gaussian model there are instances where the linear instability sets in as the variance of interactions is increased, but where abundances remain finite and the divergence only occurs at a later point at even higher variance of the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This leads to a phase with multiple attractors between the two tran- sitions [14, 16, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Our analysis indicates that the model with Hebbian couplings does not have such a multiple-attractor regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' PHASE DIAGRAM AND FURTHER BEHAVIOUR OF THE MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Phase diagram for the fully connected system (c = 1) The phase diagram of the fully connected model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We recall that, for c = 1, the only model parameters are the self-interaction co- efficient u and the ratio of the number of traits to the number of interspecies interactions in the origi- nal pool (α = P/cN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' For a fixed value of α, the sys- tem shows a unique stable fixed point for u > uc(α), where uc(α) marks the onset of instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 3(a), obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (B1), shows the phase boundary between the stable and unstable regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' At this boundary M and q diverge, and if uc < 1/2 we also observe a divergence of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The two types of trajectory in the stable and di- vergent phases are illustrated in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In the stable phase the system reaches a fixed point, for any one realisation of the interaction matrix (two examples are illustrated in green and red respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 3(b) also shows two examples in which the species abundances diverge (blue and orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The divergence occurs at a finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We will discuss this further in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' IV C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Phase diagram for connectivity c < 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 4 shows how the phase diagram for the system with c < 1 depends on the connectivity c and the symmetry parameter Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In all cases there is a sin- gle phase boundary, where the divergence of M and q and the onset of linear instabilities coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This phase boundary separates a region where trajectories converge to a single globally stable fixed point (phase to the right of the line), from a region where trajecto- ries are unbounded and diverge in finite time (phase on the left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The phase diagrams in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 3 and 4 show that the system is in the stable phase for small values of α (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' a small number of traits relative to the number of species in the initial pool), or large val- ues of u (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' large negative self-interaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This is the consequence of two competing effects, the self- interaction (parametrised by u) which stabilises the system, and the interaction between species (induced by competition of similar species) which promotes in- stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' When u is large and/or α is small, the stabi- lizing effect of the intra-species interaction dominates over the interactions across species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In the extreme limit α = 0 (no interaction between different species), each abundance follows a separate logistic equation, ˙xi = xi(1−uxi), and converges to xi = 1/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' When α is small but non-zero, the system consists of weakly interacting species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The effect of the interactions be- tween species is then a small perturbation to the lo- gistic behaviour of individual species, and does not change the convergence to a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This can be confirmed from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (21) and (22) by taking the limit α → 0, which results in all species surviving with fixed point abundance x∗ i = 1/u (φ → 1, M → 1 u, and Var[x] → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' A similar result is obtained for u ≫ 1 at fixed value of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Conversely, for low values of u or large values of α the system is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In this situation the sta- bilising self-interaction is not sufficient to overcome the destabilising effect of the random interactions be- tween species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The most interesting behaviour takes place at the phase boundary, where the effect of the intraspe- cific and interspecific nonlinearities are of comparable magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (21) and (22) we can conclude that φ → 1/2, and M ∼ (αc − α)−1 as the system approaches the instability (from the stable phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='00 u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 α a) 0 2 4 6 8 10 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 x(t) b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Phase behaviour of the fully connected model (c = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Panel (a): Phase diagram for the model with c = 1, the only model parameters are then u and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The system is stable to the right of the lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' At the dot-dashed line (u < 1/2) q, M and χ all diverge, and at the dashed line M and q diverge, but χ remains finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Panel (b): Illustration of the behaviour of the abundances of individual species in the two different phases (convergence to fixed point shown in green and red, diverging abundances in orange and blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 α a) Γ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 c = 1 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='33 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 b) Γ = 0 c = 1 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='25 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='75 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 c) Γ = 1 c = 1 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='25 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='75 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Phase diagram for different choices of the connectivity c, and the symmetry parameter Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The coloured lines in each panel indicate where the linear instability occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The instability coincides with the divergence of M and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The system is stable to the right of the line, abundances diverge on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Further details can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We further note that decreasing the value of the symmetry parameter Γ, increases the range of the stable region in the phase diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This is similar to the effect of increasing the fraction of predator-prey interactions in Lotka–Volterra models with Gaussian interactions [10, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Indeed, the effect of a reduction of Γ is to increase the fraction of species pairs i, j with cij = 1 and cji = 0, that is the proportion of uni-directional interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Interestingly, the effect of varying the ‘connectance’ c is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 4 an increased connectivity can, depending on the other model parameters, turn a previously stable system into an unstable one, or vice versa, stabilise a previ- ously unstable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Finite-time divergence of the mean abundance As mentioned earlier, the divergence of the abun- dances in the divergent phase occurs at finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This has previously been reported in the model with Gaussian interactions [17], and can be justified heuristically from the Lotka-Volterra equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In- deed, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (1) has a second order non-linearity in the abundances xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This can lead to dynamics of the form ˙x ∼ x2, which in turn implies a solution of the form x(t) = (c − t)−1, where c is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This results in a divergence at finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 5 shows the time, tdiv, at which the divergence occurs for different choices of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This time grows as one approaches the stability line (from inside the unstable phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' When the stability line is crossed (into the stable phase), the time-to- divergence diverges itself (tdiv → ∞), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' the diver- gence no longer occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Results from the numerical integration of the gLVE suggest that the divergence of 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 α a) c = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 u b) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5, Γ = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 u c) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5, Γ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 10−2 10−1 100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Finite-time divergence of abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The heatmaps indicate the time, tdiv, at which abundances diverge, for initial conditions xi(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Data is obtained from numerical integration of the gLVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The dotted line is the phase boundary predicted by the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' To the right of the phase boundary the system is in the stable phase, so that no divergence occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' the abundances is of the form M ∼ (tdiv − t)ν, where ν ≈ 1, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This behaviour appears to be independent of initial conditions, the values of the parameters u and α, and the initial number of species N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 10−13 10−11 10−9 10−7 10−5 10−3 10−1 101 t∞ − t 10−2 101 104 107 1010 1013 M α = 2 α = 5 α = 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Divergence of M for initial conditions uniformly distributed in (0, 10−3) and parameters N = 1000, u = 1, and different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The dotted black line corre- sponds to (t∞ −t)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The deviation from M ∼ (t∞ −t)−1 close to the divergence is attributed to numerical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' REDUCED INTERACTION MATRIX AND ITS EIGENVALUE SPECTRUM Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' [25] recently established a close connection be- tween different instabilities in the Gaussian random Lotka–Volterra model and the eigenvalue spectrum of the interaction matrix of the surviving species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' More specifically, the spectrum of this reduced interaction matrix is composed of a bulk region and a potential outlier eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' As parameters are changed (start- ing from within the stable phase) either the bulk spec- trum or the outlier eigenvalue can cross into the right half of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In the Gaussian model, the crossing of the outlier is associated with a transition marked by the divergence of abundances, and a cross- ing of the bulk with a linear instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In this section we explore in numerical simulations how the different transitions in the gLVE model with Hebbian interactions relate to the eigenvalue spec- trum of the matrix of interactions between surviving species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Spectrum of the original interaction matrix Before we discuss spectra of the reduced interac- tion matrix for the Hebbian model, we make a few remarks on the initial interaction matrix αij = cijJij among all species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Throughout this section we set the diagonal elements of this matrix to zero, the only effect of self-interaction (the term −uxi) is a simple shift of this spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In the large-N limit the cen- tral limit theorem applies to Jij = − 1 cN �αcN µ=1 ξµ i ξµ j , so each off-diagonal entry αij of the interaction ma- trix is either a Gaussian random variable (if cij = 1) or equals zero (if cij = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The variance of αij is Var(αij) = 1 cN 2 αcN � µ,µ′ � ξµ i ξµ′ i ξµ j ξµ′ j � = α N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (29) Calculating the correlations between pairs of ele- ments we obtain Corr[αij, αnm] = ⟨αijαnm⟩ − ⟨αij⟩⟨αnm⟩ Var(αij) = � � � � � Γ(1 − c) + c (i, j) = (m, n) 1 (i, j) = (n, m) 0 else , (30) 10 −6 −4 −2 0 2 4 6 ℜ [λ] −6 −4 −2 0 2 4 6 ℑ [λ] α = 35 α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='35 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='03 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Examples of the eigenvalue spectrum of the original interaction matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The dashed lines are the na¨ıve predictions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Model parameters are c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='4, Γ = c/(c − 1) and N = 5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' where we have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (2) and the fact that Jij is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This means that only diagonally op- posed pairs of elements are correlated, and that their correlation is determined by both, Γ and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Based on a theory that only takes into account cor- relations between diagonally opposed matrix entries, one might then expect an elliptic spectrum [34], with support given by the ellipse � x √α(1 + τ) �2 + � y √α(1 − τ) �2 = 1, (31) with τ = Γ(1 − c) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' However, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 7, this is an approximation to the true spectrum at best for large values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' For intermediate values of α (an example is shown in orange in the figure) the eigenvalue spectrum appears to have a triangu- lar shape, and for small values of α (shown in green) the spectrum becomes even more skewed, and even- tually appears to consist of two separate components (example shown in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' While we cannot fully ex- clude finite-size effects (the spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 7 are for N = 5000), we believe that the deviations from an el- liptical spectrum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (31) are due to higher-order correlations between entries of the interaction ma- trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' For example, it has been shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' [35] that cyclic correlations can result in eigenvalue spec- tra with shapes similar to the ones in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' −10 −8 −6 −4 −2 0 ℜ [λ] −4 −2 0 2 4 ℑ [λ] α/αc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='90 α/αc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='75 α/αc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='50 α/αc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='20 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Eigenvalue spectrum of the reduced interaction matrix for u = 5, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5, Γ = c/(c − 1), for different choices of the model parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The vertical dashed lines indicate the real part of the right-most eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Eigenvalues of the reduced interaction matrix We now conclude the analysis of the model with a numerical study of the spectra of the reduced interac- tion matrix, that is, the interaction matrix between species that survive at the fixed points of the gLVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 8 shows the spectra of this matrix for the case c < 1, and for a choice of Γ less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This means that the initial interaction matrix is not symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The reduced matrix is not symmetric either, and as a consequence its eigenvalues will generally be complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 8, the spectrum is not elliptic, and we have found no evidence of an outlier eigenvalue in this scenario (c < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In the figure we have fixed u, and varied α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The data suggests that the phase transition at α = αc(u) coincides with the point at which the right-most bulk eigenvalue crosses the imaginary axis into the right half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 9 we study the fully connected system for two different values of the self-interaction strength u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The original interaction matrix in the fully con- nected model is symmetric by construction, and so is the reduced interaction matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' As a consequence all eigenvalues are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Panel (a) focuses on the case u > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We find no signs of outlier eigenvalues, and again the data indi- cates that the transition to instability occurs when the leading bulk eigenvalue crosses into the positive half of the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Panel (b) shows a scenario in which u < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In contrast with the situation in (a), outlier an eigen- 11 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 λ 10−3 10−2 10−1 100 P(λ) a) α/αc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='95 α/αc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='90 α/αc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='80 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 λ 10−3 10−2 10−1 100 101 b) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='00 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Eigenvalue spectrum of the reduced interaction matrix in the fully connected system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The matrices are symmetric, and their eigenvalues are therefore real valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Panel (a) is for u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='7, panel (b) for u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' value now becomes apparent, and the transition to instability in the gLVE at α = αc(u) now appears to coincide with the point at which the outlier becomes positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The connection between the different types of tran- sition and the behaviour of the spectrum of the re- duced interaction matrix is shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We re- call that the mean abundance and the second moment of the abundances diverge at all transitions, and that the onset of the linear instability always coincides with the point of diverging abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' There are thus only two types of transition, one in which the susceptibility remains finite (χ < ∞), and another for which it diverges (χ → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The table indicates that the former transition (χ finite) appears to co- incide with the bulk spectrum of the reduced matrix crossing into the right half of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The transition at which χ → ∞ (along with the diver- gences of M and q) on the other hand seems to be seen when the outlier eigenvalue of the reduced ma- trix in the fully connected system reaches the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We stress that these are numerical observations, and that these findings should therefore be seen mostly as conjectures at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In principle, the spectrum of the reduced interaction matrix can likely be calculated in the Hebbian model, adapting c < 1 q, M diverge bulk spectrum χ remains finite crosses axis c = 1 u > 1/2 q, M diverge bulk spectrum χ remains finite crosses axis u < 1/2 q, M, χ outlier eigenvalue all diverge crosses axis (at u = α) TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Types of phase transition in the gLVE model with Hebbian interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The table summarises the dif- ferent transitions, giving details about the nature of the divergence at the transition, and the associated behaviour of the spectrum of the reduced interaction matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' the method used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' However this involves a substantial calculation and is beyond the scope of the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' DISCUSSION To summarise, we have carried out a generating functional analysis of a random generalised Lotka– Volterra system with competitive exclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Species interactions in the model are governed by Hebbian couplings subject to mild dilution (the remaining con- nectivity is still extensive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We have computed the statistics of surviving species in the stable fixed point phase, and we have analytically determined the on- set of instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Similar to the gLVE with Gaussian interactions, asymmetry in the connectivity matrix promotes stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' That is to say, the system be- comes more stable when there is a larger fraction of unidirectional interactions (cij = 1, but cji = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In contrast with the Gaussian model, the linear instabil- ity against small perturbations cannot be separated from an instability at which species abundances di- verge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' As a consequence, there is no phase with mul- tiple stable fixed points in the Hebbian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' De- spite some common features, the statistical mechan- ics of the Gaussian and Hebbian models are therefore rather distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Our analysis shows that there are two types of transitions to divergent abundances in the Hebbian model, one in which the integrated response χ re- mains finite, and another in which χ diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This raises interesting questions about the exact nature of memory onset in the system (a diverging inte- grated response indicates persistent memory of per- turbations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Future work could focus on the precise shape of the response function, where the numeri- cal methods in [17] might prove particularly useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Given that the fully connected system has symmet- ric couplings it would also be interesting to see how crossing each of the different types of transition af- fects the energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' A natural approach here 12 might be the replica method and suitable levels of replica symmetry breaking [16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Numerical simulations provide evidence that the transition at which the integrate response remains fi- nite (χ < ∞) is associated with the bulk spectrum of the reduced interaction matrix (the matrix of interac- tions between extant species) crossing the axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The transition at which χ diverges on the other hand ap- pears to be signalled by an outlier eigenvalue crossing the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' These findings in simulations reinforce the in- triguing analytical result obtained recently in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Namely, the eigenvalues of the interaction matrix in the community of surviving species can be used to de- cide the stability of feasible equilibria, that is to say fixed points with non-negative species abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In the traditional approach to ecosystem stability by Robert May [9], based on the eigenvalue spectra of random matrices, no actual dynamics is specified, and the feasibility of the assumed equilibria therefore re- mains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Any fixed point of the generalised Lotka–Volterra model on the contrary is feasible by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The study of the spectra of reduced in- teraction matrices resulting from Lotka–Volterra dy- namics can therefore contribute to establishing how May’s approach can be adapted to include feasible equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' On a broader level, our study highlights two com- mon facets of work on the statistical physics of com- plex systems, which were also seen for example 30-40 years ago when physicists studied neural networks, or 15-20 years ago when a number of physicists worked on the Minority Game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' On the one hand, tools from physics can make a difference for problems in other disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In our system (and other models of com- plex ecosystems more generally) this is the study of feasible equilibria with methods from spin glass physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' At the same time, studying problems aris- ing in other areas can reveal new types of physics and complexity, which one would perhaps not find within the strict boundaries of traditional physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In our case these are the different types of phase transition in the generalised Lotka–Volterra model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We think that this mutually beneficial relation of physics and adjacent disciplines is what makes the field of com- plex systems particularly attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Partial financial support has been received from the Agencia Estatal de Investigaci´on and Fondo Europeo de Desarrollo Regional (FEDER, UE) under project APASOS (PID2021-122256NB-C21/PID2021- 122256NB-C22), and the Maria de Maeztu program for Units of Excellence, CEX2021-001164-M funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Stein, Physical review letters 60, 1895 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' [35] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Aceituno, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Rogers, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Schomerus, Phys- ical Review E 100, 010302 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Appendix A: Details of numerical procedures For the numerical integration of the gLVE (1) we used scypi’s solve ivp function, which makes use of a RK45 integration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' To determine the fraction of survivors we counted the number of species above a threshold abundance of 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' There are two sources of systematic error asso- ciated to this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The most relevant is the over- estimation of the fraction of survivors if the system is not close enough to the equilibrium configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This can be addressed by extending the simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The second source of error comes from the fact there is no ‘gap’ between zero and the lowest non- zero abundance [see the clipped Gaussian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (15)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This implies that for any value of the threshold there is a nonzero probability of finding equilibrium abun- dances below it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' A possible solution, making use of the facts that in simulations N is finite and that we know the abundance distribution analytically, is to choose the threshold value so that the expected num- ber of surviving species with an abundance below the threshold is small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' smaller than one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The chosen value of 10−4 provides good results in the parameter ranges we have explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' As part of our measurements it is necessary to de- tect divergences in the species abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' To de- tect this divergence we have used the failure of the integration method as an indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Indeed, as the abundances grow with each iteration, so does the es- timated error used to adapt the step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This causes the solver to lower the time-step until it eventually drops below machine precision, at which point inte- gration is stopped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The agreement of the theoretical and numerical phase boundaries in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 5 confirm the validity of this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Appendix B: Order parameters at the fixed point of the fully connected system The parametric solution for the order parameters of the fully connected system (c = 1) in the fixed point phase can be obtained from the following relations, α = (u + f0)2f2 (f0 + f2)2 , M = f 2 1 (f0 + f2) (f0 − f2)(f 2 0 − uf2), q = f 2 1 f2(f0 + f2)2 (f0 − f2)2(f 2 0 − uf2)2 , χ = f0(f0 + f2) f 2 0 − uf2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (B1) The functions fn(∆) on the right provide α, M, q, χ as implicit functions of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Keeping Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (18) in mind one sees that M and q can only diverge if f0 = f2 or f 2 0 = uf2, as indicated in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Appendix C: Limiting behaviour of the order parameters 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Limit α → 0 The weak interaction limit α → 0 corresponds to ∆ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' [This can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (22), keeping in mind that f0 > 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' From the definition in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (18) we have, in this limit, f0(∆) = φ = 1 + e− ∆2 2 √ 2π � − 1 ∆ + O � ∆−3�� (C1) f1(∆) = ∆ + e− ∆2 2 √ 2π � 1 ∆2 + O � ∆−4�� (C2) f2(∆) = 1 + ∆2 + e− ∆2 2 √ 2π � − 2 ∆3 + O � ∆−5�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (C3) Next, we compute the value of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Since only f0 and f2 are present in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (21), and only f2 is divergent, we group in terms proportional to f2 to obtain 0 = 1−χ [2(1 − c) − u]−χ2(c−1)(1−2u)−χ3u(c−1), which we can check always has χ = −1/u as its negative solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Finally, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (22) we obtain M = 1/u and Var[x] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' As expected, these values are independent of c and Γ, since in the limit of absent interactions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (4) becomes a set of independent logistic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In this case all species survive with abundance 1/u, which is what we obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Limit α → αc There are two different scenarios for the limit α → αc (where αc is the location of the phase transition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (1) If c = 1, u < 1/2 the divergence takes place as uf2 → f 2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (B1) we see that αc = u, and αc − α = u − f2 (f0 + f2)2 (f 2 0 − uf2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (C4) This implies that both χ and M diverge as (αc−α)−1, and q diverges as (αc − α)−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (2) The other type of transition occurs when f0 → f2, which implies ∆ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' In this case χ remains finite and we have near ∆ = 0, f0(∆) = 1 2 + ∆ √ 2π − ∆3 6 √ 2π + O � ∆4� f1(∆) = 1 √ 2π + ∆ 2 + ∆2 2 √ 2π + O � ∆4� f2(∆) = 1 2 + � 2 π ∆ + ∆2 2 + ∆3 3 √ 2π + O � ∆4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (C5) We note from these expansions that f0 − f2 ∝ ∆ as ∆ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (22) we then find M ∼ ∆−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Similarly, we have αc − α ∝ ∆ [this can be seen from expanding f 2 0 /f2 in the third relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (22)], so we can conclude that M ∼ (α − αc)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' These results are consistent with simulations (see for example Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' S1 SUPPLEMENTAL MATERIAL Competitive exclusion and Hebbian couplings in random generalised Lotka–Volterra systems Enrique Rozas Garcia1,2, Mark J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Crumpton3,4, Tobias Galla1,4 1 Instituto de F´ısica Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain 2 Department of Physics, Gothenburg University, 41296 Gothenburg, Sweden 3 Department of Mathematics, King’s College London, London WC2R 2LS, United Kingdom 4 Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, Manchester M13 9PL, UK S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' GENERATION OF RANDOMLY DILUTED INTERACTIONS The dynamics studied in this work, of the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (1), requires the generation of pairs of identically distributed correlated Bernoulli random variables (cij, cji) as part of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' To do this we have to find their joint probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Let p(cij = 1, cji = 1) = x, p(cij = 1, cji = 0) = y, p(cij = 0, cji = 1) = w, p(cij = 0, cji = 0) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (S1) We can solve for x, y, w, z using the desired moments in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (2) and the normalization condi- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' ⟨cij⟩ = x + y = c , ⟨cji⟩ = x + w = c , (S2) ⟨cijcji⟩ = x = Γc(1 − c) + c2 , x + y + w + z = 1 , to obtain p(cij = 1, cji = 1) = Γc(1 − c) + c2, p(cij = 1, cji = 0) = p(cij = 0, cji = 1) = c − Γc(1 − c) − c2, p(cij = 0, cji = 0) = 1 − 2c + Γc(1 − c) + c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (S3) Since c ∈ [0, 1] and all probabilities need to be non-negative, we find that the values that Γ can take are restricted, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Intuitively, since our variables are identically distributed and can only take two possible values, negative correlations imply exclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' For example, it is impossible for both variables to have mean 1 if they are perfectly negatively correlated, in fact, in that case one will have mean c and the other 1 − c, making c = 1/2 the only possible choice (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' S2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='0 c FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Shaded area indicates the possible pairs of c and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' DETAILS OF GENERATING FUNCTIONAL ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The generating functional To study Lotka-Volterra equations of the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (1), we will follow [11, 15] and use the generating functional Z[ψ(t)] = � D[x(t)] p[x(0)]ei � i � dtψi(t)xi(t) (S4) N � i=1 δ � d dt ln xi − 1 + uxi − � i̸=j cijJijxj + σζi(t) + hi(t) � , where ψ is a source field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We have introduced both a Gaussian white noise ζi(t) and a perturbation field hi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' These auxiliary fields will allow us to calculate magnitudes of interest later on, but they do not play any role in the upcoming calculation and can be ignored for the most part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Writing the δ-functions in their integral representation (and absorbing any resulting factors in the integration measure) we find Z[ψ(t)] = � D[x(t), ˆx(t)]p[x(0)] ei � i � dt ψi(t)xi(t)ei � i � dt ˆxi[yi−1+uxi−� j̸=i cijJijxj+σζi(t)+hi(t)] (S5) where we use the short-hand yi = d dt ln xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Our objective is to manipulate the generating functional (S5) until we can identify a sim- plified effective dynamics in terms of a single mean-field population x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' To do this we calculate the disorder average of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' This consists of an average over both random variables {cij} and {ξµ i }: Z[ψ(t)] = � D[x(t), ˆx(t)]p[x(0)] ei � i � dt ψi(t)xi(t)ei � i � dt ˆxi[yi−1+uxi+σζi(t)+hi(t)] × e−i � i � dt ˆxi[ � j̸=i cijJijxj], (S6) where the overbar denotes the disorder average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' S3 Once the disorder-averaged generating functional is known, we can calculate magnitudes of interest by taking derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' For example, the mean abundance, correlation function, and response function can be written as M(t) = 1 N � i ⟨xi(t)⟩ = −i N � i δZ δψi(t) ���� ψ(t)=0 (S7) C(t, t′) = 1 N � i ⟨xi(t)xi(t′)⟩ = −1 N � i δ2Z δψi(t)δψi(t′) ���� ψ(t)=0 (S8) G(t, t′) = 1 N � i δ⟨xi(t)⟩ δhi(t′) = −i N � i δ2Z δψi(t)δhi(t′) ���� ψ(t)=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (S9) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Introduction of the order parameters Before we explicitly carry out the disorder average it is convenient to introduce the following order parameters a(t) = 1 N � i xi(t) k(t) = i 1 N � i ˆxi(t) Q(t, t′) = 1 N � i xi(t)xi(t′) K(t, t′)) = 1 N � i xi(t)ˆxi(t′) L(t, t′) = 1 N � i ˆxi(t)ˆxi(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (S10) To introduce the order parameters into the generating functional (S6) we use δ-functions in their exponential representation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' and insert the following expressions (all equal to one),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 1 = � D[ˆa(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' a(t)] eiN � dt ˆa(t)[a(t)− 1 N � i xi(t)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 1 = � D[ˆk(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' k(t)] eiN � dt ˆk(t)[k(t)− i N � i ˆxi(t)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 1 = � D[ ˆQ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' t′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' t′)] eiN � dt dt′ ˆQ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='t′)[Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='t′)− 1 N � i xi(t)xi(t′)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 1 = � D[ ˆK(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' t′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' K(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' t′)] eiN � dt dt′ ˆ K(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='t′)[K(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='t′)− 1 N � i xi(t)ˆxi(t′)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' 1 = � D[ˆL(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' t′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' L(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' t′)] eiN � dt dt′ ˆL(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='t′)[L(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='t′)− 1 N � i ˆxi(t)ˆxi(t′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (S11) Relevant factors of 2π have here been absorbed into the measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' By introducing these S4 expressions into (S6) we obtain Z[ψ(t)] = � D[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' ˆa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' ˆk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' ˆQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' ˆK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' ˆL] eNΨ × � D[x(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' ˆx(t)]p[x(0)] ei � i � dt ψi(t)xi(t)ei � i � dt ˆxi[yi−1+uxi+σζi(t)+hi(t)] × e−i � i � dt [ˆa(t)xi(t)+iˆk(t)ˆxi(t)]e−i � i � dt dt′ [ ˆQ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='t′)xi(t)xi(t′)+ ˆ K(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='t′)xi(t)ˆxi(t′)+ˆL(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content='t′)ˆxi(t)ˆxi(t′)] × e−i � i � dt ˆxi[ � j̸=i cijJijxj],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (S12) where Ψ[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' ˆa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' ˆk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' ˆQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' ˆK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' ˆL] =i � dt � ˆa(t)a(t) + ˆk(t)k(t) � (S13) + i � dt dt′ � ˆQ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' t′)Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' t′) + ˆK(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' t′)K(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' t′) + ˆL(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' t′)L(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' t′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' For later purposes we define Ω[ˆa, ˆk, ˆQ, ˆK, ˆL] = 1 N � i log � � D[xi(t), ˆxi(t)]p[xi(0)] exp � i � dt ψi(t)xi(t) � × exp � i � dt ˆxi [yi − 1 + uxi + σζi(t) + hi(t)] � × exp � −i � dt [ˆa(t)xi(t) + iˆk(t)ˆxi(t)] � × exp � −i � dt dt′ [ ˆQ(t, t′)xi(t)xi(t′) + ˆK(t, t′)xi(t)ˆxi(t′) + ˆL(t, t′)ˆxi(t)ˆxi(t′)] �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (S14) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' Disorder average We follow mainly [4, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' The only term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (S5) containing the disorder variables {cij} and {ξµ i } is exp � −i � i>j � dt (ˆxicijJijxj + ˆxjcjiJjixi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (S15) To simplify the calculations we will only keeps the terms that are leading order in N, since we eventually intend to take the thermodynamic limit N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' To estimate the order of each term we use their variance, thus cij ∼ � ⟨c2 ij⟩ = √c ∼ O(N 0), (S16) and ⟨J2 ij⟩ = �� 1 cN αcN � µ=1 ξµ i ξµ j �2� = 1 (cN)2 αcN � µ,µ′=1 ⟨ξµ i ξµ j ξµ′ i ξµ′ j ⟩ = 1 (cN)2 αcN � µ,µ′=1 δµ,µ′ = α cN , (S17) S5 implying that Jij ∼ O(N −1/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' (S18) We first perform the average of (S15) over the {cij}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFKT4oBgHgl3EQfBS3T/content/2301.11703v1.pdf'} +page_content=' We begin by Taylor expanding the exponential � i1 +λj +��������� , +(9b) +where {λ1, λ2, . . . } are the eigenvalues (in decreasing order) +of +� √ρρT √ρ +� 1 +2 . With this in place, we now extend this sce- +nario to the bipartite regime where we will show how Al- +ice can assist Bob (ρB) to get the target state σB with op- +timal probability. Now we use the following parameteriza- +tion: sin2 α = +� +1 − Ic +� +ρB�� +/2 and sin2 β = Ig +� +σB� +with +α, β ∈ (0, π +2). +Lemma 3. For any bipartite pure state ψAB, the optimal prob- +ability of Bob preparing a local state σB, getting assistance +from Alice, is given by +P +� +ψAB → σB� += min +�sin2 α +sin2 β +, 1 +� +. +(10) +The proof of Lemma 3 is presented in the Appendix. In +Ref. [40] the authors provided tight continuity bounds for +the geometric measure. +Using these bounds, along with +Lemma 3, we can provide an analytical expression for the +optimal probability of Bob preparing a local state with an al- +lowed error, with assistance from Alice. Similarly, we can +also find a closed expression for the optimal achievable fi- +delity, for a given probability of success. The following theo- +rem collects these results. +Theorem 4. For any bipartite pure state ψAB, the optimal +probability P f of Bob preparing a local state σB, with a fi- +delity f via assistance from Alice, is given by +Pf +� +ψAB → σB � += +����������� +1 +for α − β + γ ≥ 0 +sin2 α +sin2 ( β − γ ) +otherwise +(11) +where γ = cos−1 � +f. +The optimal achievable fidelity for a given probability of +success p, can be expressed as: +Fp +� +ψAB → σB � += +��������������� +1 +for p ≤ sin2 α +sin2 β +cos2 +� +β − sin−1 +�sin α +√p +�� +otherwise. +(12) +Details of the proof for the above theorem can be found in +the Appendix. +Imaginarity in channel discrimination—We will now dis- +cuss the role of imaginarity in channel discrimination. Specif- +ically, here we focus on the variant of channel discrimination +which we call ancilla-free, in that it does not involve an ancil- +lary system (cf. Refs. [46, 47]). It can be regarded as a game, +where one has access to a “black box” with the promise that +it implements a quantum channel Λj with probability pj. The +goal of the game is to guess Λj by choosing optimal initial +state ρ and positive operator-valued measure (POVM) +� +M j +� +, +which is used to distinguish the Λj (ρ)’s. Theoretically, the +probability of guessing the channel Λj correctly is given as +psucc +� +ρ, +� +p j, Λj +� +, +� +M j +�� += +� +j +pj Tr +� +M jΛj (ρ) +� +. +(13) +Recently, it has been shown that any quantum resource has an +operational advantage in the channel discrimination task [46, +47], namely a resource state ρ (i.e. a quantum state that is not +free) outperforms any free σ in a specific channel discrimina- +tion task. + +4 +Now we put the above protocol into imaginarity theory by +considering the task of discrimination of real channels. To see +an advantage, we need imaginarity both in the probe state and +in the measurement, since, as we show in the Appendix, this +task is equivalent to LOCC discrimination of their correspond- +ing normalized Choi states, in which we need imaginarity in +the measurements of both particles. To better illustrate this +idea, we will provide an example of two real channels that +cannot be distinguished in the ancilla-free scenario by using +only real states and measurements, but they become instead +perfectly distinguishable once we have access to imaginarity +for states and measurements. To this end, let us consider two +real qubit channels prepared with equal probability: +N : ρ �→ 1 +2 ( ρ + σx σz ρ σz σx ) , +M : ρ �→ 1 +2 ( σx ρ σx + σz ρ σz ) , +(14) +where σx and σz are Pauli matrices. If we input a real state ρ +into either of these two channels, they will produce exactly the +same output 1/2, thus we cannot distinguish them better than +making a random guess, even if we allowed imaginarity in our +measurements. On the other hand, if imaginarity is forbidden +in measurements, no matter how we choose the probe state +(even if itis non-real), we cannot still distinguish them at all, +because the only way to discriminate between the outputs of +the two channels would be to perform a measurement associ- +ated with the σy Pauli matrix. Indeed, if the probe state has an +off-diagonal entry ρ01 with non-zero imaginary part, wherever +the output of N has Im ρ01, the output of M will show −Im ρ01 +in its place. Only if we implement a projective measurement +of σy can we perfectly distinguish these two channels. There- +fore, the only way to achieve a success probability better than +random guessing is to introduce imaginarity into both the ini- +tial state ρ and the measurement. +It is worth noting that the same two channels N and M be- +come perfectly distinguishable even with no imaginarity in the +probe state and in the measurement if we remove the require- +ment of ancilla-free discrimination. If we allow an ancilla R, +we need to consider a bipartite input state ρRA and a bipartite +POVM +� +MRA +1 , MRA +2 +� +, with success probability +psucc +� +ρ, +�1 +2, Λj +� +, +� +M j +�� += 1 +2 +2 +� +j=1 +Tr +� +MRA +j +� +IR ⊗ Λj +� � +ρRA�� +, +(15) +where Λ1 = N and Λ2 = M. Now, let us take ρRA = φ+ = +|φ+⟩⟨φ+|, with |φ+⟩ = +1√ +2 (|00⟩ + |11⟩). If we feed φ+ to both +channels, we get +I ⊗ N �φ+� = 1 +2 +�|φ+⟩⟨φ+| + |ψ−⟩⟨ψ−|� , +I ⊗ M �φ+� = 1 +2 +�|φ−⟩⟨φ−| + |ψ+⟩⟨ψ+|� , +(16) +where |φ−⟩ = +1√ +2 (|00⟩ − |11⟩), |ψ+⟩ = +1√ +2 (|01⟩ + |10⟩), and +|ψ−⟩ = +1√ +2 (|01⟩ − |10⟩). As noted in Ref. [18], these two out- +put states can be perfectly distinguished by the real POVM +B +HWP +@404nm +BBO +QP +SPD +BS +M +AA +PBS +QWP +@404nm +HWP +@808nm +QWP +@808nm +A +C +Entangled +Source +State Prepara�on +& Channel Implemeta�on +Discrimina�on +& Tomography +FIG. 1. +Experimental setup. The whole experimental setup is di- +vided into three modules: A Entangled source, B state preparation & +channel implementation, and C discrimination & tomography. The +optical components include: QP, quartz plate; SPD, single photon de- +tectors; BS, beamsplitters; AA, adjustable aperture; PBS, polarizing +beamsplitter; QWP, quarter-wave plate; HWP, half-wave plate. +{M1, M2}, where +M1 = | ˆ+⟩⟨ ˆ+| ⊗ | ˆ−⟩⟨ ˆ−| + | ˆ−⟩⟨ ˆ−| ⊗ | ˆ+⟩⟨ ˆ+|, +M2 = | ˆ+⟩⟨ ˆ+| ⊗ | ˆ+⟩⟨ ˆ+| + | ˆ−⟩⟨ ˆ−| ⊗ | ˆ−⟩⟨ ˆ−|. +(17) +This shows that the two real channels can be distinguished +perfectly with the aid of an ancilla, only using real states and +real measurements. +EXPERIMENTS +We experimentally implement the aforementioned assisted +imaginarity distillation and channel discrimination protocols. +The whole experimental setup is illustrated in Fig. 1, which +consists of three modules: module A enables us to prepare +a two-qubit entangled state via spontaneous parametric down +conversion (SPDC) process: +|ψ⟩AB = a |00⟩ + b |11⟩, +(18) +with arbitrary a and b with |a|2+|b|2 = 1 which can be tuned by +changing the angles of 404 nm HWP and QWP. Note that we +have conventionally set |0⟩ := |H⟩ and |1⟩ := |V⟩. Module B +utilizes an unbalanced Mach-Zehnder interferometer together +with module A to prepare a class of Werner states: +ρAB = p |φ+⟩⟨φ+| + (1 − p) 1 +4 , +(19) +where p denotes the purity of the two-qubit state. Module B +also allow us to implement single-qubit channels in ancilla- +free scenario. Module C allows us to perform quantum-state +tomography (QST) to identify the final two-qubit polarization- +encoded states concerned, or perform assisted imaginarity dis- +tillation by performing local measurement on Alice’s photons + +5 +(a) +(b) +FIG. 2. +Experimental results for assisted imaginarity distilla- +tion. (a) Initial pure states |ψ⟩AB = a|00⟩ + b|11⟩; (b) initial Werner +states ρAB = p|φ+⟩⟨φ+| + (1 − p) 1/4. In both experiments, red disks +represent the calculated fidelity of imaginarity by assistance using +Theorem 2 for experimentally reconstructed two-qubit states, and +blue disks represent actual obtained average fidelity of imaginarity +in experiments using the optimal measurement on Alice’s system. +and identifying the exact amount of imaginarity by QST of +Bob’s state. Moreover, this module allows us to implement +channel discrimination by performing local measurement on +the polarization state of a single-photon when the other is used +as a trigger. We refer to the Appendix for more details. +We then perform proof of principle experiments of the one- +shot assisted imaginarity distillation and the ancilla-free chan- +nel discrimination tasks. Results are shown in Figs. 2 and 3 +respectively. +For assisted imaginarity distillation, we experimentally pre- +pare two classes of two-qubit states. The first class of states +as in Eq. (18). Theoretically, the upper bound for single-shot +assisted imaginarity distillation can be calculated from Theo- +rem 2 as FI +� +|ψ⟩AB� += 2 |ab|. From Fig. 2(a), we can see that +the experimentally obtained average imaginarity after assis- +tance (blue disks) approximately equals to the experimentally +obtained upper bound (red disks) within reasonable experi- +mental imperfections. The second class of states are gener- +ated as Werner states in Eq. (19). Theoretically, the maximum +average fidelity of imaginarity after assistance is calculated as +FI(ρAB) = p. Fig. 2(b) details the relevant experimental re- +sults. From both results we see that experimentally obtained +average fidelity of imaginarity data and upper bound obtained +from two-qubit state tomography agree well with theoretical +predictions. +We then show the usefulness of imaginarity in channel dis- +crimination for various discrimination tasks. Fig. 3 details +these results for two discrimination tasks. The first discrimi- +nation task involves two channels given by +M ( ρ, p ) = pρ + (1 − p) σx σz ρ σz σx, +N (ρ) = 1 +2 (σx ρ σx + σz ρ σz) . +(20) +Note that the two channels preserve real density matrices. The +experimental results of this discrimination task are shown in +Fig. 3(a). If we can use imaginarity in measurements and +initial states, we can perfectly distinguish the two channels +[orange disks in Fig. 3(a)]. However, if we allow only real +density matrices as initial states or real measurement oper- +ators, we get a theoretical optimal guessing probability of +1/2 + |2p − 1| /4 for the ancilla-free channel discrimination. +Experimental data are in agreement with the theoretical pre- +dictions [see green disks in Fig.3(a)]. +Here we note that +the two channels are exactly the same as in Eqs. (14) when +p = 1/2. +For the second discrimination task, we consider +M ( ρ, w ) = w ρ + (1 − w) 1 +2 , +N (ρ) = 1 +2 (σx ρ σx + σz ρ σz) . +(21) +The results are shown in Fig. 3(b). If non-real states and mea- +surement operators are allowed, then we get a theoretical op- +timal distinguishing probability as 3/4+w/4, which is plotted +as the upper orange line in Fig. 3(b). The relevant experimen- +tally obtained distinguishing probabilities are shown as orange +disks. If imaginarity is prohibited in this task, then the optimal +distinguishing probability reads 1/2 + w/4, and is plotted as +the lower green line, together with experimental values repre- +sented by green disks. We can draw a similar conclusion to +the first discrimination task. +DISCUSSION +The results presented above are mainly based on the new +set of LQRCC operations which was introduced and studied +in this article. We considered assisted imaginarity distillation +in this setting, and completely solved the problem for general +two-qubit states. Moreover, we discussed the task of single- +shot assisted imaginarity distillation for arbitrary pure states in +higher dimensions. The usefulness of imaginarity in channel +discrimination is both theoretically and experimentally shown +for a class of real channels. +There are in fact many scenarios of practical relevance +where the task of assisted imaginarity distillation can play a +central role. For instance, think of a remote or unaccessible +system on which imaginarity is needed as a resource (e.g., in +the task of local discrimination of quantum states): our results +give optimal prescriptions to inject such imaginarity on the +remote target by acting on an ancilla. The results provide in- +sight into both the operational characterization as well as the + +6 +(a) +(b) +FIG. 3. +Experimental results for discrimination tasks. Two +channel discrimination tasks are tested : +(a) Mp (ρ) += +pρ + +(1 − p) σxσzρσzσx, N (ρ) = (σxρσx + σzρσz) /2. Using imaginar- +ity one can perfectly distinguish the two channels. +However, if +only real operators are allowed, then the optimal guessing proba- +bility is 1/2 + |2p − 1| /2; (b) Mw (ρ) = wρ + (1 − w) 1/2, N (ρ) = +(σxρσx + σzρσz) /2. The optimal probabilities for successful guess- +ing are 3/4 + w/4 and 1/4 + w/4 for the case where imaginarity is +allowed, and where only real states and measurements are allowed, +respectively. +mathematical formalism of the resource theory of imaginar- +ity, contributing to a better understanding of this fundamental +resource. +The work at the University of Science and Technology +of China is supported by the National Key Research and +Development Program of China (No. +2018YFA0306400), +the National Natural Science Foundation of China (Grants +Nos. 12134014, 12104439, 61905234, 11974335, 11574291, +and 11774334), the Key Research Program of Frontier Sci- +ences, CAS (Grant No. +QYZDYSSW-SLH003), USTC +Research Funds of the Double First-Class Initiative (Grant +No. YD2030002007) and the Fundamental Research Funds +for the Central Universities (Grant No. +WK2470000035, +WK2030000063). The work at Poland was supported by the +National Science Centre, Poland, within the QuantERA II +Programme (No 2021/03/Y/ST2/00178, acronym ExTRaQT) +that has received funding from the European Union’s Horizon +2020 research and innovation programme under Grant Agree- +ment No 101017733 and the “Quantum Optical Technolo- +gies” project, carried out within the International Research +Agendas programme of the Foundation for Polish Science +co-financed by the European Union under the European Re- +gional Development Fund. CMS acknowledges the support +of the Natural Sciences and Engineering Research Council of +Canada (NSERC) through the Discovery Grant “The power +of quantum resources” RGPIN-2022-03025 and the Discov- +ery Launch Supplement DGECR-2022-00119. +∗ carlomaria.scandolo@ucalgary.ca +† gyxiang@ustc.edu.cn +‡ a.streltsov@cent.uw.edu.pl +[1] P. Jordan, J. von Neumann, and E. P. 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To see this, let’s +say we want to implement a quantum operation Λ on ρ with +Kraus operators given by {Kj}, such that � +j K† +j Kj = P ≤ 1. +To implement this, we construct a real quantum operation (Λr) +with Kraus operators given by {Kj ⊗ | ˆ+⟩⟨ ˆ+| + K∗ +j ⊗ | ˆ−⟩⟨ ˆ−|}. It +is easy to see that +Λr(ρ ⊗ | ˆ+⟩⟨ ˆ+|) = Λ(ρ) ⊗ | ˆ+⟩⟨ ˆ+| +(22) +and +� +j +(K† +j ⊗ | ˆ+⟩⟨ ˆ+| + KT +j ⊗ | ˆ−⟩⟨ ˆ−|)(Kj ⊗ | ˆ+⟩⟨ ˆ+| + K∗ +j ⊗ | ˆ−⟩⟨ ˆ−|) += P ⊗ | ˆ+⟩⟨ ˆ+| + PT ⊗ | ˆ−⟩⟨ ˆ−| ≤ 1 ⊗ 1 +The last inequality follows from the fact that, +P ≤ 1 ⇐⇒ PT ≤ 1. +(23) +This shows that one imbit is sufficient to implement general +quantum operations. Now we show that, there exists a quan- +tum channel, which necessarily requires one imbit, to imple- +ment via real operations. As an example, consider the follow- +ing map (Λ+) given by +Λ+(ρ) = | ˆ+⟩⟨ ˆ+| forall ρ. +(24) +We now show, by contradiction, that the above quantum map +requires one imbit to implement. Let’s say there is a imple- +mentation (with a real operation Λ′ +r) such that, +Λ′ +r(ρ ⊗ σ) = Λ+(ρ) = | ˆ+⟩⟨ ˆ+| +(25) +here, if σ is not an imbit and ρ = |0⟩⟨0|, its easy to see that +the state transformation in Eq. (25) is not possible. This is +because Ig(|0⟩⟨0| ⊗ σ) = Ig(σ) < Ig(| ˆ+⟩⟨ ˆ+|). + +8 +Properties of LRCC operations +For any real CP map Λ : R → R′, ΓΛ +RR′ is the corresponding +choi matrix of ΓΛ +RR′, given by +ΓΛ +RR′ = 11 ⊗ Λ +�������� +� +j,k +| j⟩⟨k| ⊗ | j⟩⟨k| +�������� . +(26) +Any LQRCC map (Λ) can be represented in the following way +Λ = +� +i +Λi ⊗ Λr +i. +(27) +Here, λi is a CP (trace non increasing) map acting locally on +Alice’s hilbert spapce and Λr +i is a local real CP map on Bob’s +hilbert space. The choi matrix of Λi ⊗ Λr +i is given by +Γ +Λi⊗Λr +i +AB→A′B′ += 11AB ⊗ Λi ⊗ Λr +i +�� +j, j′,k,k′ | jk⟩⟨ j′k′| ⊗ | jk⟩⟨ j′k′| +� += � +j,j′,k,k′ | j⟩⟨j′| ⊗ |k⟩⟨k′| ⊗ Λi(|j⟩⟨j′|) ⊗ Λr +i(|k⟩⟨k′|) +Let’s now take the transpose of this choi matrix over BB′ +(Γ +Λi⊗Λr +i +AB→A′B′)TBB′ = � +j, j′,k,k′ | jk′⟩⟨j′k| ⊗ Λi(| j⟩⟨j′|) ⊗ (Λr +i(|k⟩⟨k′|))T += � +j,j′,k,k′ | jk′⟩⟨j′k| ⊗ Λi(| j⟩⟨ j′|) ⊗ Λr +i(|k′⟩⟨k|) += Γ +Λi⊗Λr +i +AB→A′B′ +(28) +In the second line we used the fact that, real operations com- +mute with transpose. Since any LQRCC operation can be rep- +resented as (27), the choi matric of any LQRCC operation +is invariant under partial transpose over Bob’s systems. For +LRCC operations, additionally the choi matrix is always real. +Proof of Theorem 1 +In the following, we assume that A and B is a qubit. A +general two-qubit state ρAB can be written as +ρ = 1 +4 +��������1⊗1 + +� +k +akσk⊗1 + +� +l +bl1⊗σl + +� +k,l +Eklσk⊗σl +�������� , +(29) +where a = (a1, a2, a3) and b = (b1, b2, b3) are local Bloch +vectors of Alice and Bob, respectively, and Ekl = Tr(σk⊗σlρ). +A general single-qubit POVM element on Alice’s side can be +written as +MA +n = qn +��������1 + +� +j +αnjσ j +�������� +(30) +with probabilities 0 ≤ qn ≤ 1, � +n qn = 1, and vectors αn such +that |αn| ≤ 1 and � +n qnαn = 0. The measurement {MA +n } gives +outcome n with probability +pn = qn (1 + a · αn) , +(31) +and the Bloch vector of Bob’s post-measurement state is +bn = b + ETαn +1 + a · αn +. +(32) +After Alice communicates her measurement outcome n to +Bob, he applies a real operation Λn to his post-measurement +state ρB +i . For each measurement outcome n, Bob aims to max- +imize the fidelity between Λn[ρB +n] and the maximally imagi- +nary state | ˆ+⟩. The maximum is given by the fidelity of imag- +inarity FI which for single-qubit states ρB +n reduces to +FI(ρB +n) = 1 +2 +� +1 + +���Tr[ρB +nσ2] +��� +� +. +(33) +Using this result together with Eqs. (31) and (32) we can ex- +press our figure of merit Fa as follows: +Fa(ρAB) = max +MAn +� +n +pnFI(ρB +i ) += max +qn,αn +1 +2 +�������1 + +� +n +qn |b2 + s · αn| +������� , +(34) +where the maximization in the last expression is performed +over all vectors αn and probabilities 0 ≤ qn ≤ 1 such that +� +n qn = 1, |αn| ≤ 1 and � +n qnαn = 0. +If |b2| ≥ |s|, then using the conditions |αn| ≤ 1 and +� +n qnαn = 0 we immediately obtain +� +n +qn |b2 + s · αn| = |b2| +(35) +for any choice of qn and αn. +This directly implies that +Fa(ρAB) = 1/2+|b2|/2 in this case, in accordance with Eq. (8). +We now consider the case if |b2| < |s|. We will show that in +the maximization in Eq. (34) it is enough to consider POVMs +consisting of two elements. For a given set of vectors αn and +probabilities qn we introduce two sets, depending whether b2+ +s · αn is positive or negative: +S 0 = {n : b2 + s · αn ≥ 0}, +(36a) +S 1 = {j : b2 + s · α j < 0}. +(36b) +Using these sets, we express the sum � +n qn |b2 + s · αn| as fol- +lows: +� +n +qn |b2 + s · αn| = +�������� +� +n∈S 0 +qn +�������� +������b2 + +� +n∈S 0 qn(s · αn) +� +n∈S 0 qn +������ +(37) ++ +�������� +� +j∈S 1 +qj +�������� +������b2 + +� +j∈S 1 qj(s · α j) +� +j∈S 1 q j +������ . +In the next step, we introduce the probabilities ˜q0 = � +n∈S 0 qn, +˜q1 = � +j∈S 1 qj and vectors +˜α0 = +� +n∈S 0 qnαn +� +n∈S 0 qn +, +(38a) +˜α1 = +� +j∈S 1 q jαj +� +j∈S j qj +. +(38b) +Noting that +b2 + s · ˜α0 ≥ 0, +(39a) + +9 +b2 + s · ˜α1 < 0, +(39b) +we further obtain the following result: +� +n +qn |b2 + s · αn| = ˜q0|b2 + s · ˜α0| + ˜q1|b2 + s · ˜α1| += ˜q0(b2 + s · ˜α0) − ˜q1(b2 + s · ˜α1). +(40) +The vectors ˜αn and probabilities ˜qn fulfill the conditions +� +n ˜qn = 1, | ˜αn| ≤ 1, and � +n ˜qn ˜αn = 0. This implies that +they correspond to a two-element POVM on Alice’s side via +the relation in Eq. (30). +The arguments just presented show that the maximum in +Eq. (34) can be achieved with two vectors α0 and α1 and two +probabilities q0 and q1 having the properties 0 ≤ q0 ≤ 1, +q1 = 1 − q0, |αn| ≤ 1, � +i qnαn = 0. To complete the proof, we +will show that the optimal solution is obtained for +q0 = q1 = 1 +2, +(41a) +α0 = −α1 = s +|s|. +(41b) +Recalling that |b2| ≤ |s|, the values in Eq. (41) immediately +give a lower bound on the assisted fidelity of imaginarity: +Fa(ρAB) ≥ 1 +2(1 + |s|). +(42) +Let now qn and αn be optimal probabilities and vectors [not +necessarily coinciding with Eq. (41)]. Without loss of gener- +ality we can assume that1 +b2 + s · α0 ≥ 0, +(43a) +b2 + s · α1 < 0. +(43b) +For the assisted fidelity of imaginarity we thus obtain +Fa(ρAB) = 1 +2[q0(b2 + s · α0) − q1(b2 + s · α1)] + 1 +2. +(44) +Since q0 + q1 = 1, it must be that either q0 ≤ 1/2 or q1 ≤ 1/2. +In the first case we rewrite Eq. (44) as follows: +Fa(ρAB) = 1 − b2 +2 ++ q0(b2 + s · α0) ≤ 1 +2(1 + |s|). +(45) +In the second case (q1 ≤ 1/2), we rewrite Eq. (44) as +Fa(ρAB) = 1 + b2 +2 +− q1(b2 + s · α1) ≤ 1 +2(1 + |s|). +(46) +Thus, for |b2| < |s| the assisted fidelity of imaginarity is +bounded above as +Fa(ρAB) ≤ 1 +2(1 + |s|). +(47) +1 Otherwise, if b2 + s · αn is positive (or negative) for all n, we obtain +� +n qn |b2 + s · αn| = |b2|. Since |b2| < |s|, this means that we will not +be able to reach the maximal value. +Together with Eq. (42) this proves that Fa(ρAB) = 1/2 + |s|/2 +in this case, and the proof of the theorem is complete. +Theorem 2 has few surprising consequences. If a two-qubit +state has the property |b2| ≥ |s|, then the assisted fidelity of +imaginarity coincides with the fidelity of imaginarity of Bob’s +local state: Fa(ρAB) = (1 + |b2|)/2. Thus, in this case Bob will +not gain any advantage from assistance, as he can obtain the +maximal fidelity by performing a local real operation without +any communication. For example, let us consider a quantum +state shared by Alice and Bob +ρAB = p +2 11A ⊗ | ˆ+⟩⟨ ˆ+| + (1 − p)|φ+⟩⟨φ+| +(48) +where we have b2 = p and s = (0, p−1, 0). Then if p = 1, then +ρAB is a product pure state, then no matter what Alice does, +Bob can always get the maximal imaginary state | ˆ+⟩. If 1 +2 < +p < 1, the state ρAB has nonzero entanglement, but we have +|b2| > |s|. If Alice chooses a projective measurement along +α, then Bob will get states with Bloch vector b ± ET · α with +equal probability. Then the average fidelity with maximally +imaginary state reads 1 +2 (|p + (1 − p)α2| + |p − (1 − p)α2|). As +we have 1 +2 < p < 1, |(1 − p)α2| < p, then the average fidelity +reads p. For all other two-qubit states the proof of Theorem 2 +provides an optimal procedure for obtaining maximal fidelity +of imaginarity on Bob’s side. For this, Alice needs to perform +a von Neumann measurement in the basis {|ψ0⟩, |ψ1⟩}, where +|ψ0⟩ has the Bloch vector s/|s|. The outcome of the measure- +ment is communicated to Bob, who leaves his state untouched +if the outcome was 0, and otherwise applies the real unitary +iσ2. Needs to be checked +Proof of Lemma 1 +Note that, the geometric measure of imaginarity and the +concurrence of imaginarity are given by [40, 45] +G(ρ) = min +e +� +j +pj +1 − |⟨ψ∗ +j|ψ j⟩| +2 += 1 − +� +F(ρ, ρT) +2 +(49) +C(ρ) = min +e +� +pj|⟨ψ∗ +j|ψ j⟩| = max +�������0, λ1 − +� +k>1 +λk +������� (50) +In the above maxe and mine are maximisation and minimisa- +tion over pure state ensembles of ρ. Whereas, {λ1, λ2...} are +the eigenvalues (in decreasing order) of ( √ρρT √ρ) +1 +2 . In gen- +eral, for probabilistic transformations, the following inequal- +ity holds +p(ρ → σ) ≤ min +� G(ρ) +G(σ), 1 +� +. +(51) +It was further shown in [40], that the optimal probability of +converting a pure state ψ to a arbitrary quantum state ρ is given +by +p(ψ → ρ) = min +�G(ψ) +G(ρ) , 1 +� +. +(52) + +10 +In a one way LQRCC procedure, Alice performs a general +quantum measurement and corresponding to the outcomes +(with probabilites {p j}) of Alice, Bob’s local state is found +in the state ρj, such that, {pj, ρi} is an ensemble of ρB. Con- +ditioned on the outcome of Alice (i), Bob can perform a local +stochastic real operation on ρi, probabilistically converting it +into σB. Using Eq. (51) and Eq. (52), it follows +Pa ≤ +� +j +p j min +� G(ρj) +G(σB), 1 +� +≤ +� +jk +p jqk min +�G(ψ j,k) +G(σB) , 1 +� +. +(53) +The second inequality follows from Eq.(49), G(ρj) is cal- +culated by minimising over all pure state ensembles of ρj. +Therefore, the second inequality holds for any pure state de- +composition of ρj, like {qk, ψjk}. Note that {pjqk, ψjk} is a pure +state decomposition of ρB. Note that, any pure state decompo- +sition of ρB can be realised by a suitable local measurement by +Alice. Using this fact, along with Eq.(49) and Eq.(52) implies +that +Pa = min +�1 − mine +� +k pk|⟨ψk|ψ∗ +k⟩| +2G(σB) +, 1 +� += min +������� +1 − C(ρB) +1 − +� +F(σB, (σB)T) +, 1 +������� . +Here, mine is the minimisation over pure state ensembles of +ρB. This completes the proof. +Proof of Theorem 4 +From Lemma 1, we know that optimal probability for Bob +to locally achieve σB from a shared bipartite pure state ψAB +with unit fidelity, via LQRCC is given by +P(ψAB → σB) = min +�1 − Ic(ρB) +2Ig(ρ) +, 1 +� +. +(54) +If we want to achieve σB with fidelity at least f, the best strat- +egy is to go to a state (σ′B), within the fidelity ball around σB, +with a minimal geometric measure of imaginarity. Therefore, +Pf (ψAB → σB) = min +�1 − Ic(ρB) +2Ig(σ′B) , 1 +� +. +(55) +From [40], we know that +Ig(σ′B) = sin2 � +max +� +sin−1 � +Ig(σB) − cos−1 � +f, 0 +�� +. (56) +We now define +m = sin−1 +� +1 − Ic(ρB) +2 +−sin−1 � +Ig(σB)+cos−1 � +f. (57) +First, consider the case when m ≥ 0, which implies +sin−1 � +Ig(σB) − cos−1 � +f ≤ sin−1 +� +1 − Ic(ρB) +2 +. +(58) +We know that +sin−1 � +Ig(σB) − cos−1 � +f ∈ [−π/2, π/4] +(59) +and sin−1 +� +1−Ic(ρB) +2 +∈ [0, π/4]. Therefore, +max +� +sin−1 � +Ig(σB) − cos−1 � +f, 0 +� +≤ sin−1 +� +1 − Ic(ρB) +2 +. +(60) +Using these results, we get +Ig(σB′) = sin2 � +max +� +sin−1 � +Ig(σB) − cos−1 � +f, 0 +�� +(61) +≤ sin2 +�������sin−1 +� +1 − Ic(ρB) +2 +������� = 1 − Ic(ρB) +2 +. +For the case when 1−Ic(ρB) +2 +> 0, the above inequality implies +1 − Ic(ρB) +2Ig(σB′) ≥ 1. +(62) +This shows that Pf (ψAB → σB) = 1 when m ≥ 0. +Now, we look at the other case when m < 0, i.e., +sin−1 � +Ig(σB) − cos−1 � +f > sin−1 +� +1 − Ic(ρB) +2 +> 0. +(63) +From the above inequality and Lemma 1, we have +Pf (ψAB → σB) = +1 − Ic(ρB) +2 sin2(sin−1 � +Ig(σB) − cos−1 � +f) +. (64) +Using the above result, a closed expression can also be found +for Fp. Let’s first consider the case when p ≤ +1−Ic(ψAB) +2Ig(σB) +< +1, in this case Fp(ψ → σB) = 1 (follows from Lemma 1). +When 1 ≥ p > +Ig(ψ) +Ig(σB), the optimal achievable fidelity can be +obtained by solving Eq. (11) for f, which gives +Fp(ψAB → σB) = cos2 +���������sin−1 � +Ig(σB) − sin−1 +� +1 − Ic(ρB) +2p +��������� . +This completes the proof. +SDP upperbounds for state transformations +As we already mentioned, for any real CP map Λ : R → R′, +ΓΛ +RR′ is the corresponding choi matrix of ΓΛ +RR′, given by +ΓΛ +RR′ = 11 ⊗ Λ +�������� +� +j,k +| j⟩⟨k| ⊗ | j⟩⟨k| +�������� . +(65) +It follows that (see Eq. (4.2.12) of [48]), +Λ(ρR) = TrR(ΓΛ +RR′(ρT +R ⊗ 11R′)). +(66) + +11 +For any pure state |ψR′⟩ +⟨ψR′|Λ(ρR)|ψR′⟩ = Tr(ΓΛ +RR′(ρT +R ⊗ |ψR′⟩⟨ψR′|)). +(67) +Using the fact that, choi matrices of LQRCC operations are +invariant under partial transpose, one can give a SDP com- +putable upperbound for the optimal achievable fidelity for a +given probability Fp(ρAB → |ψAB⟩): +Maximise: +1 +p Tr(XABA′B′ρT +AB ⊗ |ψA′B′⟩⟨ψA′B′|) +(68) +under the constraints, +XABA′B′ ≥ 0, XTBB′ +ABA′B′ = XABA′B′, TrA′B′ XABA′B′ ≤ 1AB and +Tr(XABA′B′ρT +AB ⊗ 1B′) = p. +(69) +Quantum Chernoff divergence and scaling of asymptotic +imaginarity distillation +Fidelity of imaginarity FI, quantifies the maximum achiev- +able fidelity between a state ρ and the maximally imaginary +state. It can be expressed as +FI(ρ) = max +Λ F(Λ(ρ), | ˆ+⟩⟨ ˆ+|) = 1 +2 + 1 +4||ρ − ρT||1. +(70) +Here, the maximisation is performed over all real CPTP maps. +If we have n copies of ρ, we can write +FI(ρ⊗n) = 1 +2 + 1 +4||ρ⊗n − (ρT)⊗n||1. +(71) +If ρ is a pure state, i.e., ρ = |ψ⟩⟨ψ|, then we can calculate +fidelity of imaginarity of multiple copies as +FI(|ψ⟩⟨ψ|⊗n) = 1 +2 + 1 +4|||ψ⟩⟨ψ|⊗n − (|ψ⟩⟨ψ|T)⊗n||1 += 1 +2 + 1 +4|||ψ⟩⟨ψ|⊗n − (|ψ∗⟩⟨ψ∗|)⊗n||1 += 1 +2 + 1 +2 +� +1 − |⟨ψ∗|ψ⟩|2n. +(72) +For general states, to see the behaviour of FI(ρ⊗n), with in- +creasing n, consider the quantity P = 1 − FI(ρ⊗n). +From +Ref. [49], it follows that the following limit exists and is equal +to the quantum Chernoff divergence between ρ and ρT: +lim +n→∞ +− log P +n += χ(ρ, ρT) = − log( min +0≤s≤1 Tr(ρs(ρT)1−s)). (73) +One can analytically perform this minimisation and show that +minimum value is attained at s = 1/2. In order to show this +fact, let’s assume that the spectral decomposition of ρ is given +by +ρ = +� +j +p j|ψj⟩⟨ψj|, +(74) +and therefore +ρT = +� +j +p j|ψ∗ +j⟩⟨ψ∗ +j|. +(75) +The Chernoff divergence is given by +χ(ρ, ρT) = − log(min0≤s≤1 Tr(� +j ps +j|ψ j⟩⟨ψ j|)(� +k p1−s +k +|ψ∗ +j⟩⟨ψ∗ +j|)) += − log(min0≤s≤1 +� +j,k ps +i p1−s +k +|⟨ψ j|ψ∗ +k⟩|2). +(76) +Note that, |⟨ψ j|ψ∗ +k⟩| = |⟨ψk|ψ∗ +j⟩|. This implies that +χ(ρ, ρT) = − log( min +0≤s≤1 +� +j≤k +(ps +jp1−s +k ++ ps +kp1−s +j +)|⟨ψ j|ψ∗ +k⟩|2) +(77) +here, ps +jp1−s +k ++ ps +kp1−s +j +≥ 2 √pjpk. This follows from AM-GM +inequality, which says a+b +2 +≥ +√ +ab for all a, b ≥ 0. This lower +bound (minimum value) is attained at s = 1/2. This proves +that +χ(ρ, ρT) = − log(Tr +� +ρρT). +(78) +Therefore, from Eq. (73), it follows that asymptotically the +fidelity of imaginarity behaves as +FI(ρ⊗n) ∼ 1 − exp(−n · χ(ρ, ρT)) +(79) += 1 − (Tr +� +ρρT)n +Proof of the relation between channel discrimination and state +discrimination +Here we demonstrate a clear link between the task of +ancilla-free channel discrimination and the task of LOCC dis- +crimination of bipartite states, the latter studied in Refs. [4, 18, +20]. Specifically, we consider the following two scenarios: +1. Let N and M be two real channels from A to B, cho- +sen with equal probability 1 +2. If we want to discrim- +inate between them in an ancilla-free scenario better +than with a random guess, we must find a real state +ρ of A and a real POVM element E of B such that +Tr �EN (ρ)� � Tr �EM (ρ)�. Notice that this protocol +does not involve any bipartite input states and bipartite +effects. +2. Let N and M be two real channels from A to B. This +time, we bring in the maximally entangled state φ+ = +|φ+⟩⟨φ+|AA′, between systems A and A′ (A′ is a copy of +A), where |φ+⟩ = � +j | jj⟩/ √dA, and dA is the dimen- +sion of A. We apply N and M only to the A′ part of +this maximally entangled state. This results in two bi- +partite states between systems A and B, NAB and MAB, +respectively, which are the normalized Choi states of +the two channels N and M. Now consider the task of +discriminating between these two bipartite states of AB +using only local real measurements. Again, if we want +to discriminate between them better than with a random +guess, we must find a real POVM element E of system +A and a real POVM element F of system B such that +Tr +� +(E ⊗ F) NAB� +� Tr +� +(E ⊗ F) MAB� +. + +12 +In the following we show that these two scenarios produce +the same probabilites when POVMs are applied to states. Note +that we can reconstruct the action of a channel on a state from +its normalized Choi state: if N is a channel from A to B, ρ is +a state of A, we have that N (ρ) can be written in terms of the +normalized Choi state NAB as +N +� +ρA� += dA TrA +��� +ρA�T ⊗ 1B� +NAB� +, +(80) +where dA is the dimension of the input system A. Thus, if E +is a (real) POVM element on B, omitting system superscripts +for simplicity, we have +Tr �EN (ρ)� = dA Tr +�� +ρT ⊗ E +� +N +� += Tr +�� +1 +√dA +ρT ⊗ +1 +√dA +E +� +N +� +. +(81) +Note that 0 ≤ +1 +√dA ρT ≤ 1 and 0 ≤ +1 +√dA E ≤ 1, then +1 +√dA ρT and +1 +√dA E are both valid (real) POVM elements on A and B, re- +spectively. So now we have an LOCC discrimination scenario +on the normalized Choi state NAB that yields exactly the same +probability as the original ancilla-free channel discrimination +scenario. +Conversely, let us consider the LOCC discrimination sce- +nario of normalized Choi states. Let NAB be the normalized +Choi state of a channel N from A to B. If E and F are POVM +elements on A and B, respectively, we want to calculate the +probability Tr +� +(E ⊗ F) NAB� +. +Note that, assuming E � 0, +ρ := +1 +Tr E E is a valid quantum state, so Tr +� +(E ⊗ F) NAB� += +Tr E Tr +� +(ρ ⊗ F) NAB� +. Then, we have +Tr E TrAB +� +(ρ ⊗ F) NAB� += Tr E TrB +� +F TrA +� +(ρ ⊗ 1) NAB�� += Tr E +dA +TrB +� +FN +� +ρT�� += Tr +� +F′N +� +ρT�� +, +(82) +where we have used Eq. (80), and we have defined F′ := +Tr E +dA F. Now, ρT is still a valid quantum state of A, and F′ +is still a valid POVM element on B because Tr E +dA ≤ 1. So now +we have an ancilla-free discrimination scenario on the chan- +nels associated with the bipartite normalized state that yields +exactly the same probability as the original bipartite LOCC +discrimination scenario. In this way, we have proven that all +probabilities arising in one of the two scenarios can be com- +pletely reproduced by the other scenario, so they are in some +sense equivalent in terms of the probabilities they can gener- +ate. +Having established the relation of channel discrimination +and local discrimination of their corresponding Choi states, +we can see that the advantage of imaginarity in real chan- +nel discrimination shows up when both initial probe state and +measurement contain imaginarity. +We accomplish this by +mapping the ancilla-free channel discrimination scenario into +the LOCC state discrimination scenario, using (normalized) +Choi matrices, as discussed above. Let us consider the exam- +ple of a qubit channel N. Note that its (normalized) Choi state +can be written as +NAB += 1 +2 +��������1 + +� +j +ajσA +j ⊗ 1B + 1A ⊗ +� +j +b jσB +j + +� +j,k +E jkσA +j ⊗ σB +k +�������� , +(83) +where i, j ∈ {x, y, z}, and the σ j’s are Pauli matrices. If N is +a real operation, then we can conclude that the only term con- +taining σy must only be σy ⊗ σy. Recall that Tr +� +S σy +� += 0 for +any real symmetric 2×2 matrix S (cf. Ref. [18]). For this rea- +son, any POVM element MAB = EA⊗FB, with real symmetric +matrices E or F, cannot be used to detect the presence of the +σy ⊗ σy term in a Choi matrix of a real operation. Conse- +quently, there are some real operations that are perfectly dis- +tinguishable, but become indistinguishable using an ancilla- +free protocol if we only use real states and measurements. +However, if we are still restricted to real probe states and mea- +surements, but we allow an ancilla, then the same real opera- +tions become perfectly distinguishable again. To understand +why, notice that when we allow an ancilla, we can use the state +φ+ as probe state for all real operations, thus producing their +normalized Choi states. Then the task becomes distinguishing +between their Choi states, but without any LOCC constraints +(recall that the LOCC constraint comes from the ancilla-free +scenario). Removing the LOCC constraint from the discrim- +ination of the Choi states makes the advantage provided by +imaginarity disappear. Consequently, with an ancilla, we can +perform as well with just real states and measurements as we +do with non-real ones. +Experimental details +In Module A, two type-I phase-matched β-barium borate +(BBO) crystals, whose optical axes are normal to each other, +are pumped by a continuous laser at 404 nm, with a power of +80 mW, for the generation of photon pairs with a central wave- +length at λ = 808 nm via a spontaneous parametric down- +conversion process (SPDC). A half-wave plate (HWP) and a +quarter-wave plate (QWP) working at 404 nm set before the +lens and BBO crystals is used to control the polarization of +the pump laser. Two polarization-entangled photons are gen- +erated and then distributed through two single-mode fibers +(SMF), where one represents Bob and the other Alice. Two +interference filters (IF) with a 3 nm full width at half max- +imum (FWHM) are placed to filter out proper transmission +peaks. HWPs at both ends of the SMFs are used to control the +polarization of both photons. +In Module B for preparing Werner states, two 50/50 beam +splitters (BSs) are inserted into one branch. In the transmis- +sion path, the two-photon state is still a Bell state. In the re- +flected path, three 400λ quartz crystals and a HWP with an- +gles set to 22.5◦ are used to dephase the two-photon state into + +13 +a completely mixed-state 11AB/4. The ratio of the two states +mixed at the output port of the second BS can be changed +by the two adjustable apertures (AA) for the generation of +Werner states. This setup also allows us to implement a class +of quantum channels which are specified in the main text. + diff --git a/kdE3T4oBgHgl3EQf5gub/content/tmp_files/load_file.txt b/kdE3T4oBgHgl3EQf5gub/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd010849a38d55927787198e6f07a473b85cabf9 --- /dev/null +++ b/kdE3T4oBgHgl3EQf5gub/content/tmp_files/load_file.txt @@ -0,0 +1,882 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf,len=881 +page_content='Resource Theory of Imaginarity: New Distributed Scenarios Kang-Da Wu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 2 Tulja Varun Kondra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='3 Carlo Maria Scandolo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' ∗ Swapan Rana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='6 Guo-Yong Xiang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' † Chuan-Feng Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 2 Guang-Can Guo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 2 and Alexander Streltsov3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' ‡ 1CAS Key Laboratory of Quantum Information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Hefei 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' People’s Republic of China 2CAS Center For Excellence in Quantum Information and Quantum Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' People’s Republic of China 3Centre for Quantum Optical Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Centre of New Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' University of Warsaw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Banacha 2c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 02-097 Warsaw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Poland 4Department of Mathematics and Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' University of Calgary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' AB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Canada T2N 1N4 5Institute for Quantum Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' University of Calgary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' AB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Canada T2N 1N4 6Physics and Applied Mathematics Unit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Indian Statistical Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 203 B T Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Kolkata 700108,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' India (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 2023) The resource theory of imaginarity studies the operational value of imaginary parts in quantum states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' op- erations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' and measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Here we introduce and study the distillation and conversion of imaginarity in distributed scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This arises naturally in bipartite systems where both parties work together to generate the maximum possible imaginarity on one of the subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We give exact solutions to this problem for general qubit states and pure states of arbitrary dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We present a scenario that demonstrates the operational advantage of imaginarity: the discrimination of quantum channels without the aid of an ancillary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We then link this scenario to LOCC discrimination of bipartite states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We experimentally demonstrate the relevant assisted distillation protocol, and show the usefulness of imaginarity in the aforementioned two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' INTRODUCTION Standard quantum theory describes physical reality with complex states, operators, and Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' However, there have always been lots of questions on the role of com- plex numbers since the early days of quantum physics [1– 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Recently the necessity and usefulness of the imagi- nary part of quantum mechanics has received significant at- tention [16–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Today, quantum mechanics with imagi- nary numbers seems to be the most successful theory to de- scribe the microscopic world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' These research contributions have shown that complex quantum mechanics is fundamen- tally different from the corresponding real version in many aspects [2, 10, 13, 15, 19, 24–28], revealing that the imagi- nary part is not only necessary for the formulation of quantum theory but also plays an important role in many quantum in- formation tasks [9, 11, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The development of quantum information science over the last two decades has led to a reassessment of quantum proper- ties, such as entanglement [30, 31] and coherence [32, 33], as resources, which led to the development of quantitative theories that captured these phenomena in a mathematically rigorous fashion [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Nevertheless, imaginarity had not been studied in this framework until the last few years [16, 18, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In this setting, imaginarity is regarded as a valuable resource that cannot be generated or increased under a restricted class of operations known as real operations (RO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Quantum states whose density matrices (in a fixed basis) con- tain imaginary parts are viewed as resource states, and thus cannot be created freely by RO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In this Letter, we study the resource theory of imaginar- ity in distributed scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (At least) two parties, Alice (A) and Bob (B) are involved, who share a bipartite state ρAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In this setting, imaginarity is considered a resource only in Bob’s system, while Alice can perform arbitrary quantum operations on her system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The duo is further allowed to communicate classically with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Overall, we refer to the allowed set of operations in this protocol as Local Quantum-Real op- erations and Classical Communication (LQRCC) borrowing the notion from the theory of entanglement [30] and quantum coherence [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This framework leads to a variety of prob- lems, which we address and solve in this Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In particu- lar, we consider assisted imaginarity distillation, where Alice assists Bob in extracting local imaginarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If only one-way classical communication is used, we provide a solution of this problem for arbitrary two qubit states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We also study assisted state conversion, where the goal is to obtain a specific tar- get state on Bob’s side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We solve this problem for any target state, if Alice and Bob share a pure state initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Further- more, we study the role of imaginarity in ancilla-free chan- nel discrimination, showing two real channels that are per- fectly distinguishable in the ancilla-free scenario once we al- low imaginarity, but become completely indistinguishable if we have access only to real states and real measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Ad- ditionally, we prove how this task is related to LOCC (Local Operations and Classical Communication) discrimination of quantum states, specifically to the LOCC discrimination of their normalized Choi matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Finally, we experimentally implement the above protocols in a quantum photonic setup, performing the proof of principle experiment testing the use- fulness of imaginarity in such quantum tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Our work opens new avenues towards both theoretical and experimental explo- ration of imaginarity as a quantum resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='04782v1 [quant-ph] 12 Jan 2023 2 RESOURCE THEORY OF IMAGINARITY The starting point of our work is the resource theory of imaginarity, introduced very recently in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' [16, 18, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The free states in imaginarity theory are identified as real states, which are real density matrices in a given basis {|j⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The set of all real states is denoted by R, which can be de- scribed by R = {ρ : ⟨j|ρ|k⟩ ∈ R for all j, k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' A quantum oper- ation specified by Kraus operators {Kj} satisfying � j K† j Kj = 1, is considered to be free, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=', real, if it contains only real elements in the chosen basis: ⟨m|K j|n⟩ ∈ R for all j, m, n [16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' It is known that the set RO coincides with the set of completely non-imaginarity creating operations [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' More- over, RO coincides with the set of operations which have a real dilation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The golden unit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' the maximally re- sourceful state, is the same in any Hilbert space, regardless of its dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In particular, the maximally imaginary states are the two eigenstates of Pauli matrix σy, | ˆ±⟩ = ( |0⟩ ± i |1⟩ ) √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (1) One maximally imaginary qubit is referred to as an imbit in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Within the framework of quantum resource distillation [35– 38], general quantum states can be used for single-shot or asymptotic distillation of imbits via ROs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In the single-shot regime, the answer was already given in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' [18, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In particular, the fidelity of imaginarity FI, which quantifies the maximum achievable fidelity between a state ρ and the imbit FI (ρ) = max Λ F � Λ �ρ� , | ˆ+⟩⟨ ˆ+| � , (2) was used as the figure of merit for single-shot distillation, where F (ρ, σ) = � Tr � √σρ √σ � 1 2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The exact value of fi- delity of imaginarity for general ρ was shown to be equal to FI (ρ) = 1 + IR (ρ) 2 , (3) where IR (ρ) = minτ {s ≥ 0 : (ρ + sτ) / (1 + s) ∈ R} is the ro- bustness of imaginarity [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' When we consider the asymp- totic setting, for large n, the fidelity of imaginarity exponen- tially converges to 1 (for any non-real states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The exponent, for large n, is given by − log � Tr � ρρT� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For real states, the fidelity of imaginarity is independent of n, and is 1/2 [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Details of the proof can be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' One of the key motivations for us to study the resource of imaginarity is that we can simulate arbitrary operations or measurements with one imbit at hand, even if all devices al- low only real ones in our lab, as we show explicitly in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In entanglement theory, one maximally entangled qubit state (ebit) has a clear operational meaning: it can be used to teleport the state of an unknown qubit deterministi- cally to a remote lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In imaginarity theory, if all the devices are restricted to implement ROs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=', we have only half-wave plate in an optical setup [18, 20], we can still prepare arbitrary states or implement arbitrary measurements if we get one im- bit at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We refer to the Appendix for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' BIPARTITE IMAGINARITY THEORY The results studied so far concern imaginarity as resource in a single physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We now extend our considerations to the bipartite setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' As mentioned earlier, the task involves a bipartite state ρAB shared by Alice and Bob, and the goal is to maximize imaginarity on Bob’s side under LQRCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If both parties are restricted to real operations, the corresponding set is called local real operations and classical communication (LRCC) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' It is clear that via LQRCC it is possible to create only states of the form ρqr = � j pj ρA j ⊗ σB j , (4) where ρA j is an arbitrary state on Alice’s side, and σB j is a real state on Bob’s side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' States of this form will be called Quantum-Real (QR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In the appendix, we show that the choi matrices corresponding to LQRCC are "invariant" under par- tial transpose over Bob (Bob is restricted to real operations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This also holds for more general LQRCC maps, which are trace non-increasing (similar to SLOCC in entanglement the- ory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Using this, we now show that, for arbitrary initial state ρAB and the target pure state |ψA′B′⟩, the optimal achievable fi- delity for a given probability of success p (given by Fp), can be upperbounded by a SDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Achievable fidelity for a given probablity of suc- cess (Fp(ρAB LQRCC −−−−−→ |ψA′B′⟩), of transforming ρAB into |ψA′B′⟩ via LQRCC operations can upper bounded by the following semidefinite programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Maximise: 1 p Tr � XABA′B′ ρT AB ⊗ |ψA′B′⟩⟨ψA′B′| � (5) under the constraints, XABA′B′ ≥ 0, XTBB′ ABA′B′ = XABA′B′, TrA′B′ XABA′B′ ≤ 1AB and Tr � XABA′B′ ρT AB ⊗ 1B′ � = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (6) In the case of LRCC operations, one has to add an additonal constraint, given by XTAA′ ABA′B′ = XABA′B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For the details about the proof, please refer to the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In the special case when the target state is a local pure state of Bob |ψB′⟩, one can replace |ψA′B′⟩ by |0⟩ ⊗ |ψB′⟩, in the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' ASSISTED IMAGINARITY DISTILLATION Having extended the theory of imaginarity to multipartite systems, we are now ready to present assisted imaginarity dis- tillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In this task, Alice and Bob aim to extract imaginar- ity on Bob’s side by applying LQRCC operations, which is 3 in analogy to assisted entanglement distillation [41–43] and assisted distillation of quantum coherence [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We assume that Alice and Bob share an arbitrary mixed state ρAB, and the process is performed on a single copy of the state and only one-way classical communication from Alice to Bob is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If Alice performs a general measurement � MA j � on her side, the probability pj and the corresponding post-measurement state of Bob ρB j are given respectively by pj = Tr �� MA j ⊗ 1B� ρAB� , ρB j = 1/p j TrA �� MA j ⊗ 1B� ρAB� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' As a figure of merit we now introduce the assisted fidelity of imaginarity, quantifying the maximal single-shot fidelity between Bob’s final state and the maximally imaginary state | ˆ+⟩: Fa � ρAB� = max � MA j , Λj � � j p jF � Λj � ρB j � , | ˆ+⟩⟨ ˆ+| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (7) The maximum is taken over all POVMs on Alice’s side, and all real operations Λj on Bob’s side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For two-qubit states, we can derive the exact analytic expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Con- sider a two-qubit state ρAB, which can be written as ρ = � 14 + a · σ ⊗ 1 + 1 ⊗ b · σ + � k,l Eklσk ⊗ σl � /4, where the σk’s are Pauli matrices, a = (a1, a2, a3) and b = (b1, b2, b3) describe local Bloch vectors of Alice and Bob, respectively, and Ekl = Tr (σk ⊗ σlρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Equipped with these tools, we are now ready to give a closed expression for the assisted fidelity of imaginarity for all two-qubit states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For any two-qubit state ρAB the assisted fidelity of imaginarity is given by Fa � ρAB� = 1 2 (1 + max {|b2| , |s|}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (8) where the vector s = (E12, E22, E32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The proof is presented in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We will now extend our results to stochastic state transfor- mations, where the goal is to achieve a transformation with the maximum possible probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' To this end, we introduce the geometric measure of imaginarity and the concurrence of imaginarity, presented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' [40, 45] respectively as Ig (ρ) = 1 − � F �ρ, ρT� 2 , (9a) Ic (ρ) = max ���������0, λ1 − � j>1 λj ��������� , (9b) where {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' } are the eigenvalues (in decreasing order) of � √ρρT √ρ � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' With this in place, we now extend this sce- nario to the bipartite regime where we will show how Al- ice can assist Bob (ρB) to get the target state σB with op- timal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Now we use the following parameteriza- tion: sin2 α = � 1 − Ic � ρB�� /2 and sin2 β = Ig � σB� with α, β ∈ (0, π 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For any bipartite pure state ψAB, the optimal prob- ability of Bob preparing a local state σB, getting assistance from Alice, is given by P � ψAB → σB� = min �sin2 α sin2 β , 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (10) The proof of Lemma 3 is presented in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' [40] the authors provided tight continuity bounds for the geometric measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Using these bounds, along with Lemma 3, we can provide an analytical expression for the optimal probability of Bob preparing a local state with an al- lowed error, with assistance from Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Similarly, we can also find a closed expression for the optimal achievable fi- delity, for a given probability of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The following theo- rem collects these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For any bipartite pure state ψAB, the optimal probability P f of Bob preparing a local state σB, with a fi- delity f via assistance from Alice, is given by Pf � ψAB → σB � = ����������� 1 for α − β + γ ≥ 0 sin2 α sin2 ( β − γ ) otherwise (11) where γ = cos−1 � f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The optimal achievable fidelity for a given probability of success p, can be expressed as: Fp � ψAB → σB � = ��������������� 1 for p ≤ sin2 α sin2 β cos2 � β − sin−1 �sin α √p �� otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (12) Details of the proof for the above theorem can be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Imaginarity in channel discrimination—We will now dis- cuss the role of imaginarity in channel discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Specif- ically, here we focus on the variant of channel discrimination which we call ancilla-free, in that it does not involve an ancil- lary system (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' [46, 47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' It can be regarded as a game, where one has access to a “black box” with the promise that it implements a quantum channel Λj with probability pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The goal of the game is to guess Λj by choosing optimal initial state ρ and positive operator-valued measure (POVM) � M j � , which is used to distinguish the Λj (ρ)’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Theoretically, the probability of guessing the channel Λj correctly is given as psucc � ρ, � p j, Λj � , � M j �� = � j pj Tr � M jΛj (ρ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (13) Recently, it has been shown that any quantum resource has an operational advantage in the channel discrimination task [46, 47], namely a resource state ρ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' a quantum state that is not free) outperforms any free σ in a specific channel discrimina- tion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 4 Now we put the above protocol into imaginarity theory by considering the task of discrimination of real channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' To see an advantage, we need imaginarity both in the probe state and in the measurement, since, as we show in the Appendix, this task is equivalent to LOCC discrimination of their correspond- ing normalized Choi states, in which we need imaginarity in the measurements of both particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' To better illustrate this idea, we will provide an example of two real channels that cannot be distinguished in the ancilla-free scenario by using only real states and measurements, but they become instead perfectly distinguishable once we have access to imaginarity for states and measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' To this end, let us consider two real qubit channels prepared with equal probability: N : ρ �→ 1 2 ( ρ + σx σz ρ σz σx ) , M : ρ �→ 1 2 ( σx ρ σx + σz ρ σz ) , (14) where σx and σz are Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If we input a real state ρ into either of these two channels, they will produce exactly the same output 1/2, thus we cannot distinguish them better than making a random guess, even if we allowed imaginarity in our measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' On the other hand, if imaginarity is forbidden in measurements, no matter how we choose the probe state (even if itis non-real), we cannot still distinguish them at all, because the only way to discriminate between the outputs of the two channels would be to perform a measurement associ- ated with the σy Pauli matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Indeed, if the probe state has an off-diagonal entry ρ01 with non-zero imaginary part, wherever the output of N has Im ρ01, the output of M will show −Im ρ01 in its place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Only if we implement a projective measurement of σy can we perfectly distinguish these two channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' There- fore, the only way to achieve a success probability better than random guessing is to introduce imaginarity into both the ini- tial state ρ and the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' It is worth noting that the same two channels N and M be- come perfectly distinguishable even with no imaginarity in the probe state and in the measurement if we remove the require- ment of ancilla-free discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If we allow an ancilla R, we need to consider a bipartite input state ρRA and a bipartite POVM � MRA 1 , MRA 2 � , with success probability psucc � ρ, �1 2, Λj � , � M j �� = 1 2 2 � j=1 Tr � MRA j � IR ⊗ Λj � � ρRA�� , (15) where Λ1 = N and Λ2 = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Now, let us take ρRA = φ+ = |φ+⟩⟨φ+|, with |φ+⟩ = 1√ 2 (|00⟩ + |11⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If we feed φ+ to both channels, we get I ⊗ N �φ+� = 1 2 �|φ+⟩⟨φ+| + |ψ−⟩⟨ψ−|� , I ⊗ M �φ+� = 1 2 �|φ−⟩⟨φ−| + |ψ+⟩⟨ψ+|� , (16) where |φ−⟩ = 1√ 2 (|00⟩ − |11⟩), |ψ+⟩ = 1√ 2 (|01⟩ + |10⟩), and |ψ−⟩ = 1√ 2 (|01⟩ − |10⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' As noted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' [18], these two out- put states can be perfectly distinguished by the real POVM B HWP @404nm BBO QP SPD BS M AA PBS QWP @404nm HWP @808nm QWP @808nm A C Entangled Source State Prepara�on & Channel Implemeta�on Discrimina�on & Tomography FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The whole experimental setup is di- vided into three modules: A Entangled source, B state preparation & channel implementation, and C discrimination & tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The optical components include: QP, quartz plate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' SPD, single photon de- tectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' BS, beamsplitters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' AA, adjustable aperture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' PBS, polarizing beamsplitter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' QWP, quarter-wave plate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' HWP, half-wave plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' {M1, M2}, where M1 = | ˆ+⟩⟨ ˆ+| ⊗ | ˆ−⟩⟨ ˆ−| + | ˆ−⟩⟨ ˆ−| ⊗ | ˆ+⟩⟨ ˆ+|, M2 = | ˆ+⟩⟨ ˆ+| ⊗ | ˆ+⟩⟨ ˆ+| + | ˆ−⟩⟨ ˆ−| ⊗ | ˆ−⟩⟨ ˆ−|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (17) This shows that the two real channels can be distinguished perfectly with the aid of an ancilla, only using real states and real measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' EXPERIMENTS We experimentally implement the aforementioned assisted imaginarity distillation and channel discrimination protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The whole experimental setup is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 1, which consists of three modules: module A enables us to prepare a two-qubit entangled state via spontaneous parametric down conversion (SPDC) process: |ψ⟩AB = a |00⟩ + b |11⟩, (18) with arbitrary a and b with |a|2+|b|2 = 1 which can be tuned by changing the angles of 404 nm HWP and QWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Note that we have conventionally set |0⟩ := |H⟩ and |1⟩ := |V⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Module B utilizes an unbalanced Mach-Zehnder interferometer together with module A to prepare a class of Werner states: ρAB = p |φ+⟩⟨φ+| + (1 − p) 1 4 , (19) where p denotes the purity of the two-qubit state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Module B also allow us to implement single-qubit channels in ancilla- free scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Module C allows us to perform quantum-state tomography (QST) to identify the final two-qubit polarization- encoded states concerned, or perform assisted imaginarity dis- tillation by performing local measurement on Alice’s photons 5 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Experimental results for assisted imaginarity distilla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (a) Initial pure states |ψ⟩AB = a|00⟩ + b|11⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (b) initial Werner states ρAB = p|φ+⟩⟨φ+| + (1 − p) 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In both experiments, red disks represent the calculated fidelity of imaginarity by assistance using Theorem 2 for experimentally reconstructed two-qubit states, and blue disks represent actual obtained average fidelity of imaginarity in experiments using the optimal measurement on Alice’s system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' and identifying the exact amount of imaginarity by QST of Bob’s state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Moreover, this module allows us to implement channel discrimination by performing local measurement on the polarization state of a single-photon when the other is used as a trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We refer to the Appendix for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We then perform proof of principle experiments of the one- shot assisted imaginarity distillation and the ancilla-free chan- nel discrimination tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Results are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 2 and 3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For assisted imaginarity distillation, we experimentally pre- pare two classes of two-qubit states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The first class of states as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Theoretically, the upper bound for single-shot assisted imaginarity distillation can be calculated from Theo- rem 2 as FI � |ψ⟩AB� = 2 |ab|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 2(a), we can see that the experimentally obtained average imaginarity after assis- tance (blue disks) approximately equals to the experimentally obtained upper bound (red disks) within reasonable experi- mental imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The second class of states are gener- ated as Werner states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Theoretically, the maximum average fidelity of imaginarity after assistance is calculated as FI(ρAB) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 2(b) details the relevant experimental re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' From both results we see that experimentally obtained average fidelity of imaginarity data and upper bound obtained from two-qubit state tomography agree well with theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We then show the usefulness of imaginarity in channel dis- crimination for various discrimination tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 3 details these results for two discrimination tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The first discrimi- nation task involves two channels given by M ( ρ, p ) = pρ + (1 − p) σx σz ρ σz σx, N (ρ) = 1 2 (σx ρ σx + σz ρ σz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (20) Note that the two channels preserve real density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The experimental results of this discrimination task are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If we can use imaginarity in measurements and initial states, we can perfectly distinguish the two channels [orange disks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' However, if we allow only real density matrices as initial states or real measurement oper- ators, we get a theoretical optimal guessing probability of 1/2 + |2p − 1| /4 for the ancilla-free channel discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Experimental data are in agreement with the theoretical pre- dictions [see green disks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Here we note that the two channels are exactly the same as in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (14) when p = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For the second discrimination task, we consider M ( ρ, w ) = w ρ + (1 − w) 1 2 , N (ρ) = 1 2 (σx ρ σx + σz ρ σz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (21) The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If non-real states and mea- surement operators are allowed, then we get a theoretical op- timal distinguishing probability as 3/4+w/4, which is plotted as the upper orange line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The relevant experimen- tally obtained distinguishing probabilities are shown as orange disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If imaginarity is prohibited in this task, then the optimal distinguishing probability reads 1/2 + w/4, and is plotted as the lower green line, together with experimental values repre- sented by green disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We can draw a similar conclusion to the first discrimination task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' DISCUSSION The results presented above are mainly based on the new set of LQRCC operations which was introduced and studied in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We considered assisted imaginarity distillation in this setting, and completely solved the problem for general two-qubit states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Moreover, we discussed the task of single- shot assisted imaginarity distillation for arbitrary pure states in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The usefulness of imaginarity in channel discrimination is both theoretically and experimentally shown for a class of real channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' There are in fact many scenarios of practical relevance where the task of assisted imaginarity distillation can play a central role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For instance, think of a remote or unaccessible system on which imaginarity is needed as a resource (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=', in the task of local discrimination of quantum states): our results give optimal prescriptions to inject such imaginarity on the remote target by acting on an ancilla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The results provide in- sight into both the operational characterization as well as the 6 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Experimental results for discrimination tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Two channel discrimination tasks are tested : (a) Mp (ρ) = pρ + (1 − p) σxσzρσzσx, N (ρ) = (σxρσx + σzρσz) /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Using imaginar- ity one can perfectly distinguish the two channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' However, if only real operators are allowed, then the optimal guessing proba- bility is 1/2 + |2p − 1| /2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (b) Mw (ρ) = wρ + (1 − w) 1/2, N (ρ) = (σxρσx + σzρσz) /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The optimal probabilities for successful guess- ing are 3/4 + w/4 and 1/4 + w/4 for the case where imaginarity is allowed, and where only real states and measurements are allowed, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' mathematical formalism of the resource theory of imaginar- ity, contributing to a better understanding of this fundamental resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The work at the University of Science and Technology of China is supported by the National Key Research and Development Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 2018YFA0306400), the National Natural Science Foundation of China (Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 12134014, 12104439, 61905234, 11974335, 11574291, and 11774334), the Key Research Program of Frontier Sci- ences, CAS (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' QYZDYSSW-SLH003), USTC Research Funds of the Double First-Class Initiative (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' YD2030002007) and the Fundamental Research Funds for the Central Universities (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' WK2470000035, WK2030000063).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The work at Poland was supported by the National Science Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Poland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' within the QuantERA II Programme (No 2021/03/Y/ST2/00178,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' acronym ExTRaQT) that has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agree- ment No 101017733 and the “Quantum Optical Technolo- gies” project,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' carried out within the International Research Agendas programme of the Foundation for Polish Science co-financed by the European Union under the European Re- gional Development Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' CMS acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Discovery Grant “The power of quantum resources” RGPIN-2022-03025 and the Discov- ery Launch Supplement DGECR-2022-00119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' ∗ carlomaria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Wilde, Principles of quantum communica- tion theory: A modern approach (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' [49] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Audenaert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Calsamiglia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Muñoz Tapia, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Bagan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Masanes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Acin, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Verstraete, Discriminating states: The quantum chernoff bound, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 98, 160501 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Implementing general quantum operations Here, we show that one imbit is necessary and sufficient to implement arbitrary quantum operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' To see this, let’s say we want to implement a quantum operation Λ on ρ with Kraus operators given by {Kj}, such that � j K† j Kj = P ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' To implement this, we construct a real quantum operation (Λr) with Kraus operators given by {Kj ⊗ | ˆ+⟩⟨ ˆ+| + K∗ j ⊗ | ˆ−⟩⟨ ˆ−|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' It is easy to see that Λr(ρ ⊗ | ˆ+⟩⟨ ˆ+|) = Λ(ρ) ⊗ | ˆ+⟩⟨ ˆ+| (22) and � j (K† j ⊗ | ˆ+⟩⟨ ˆ+| + KT j ⊗ | ˆ−⟩⟨ ˆ−|)(Kj ⊗ | ˆ+⟩⟨ ˆ+| + K∗ j ⊗ | ˆ−⟩⟨ ˆ−|) = P ⊗ | ˆ+⟩⟨ ˆ+| + PT ⊗ | ˆ−⟩⟨ ˆ−| ≤ 1 ⊗ 1 The last inequality follows from the fact that, P ≤ 1 ⇐⇒ PT ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (23) This shows that one imbit is sufficient to implement general quantum operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Now we show that, there exists a quan- tum channel, which necessarily requires one imbit, to imple- ment via real operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' As an example, consider the follow- ing map (Λ+) given by Λ+(ρ) = | ˆ+⟩⟨ ˆ+| forall ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (24) We now show, by contradiction, that the above quantum map requires one imbit to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Let’s say there is a imple- mentation (with a real operation Λ′ r) such that, Λ′ r(ρ ⊗ σ) = Λ+(ρ) = | ˆ+⟩⟨ ˆ+| (25) here, if σ is not an imbit and ρ = |0⟩⟨0|, its easy to see that the state transformation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (25) is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This is because Ig(|0⟩⟨0| ⊗ σ) = Ig(σ) < Ig(| ˆ+⟩⟨ ˆ+|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 8 Properties of LRCC operations For any real CP map Λ : R → R′, ΓΛ RR′ is the corresponding choi matrix of ΓΛ RR′, given by ΓΛ RR′ = 11 ⊗ Λ �������� � j,k | j⟩⟨k| ⊗ | j⟩⟨k| �������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (26) Any LQRCC map (Λ) can be represented in the following way Λ = � i Λi ⊗ Λr i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (27) Here, λi is a CP (trace non increasing) map acting locally on Alice’s hilbert spapce and Λr i is a local real CP map on Bob’s hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The choi matrix of Λi ⊗ Λr i is given by Γ Λi⊗Λr i AB→A′B′ = 11AB ⊗ Λi ⊗ Λr i �� j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' j′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='k′ | jk⟩⟨ j′k′| ⊗ | jk⟩⟨ j′k′| � = � j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='j′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='k′ | j⟩⟨j′| ⊗ |k⟩⟨k′| ⊗ Λi(|j⟩⟨j′|) ⊗ Λr i(|k⟩⟨k′|) Let’s now take the transpose of this choi matrix over BB′ (Γ Λi⊗Λr i AB→A′B′)TBB′ = � j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' j′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='k′ | jk′⟩⟨j′k| ⊗ Λi(| j⟩⟨j′|) ⊗ (Λr i(|k⟩⟨k′|))T = � j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='j′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='k′ | jk′⟩⟨j′k| ⊗ Λi(| j⟩⟨ j′|) ⊗ Λr i(|k′⟩⟨k|) = Γ Λi⊗Λr i AB→A′B′ (28) In the second line we used the fact that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' real operations com- mute with transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Since any LQRCC operation can be rep- resented as (27), the choi matric of any LQRCC operation is invariant under partial transpose over Bob’s systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For LRCC operations, additionally the choi matrix is always real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Proof of Theorem 1 In the following, we assume that A and B is a qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' A general two-qubit state ρAB can be written as ρ = 1 4 ��������1⊗1 + � k akσk⊗1 + � l bl1⊗σl + � k,l Eklσk⊗σl �������� , (29) where a = (a1, a2, a3) and b = (b1, b2, b3) are local Bloch vectors of Alice and Bob, respectively, and Ekl = Tr(σk⊗σlρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' A general single-qubit POVM element on Alice’s side can be written as MA n = qn ��������1 + � j αnjσ j �������� (30) with probabilities 0 ≤ qn ≤ 1, � n qn = 1, and vectors αn such that |αn| ≤ 1 and � n qnαn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The measurement {MA n } gives outcome n with probability pn = qn (1 + a · αn) , (31) and the Bloch vector of Bob’s post-measurement state is bn = b + ETαn 1 + a · αn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (32) After Alice communicates her measurement outcome n to Bob, he applies a real operation Λn to his post-measurement state ρB i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For each measurement outcome n, Bob aims to max- imize the fidelity between Λn[ρB n] and the maximally imagi- nary state | ˆ+⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The maximum is given by the fidelity of imag- inarity FI which for single-qubit states ρB n reduces to FI(ρB n) = 1 2 � 1 + ���Tr[ρB nσ2] ��� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (33) Using this result together with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (31) and (32) we can ex- press our figure of merit Fa as follows: Fa(ρAB) = max MAn � n pnFI(ρB i ) = max qn,αn 1 2 �������1 + � n qn |b2 + s · αn| ������� , (34) where the maximization in the last expression is performed over all vectors αn and probabilities 0 ≤ qn ≤ 1 such that � n qn = 1, |αn| ≤ 1 and � n qnαn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If |b2| ≥ |s|, then using the conditions |αn| ≤ 1 and � n qnαn = 0 we immediately obtain � n qn |b2 + s · αn| = |b2| (35) for any choice of qn and αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This directly implies that Fa(ρAB) = 1/2+|b2|/2 in this case, in accordance with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We now consider the case if |b2| < |s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We will show that in the maximization in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (34) it is enough to consider POVMs consisting of two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For a given set of vectors αn and probabilities qn we introduce two sets, depending whether b2+ s · αn is positive or negative: S 0 = {n : b2 + s · αn ≥ 0}, (36a) S 1 = {j : b2 + s · α j < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (36b) Using these sets, we express the sum � n qn |b2 + s · αn| as fol- lows: � n qn |b2 + s · αn| = �������� � n∈S 0 qn �������� ������b2 + � n∈S 0 qn(s · αn) � n∈S 0 qn ������ (37) + �������� � j∈S 1 qj �������� ������b2 + � j∈S 1 qj(s · α j) � j∈S 1 q j ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In the next step, we introduce the probabilities ˜q0 = � n∈S 0 qn, ˜q1 = � j∈S 1 qj and vectors ˜α0 = � n∈S 0 qnαn � n∈S 0 qn , (38a) ˜α1 = � j∈S 1 q jαj � j∈S j qj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (38b) Noting that b2 + s · ˜α0 ≥ 0, (39a) 9 b2 + s · ˜α1 < 0, (39b) we further obtain the following result: � n qn |b2 + s · αn| = ˜q0|b2 + s · ˜α0| + ˜q1|b2 + s · ˜α1| = ˜q0(b2 + s · ˜α0) − ˜q1(b2 + s · ˜α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (40) The vectors ˜αn and probabilities ˜qn fulfill the conditions � n ˜qn = 1, | ˜αn| ≤ 1, and � n ˜qn ˜αn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This implies that they correspond to a two-element POVM on Alice’s side via the relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The arguments just presented show that the maximum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (34) can be achieved with two vectors α0 and α1 and two probabilities q0 and q1 having the properties 0 ≤ q0 ≤ 1, q1 = 1 − q0, |αn| ≤ 1, � i qnαn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' To complete the proof, we will show that the optimal solution is obtained for q0 = q1 = 1 2, (41a) α0 = −α1 = s |s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (41b) Recalling that |b2| ≤ |s|, the values in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (41) immediately give a lower bound on the assisted fidelity of imaginarity: Fa(ρAB) ≥ 1 2(1 + |s|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (42) Let now qn and αn be optimal probabilities and vectors [not necessarily coinciding with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (41)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Without loss of gener- ality we can assume that1 b2 + s · α0 ≥ 0, (43a) b2 + s · α1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (43b) For the assisted fidelity of imaginarity we thus obtain Fa(ρAB) = 1 2[q0(b2 + s · α0) − q1(b2 + s · α1)] + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (44) Since q0 + q1 = 1, it must be that either q0 ≤ 1/2 or q1 ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In the first case we rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (44) as follows: Fa(ρAB) = 1 − b2 2 + q0(b2 + s · α0) ≤ 1 2(1 + |s|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (45) In the second case (q1 ≤ 1/2), we rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (44) as Fa(ρAB) = 1 + b2 2 − q1(b2 + s · α1) ≤ 1 2(1 + |s|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (46) Thus, for |b2| < |s| the assisted fidelity of imaginarity is bounded above as Fa(ρAB) ≤ 1 2(1 + |s|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (47) 1 Otherwise, if b2 + s · αn is positive (or negative) for all n, we obtain � n qn |b2 + s · αn| = |b2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Since |b2| < |s|, this means that we will not be able to reach the maximal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Together with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (42) this proves that Fa(ρAB) = 1/2 + |s|/2 in this case, and the proof of the theorem is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Theorem 2 has few surprising consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If a two-qubit state has the property |b2| ≥ |s|, then the assisted fidelity of imaginarity coincides with the fidelity of imaginarity of Bob’s local state: Fa(ρAB) = (1 + |b2|)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Thus, in this case Bob will not gain any advantage from assistance, as he can obtain the maximal fidelity by performing a local real operation without any communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For example, let us consider a quantum state shared by Alice and Bob ρAB = p 2 11A ⊗ | ˆ+⟩⟨ ˆ+| + (1 − p)|φ+⟩⟨φ+| (48) where we have b2 = p and s = (0, p−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Then if p = 1, then ρAB is a product pure state, then no matter what Alice does, Bob can always get the maximal imaginary state | ˆ+⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If 1 2 < p < 1, the state ρAB has nonzero entanglement, but we have |b2| > |s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If Alice chooses a projective measurement along α, then Bob will get states with Bloch vector b ± ET · α with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Then the average fidelity with maximally imaginary state reads 1 2 (|p + (1 − p)α2| + |p − (1 − p)α2|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' As we have 1 2 < p < 1, |(1 − p)α2| < p, then the average fidelity reads p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For all other two-qubit states the proof of Theorem 2 provides an optimal procedure for obtaining maximal fidelity of imaginarity on Bob’s side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For this, Alice needs to perform a von Neumann measurement in the basis {|ψ0⟩, |ψ1⟩}, where |ψ0⟩ has the Bloch vector s/|s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The outcome of the measure- ment is communicated to Bob, who leaves his state untouched if the outcome was 0, and otherwise applies the real unitary iσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Needs to be checked Proof of Lemma 1 Note that, the geometric measure of imaginarity and the concurrence of imaginarity are given by [40, 45] G(ρ) = min e � j pj 1 − |⟨ψ∗ j|ψ j⟩| 2 = 1 − � F(ρ, ρT) 2 (49) C(ρ) = min e � pj|⟨ψ∗ j|ψ j⟩| = max �������0, λ1 − � k>1 λk ������� (50) In the above maxe and mine are maximisation and minimisa- tion over pure state ensembles of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Whereas, {λ1, λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='} are the eigenvalues (in decreasing order) of ( √ρρT √ρ) 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In gen- eral, for probabilistic transformations, the following inequal- ity holds p(ρ → σ) ≤ min � G(ρ) G(σ), 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (51) It was further shown in [40], that the optimal probability of converting a pure state ψ to a arbitrary quantum state ρ is given by p(ψ → ρ) = min �G(ψ) G(ρ) , 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (52) 10 In a one way LQRCC procedure, Alice performs a general quantum measurement and corresponding to the outcomes (with probabilites {p j}) of Alice, Bob’s local state is found in the state ρj, such that, {pj, ρi} is an ensemble of ρB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Con- ditioned on the outcome of Alice (i), Bob can perform a local stochastic real operation on ρi, probabilistically converting it into σB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (51) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (52), it follows Pa ≤ � j p j min � G(ρj) G(σB), 1 � ≤ � jk p jqk min �G(ψ j,k) G(σB) , 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (53) The second inequality follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (49), G(ρj) is cal- culated by minimising over all pure state ensembles of ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Therefore, the second inequality holds for any pure state de- composition of ρj, like {qk, ψjk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Note that {pjqk, ψjk} is a pure state decomposition of ρB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Note that, any pure state decompo- sition of ρB can be realised by a suitable local measurement by Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Using this fact, along with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (49) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (52) implies that Pa = min �1 − mine � k pk|⟨ψk|ψ∗ k⟩| 2G(σB) , 1 � = min ������� 1 − C(ρB) 1 − � F(σB, (σB)T) , 1 ������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Here, mine is the minimisation over pure state ensembles of ρB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Proof of Theorem 4 From Lemma 1, we know that optimal probability for Bob to locally achieve σB from a shared bipartite pure state ψAB with unit fidelity, via LQRCC is given by P(ψAB → σB) = min �1 − Ic(ρB) 2Ig(ρ) , 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (54) If we want to achieve σB with fidelity at least f, the best strat- egy is to go to a state (σ′B), within the fidelity ball around σB, with a minimal geometric measure of imaginarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Therefore, Pf (ψAB → σB) = min �1 − Ic(ρB) 2Ig(σ′B) , 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (55) From [40], we know that Ig(σ′B) = sin2 � max � sin−1 � Ig(σB) − cos−1 � f, 0 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (56) We now define m = sin−1 � 1 − Ic(ρB) 2 −sin−1 � Ig(σB)+cos−1 � f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (57) First, consider the case when m ≥ 0, which implies sin−1 � Ig(σB) − cos−1 � f ≤ sin−1 � 1 − Ic(ρB) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (58) We know that sin−1 � Ig(σB) − cos−1 � f ∈ [−π/2, π/4] (59) and sin−1 � 1−Ic(ρB) 2 ∈ [0, π/4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Therefore, max � sin−1 � Ig(σB) − cos−1 � f, 0 � ≤ sin−1 � 1 − Ic(ρB) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (60) Using these results, we get Ig(σB′) = sin2 � max � sin−1 � Ig(σB) − cos−1 � f, 0 �� (61) ≤ sin2 �������sin−1 � 1 − Ic(ρB) 2 ������� = 1 − Ic(ρB) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For the case when 1−Ic(ρB) 2 > 0, the above inequality implies 1 − Ic(ρB) 2Ig(σB′) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (62) This shows that Pf (ψAB → σB) = 1 when m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Now, we look at the other case when m < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=', sin−1 � Ig(σB) − cos−1 � f > sin−1 � 1 − Ic(ρB) 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (63) From the above inequality and Lemma 1, we have Pf (ψAB → σB) = 1 − Ic(ρB) 2 sin2(sin−1 � Ig(σB) − cos−1 � f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (64) Using the above result, a closed expression can also be found for Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Let’s first consider the case when p ≤ 1−Ic(ψAB) 2Ig(σB) < 1, in this case Fp(ψ → σB) = 1 (follows from Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' When 1 ≥ p > Ig(ψ) Ig(σB), the optimal achievable fidelity can be obtained by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (11) for f, which gives Fp(ψAB → σB) = cos2 ���������sin−1 � Ig(σB) − sin−1 � 1 − Ic(ρB) 2p ��������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' SDP upperbounds for state transformations As we already mentioned, for any real CP map Λ : R → R′, ΓΛ RR′ is the corresponding choi matrix of ΓΛ RR′, given by ΓΛ RR′ = 11 ⊗ Λ �������� � j,k | j⟩⟨k| ⊗ | j⟩⟨k| �������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (65) It follows that (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='12) of [48]), Λ(ρR) = TrR(ΓΛ RR′(ρT R ⊗ 11R′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (66) 11 For any pure state |ψR′⟩ ⟨ψR′|Λ(ρR)|ψR′⟩ = Tr(ΓΛ RR′(ρT R ⊗ |ψR′⟩⟨ψR′|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (67) Using the fact that, choi matrices of LQRCC operations are invariant under partial transpose, one can give a SDP com- putable upperbound for the optimal achievable fidelity for a given probability Fp(ρAB → |ψAB⟩): Maximise: 1 p Tr(XABA′B′ρT AB ⊗ |ψA′B′⟩⟨ψA′B′|) (68) under the constraints, XABA′B′ ≥ 0, XTBB′ ABA′B′ = XABA′B′, TrA′B′ XABA′B′ ≤ 1AB and Tr(XABA′B′ρT AB ⊗ 1B′) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (69) Quantum Chernoff divergence and scaling of asymptotic imaginarity distillation Fidelity of imaginarity FI, quantifies the maximum achiev- able fidelity between a state ρ and the maximally imaginary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' It can be expressed as FI(ρ) = max Λ F(Λ(ρ), | ˆ+⟩⟨ ˆ+|) = 1 2 + 1 4||ρ − ρT||1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (70) Here, the maximisation is performed over all real CPTP maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If we have n copies of ρ, we can write FI(ρ⊗n) = 1 2 + 1 4||ρ⊗n − (ρT)⊗n||1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (71) If ρ is a pure state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=', ρ = |ψ⟩⟨ψ|, then we can calculate fidelity of imaginarity of multiple copies as FI(|ψ⟩⟨ψ|⊗n) = 1 2 + 1 4|||ψ⟩⟨ψ|⊗n − (|ψ⟩⟨ψ|T)⊗n||1 = 1 2 + 1 4|||ψ⟩⟨ψ|⊗n − (|ψ∗⟩⟨ψ∗|)⊗n||1 = 1 2 + 1 2 � 1 − |⟨ψ∗|ψ⟩|2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (72) For general states, to see the behaviour of FI(ρ⊗n), with in- creasing n, consider the quantity P = 1 − FI(ρ⊗n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' From Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' [49], it follows that the following limit exists and is equal to the quantum Chernoff divergence between ρ and ρT: lim n→∞ − log P n = χ(ρ, ρT) = − log( min 0≤s≤1 Tr(ρs(ρT)1−s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (73) One can analytically perform this minimisation and show that minimum value is attained at s = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In order to show this fact, let’s assume that the spectral decomposition of ρ is given by ρ = � j p j|ψj⟩⟨ψj|, (74) and therefore ρT = � j p j|ψ∗ j⟩⟨ψ∗ j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (75) The Chernoff divergence is given by χ(ρ, ρT) = − log(min0≤s≤1 Tr(� j ps j|ψ j⟩⟨ψ j|)(� k p1−s k |ψ∗ j⟩⟨ψ∗ j|)) = − log(min0≤s≤1 � j,k ps i p1−s k |⟨ψ j|ψ∗ k⟩|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (76) Note that, |⟨ψ j|ψ∗ k⟩| = |⟨ψk|ψ∗ j⟩|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This implies that χ(ρ, ρT) = − log( min 0≤s≤1 � j≤k (ps jp1−s k + ps kp1−s j )|⟨ψ j|ψ∗ k⟩|2) (77) here, ps jp1−s k + ps kp1−s j ≥ 2 √pjpk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This follows from AM-GM inequality, which says a+b 2 ≥ √ ab for all a, b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This lower bound (minimum value) is attained at s = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This proves that χ(ρ, ρT) = − log(Tr � ρρT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (78) Therefore, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (73), it follows that asymptotically the fidelity of imaginarity behaves as FI(ρ⊗n) ∼ 1 − exp(−n · χ(ρ, ρT)) (79) = 1 − (Tr � ρρT)n Proof of the relation between channel discrimination and state discrimination Here we demonstrate a clear link between the task of ancilla-free channel discrimination and the task of LOCC dis- crimination of bipartite states, the latter studied in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' [4, 18, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Specifically, we consider the following two scenarios: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Let N and M be two real channels from A to B, cho- sen with equal probability 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If we want to discrim- inate between them in an ancilla-free scenario better than with a random guess, we must find a real state ρ of A and a real POVM element E of B such that Tr �EN (ρ)� � Tr �EM (ρ)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Notice that this protocol does not involve any bipartite input states and bipartite effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Let N and M be two real channels from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This time, we bring in the maximally entangled state φ+ = |φ+⟩⟨φ+|AA′, between systems A and A′ (A′ is a copy of A), where |φ+⟩ = � j | jj⟩/ √dA, and dA is the dimen- sion of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We apply N and M only to the A′ part of this maximally entangled state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This results in two bi- partite states between systems A and B, NAB and MAB, respectively, which are the normalized Choi states of the two channels N and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Now consider the task of discriminating between these two bipartite states of AB using only local real measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Again, if we want to discriminate between them better than with a random guess, we must find a real POVM element E of system A and a real POVM element F of system B such that Tr � (E ⊗ F) NAB� � Tr � (E ⊗ F) MAB� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' 12 In the following we show that these two scenarios produce the same probabilites when POVMs are applied to states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Note that we can reconstruct the action of a channel on a state from its normalized Choi state: if N is a channel from A to B, ρ is a state of A, we have that N (ρ) can be written in terms of the normalized Choi state NAB as N � ρA� = dA TrA ��� ρA�T ⊗ 1B� NAB� , (80) where dA is the dimension of the input system A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Thus, if E is a (real) POVM element on B, omitting system superscripts for simplicity, we have Tr �EN (ρ)� = dA Tr �� ρT ⊗ E � N � = Tr �� 1 √dA ρT ⊗ 1 √dA E � N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (81) Note that 0 ≤ 1 √dA ρT ≤ 1 and 0 ≤ 1 √dA E ≤ 1, then 1 √dA ρT and 1 √dA E are both valid (real) POVM elements on A and B, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' So now we have an LOCC discrimination scenario on the normalized Choi state NAB that yields exactly the same probability as the original ancilla-free channel discrimination scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Conversely, let us consider the LOCC discrimination sce- nario of normalized Choi states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Let NAB be the normalized Choi state of a channel N from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If E and F are POVM elements on A and B, respectively, we want to calculate the probability Tr � (E ⊗ F) NAB� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Note that, assuming E � 0, ρ := 1 Tr E E is a valid quantum state, so Tr � (E ⊗ F) NAB� = Tr E Tr � (ρ ⊗ F) NAB� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Then, we have Tr E TrAB � (ρ ⊗ F) NAB� = Tr E TrB � F TrA � (ρ ⊗ 1) NAB�� = Tr E dA TrB � FN � ρT�� = Tr � F′N � ρT�� , (82) where we have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' (80), and we have defined F′ := Tr E dA F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Now, ρT is still a valid quantum state of A, and F′ is still a valid POVM element on B because Tr E dA ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' So now we have an ancilla-free discrimination scenario on the chan- nels associated with the bipartite normalized state that yields exactly the same probability as the original bipartite LOCC discrimination scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In this way, we have proven that all probabilities arising in one of the two scenarios can be com- pletely reproduced by the other scenario, so they are in some sense equivalent in terms of the probabilities they can gener- ate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Having established the relation of channel discrimination and local discrimination of their corresponding Choi states, we can see that the advantage of imaginarity in real chan- nel discrimination shows up when both initial probe state and measurement contain imaginarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' We accomplish this by mapping the ancilla-free channel discrimination scenario into the LOCC state discrimination scenario, using (normalized) Choi matrices, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Let us consider the exam- ple of a qubit channel N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Note that its (normalized) Choi state can be written as NAB = 1 2 ��������1 + � j ajσA j ⊗ 1B + 1A ⊗ � j b jσB j + � j,k E jkσA j ⊗ σB k �������� , (83) where i, j ∈ {x, y, z}, and the σ j’s are Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' If N is a real operation, then we can conclude that the only term con- taining σy must only be σy ⊗ σy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Recall that Tr � S σy � = 0 for any real symmetric 2×2 matrix S (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' For this rea- son, any POVM element MAB = EA⊗FB, with real symmetric matrices E or F, cannot be used to detect the presence of the σy ⊗ σy term in a Choi matrix of a real operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Conse- quently, there are some real operations that are perfectly dis- tinguishable, but become indistinguishable using an ancilla- free protocol if we only use real states and measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' However, if we are still restricted to real probe states and mea- surements, but we allow an ancilla, then the same real opera- tions become perfectly distinguishable again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' To understand why, notice that when we allow an ancilla, we can use the state φ+ as probe state for all real operations, thus producing their normalized Choi states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Then the task becomes distinguishing between their Choi states, but without any LOCC constraints (recall that the LOCC constraint comes from the ancilla-free scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Removing the LOCC constraint from the discrim- ination of the Choi states makes the advantage provided by imaginarity disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Consequently, with an ancilla, we can perform as well with just real states and measurements as we do with non-real ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Experimental details In Module A, two type-I phase-matched β-barium borate (BBO) crystals, whose optical axes are normal to each other, are pumped by a continuous laser at 404 nm, with a power of 80 mW, for the generation of photon pairs with a central wave- length at λ = 808 nm via a spontaneous parametric down- conversion process (SPDC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' A half-wave plate (HWP) and a quarter-wave plate (QWP) working at 404 nm set before the lens and BBO crystals is used to control the polarization of the pump laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Two polarization-entangled photons are gen- erated and then distributed through two single-mode fibers (SMF), where one represents Bob and the other Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' Two interference filters (IF) with a 3 nm full width at half max- imum (FWHM) are placed to filter out proper transmission peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' HWPs at both ends of the SMFs are used to control the polarization of both photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In Module B for preparing Werner states, two 50/50 beam splitters (BSs) are inserted into one branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In the transmis- sion path, the two-photon state is still a Bell state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' In the re- flected path, three 400λ quartz crystals and a HWP with an- gles set to 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content='5◦ are used to dephase the two-photon state into 13 a completely mixed-state 11AB/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' The ratio of the two states mixed at the output port of the second BS can be changed by the two adjustable apertures (AA) for the generation of Werner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} +page_content=' This setup also allows us to implement a class of quantum channels which are specified in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE3T4oBgHgl3EQf5gub/content/2301.04782v1.pdf'} diff --git a/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf b/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4e57f52f2fd21e6fd3ec20d01f8278215f3579d7 --- /dev/null +++ b/ktE0T4oBgHgl3EQfYwAL/content/2301.02310v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:dc08ba0ba7173dcf3f4f5590b45b20a56da349e3ef1ab8307c241302ee38e30e +size 5153267 diff --git a/ktE0T4oBgHgl3EQfYwAL/vector_store/index.faiss b/ktE0T4oBgHgl3EQfYwAL/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..3819f5fa67bebe71117e051ab5a8f82c45787c75 --- /dev/null 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a/n9E2T4oBgHgl3EQfJwZ7/content/tmp_files/2301.03695v1.pdf.txt b/n9E2T4oBgHgl3EQfJwZ7/content/tmp_files/2301.03695v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..50c6a5420f0b7db2cdc8a85bb319237baa7e997f --- /dev/null +++ b/n9E2T4oBgHgl3EQfJwZ7/content/tmp_files/2301.03695v1.pdf.txt @@ -0,0 +1,234 @@ +A simpler proof for the reflective properties of conic sections +Rajeev D. S. Raizada +January 11, 2023 +Abstract +Ellipses, parabolas and hyperbolas all have beautiful reflective properties, sending beams of light +either towards or away from the foci or directrix. However, the proofs of these properties tend +to involve multiple steps, making them less intuitive, and harder for students to grasp. Here, a +simpler proof is presented which only requires examining the properties of an isosceles triangle. +First, it is noted if a light-beam runs along one of the legs of an isosceles triangle, and then reflects +off a mirror that is parallel to the triangle’s base and that touches the apex, then the light will bounce +back along the path of the other leg. With that in mind, a short journey is made, starting from any +point on a conic section and moving to a nearby point on the curve by taking two equal-length +straight line steps. Each step points either towards or away from the foci or directrix of the curve, +depending on which variety of conic section is being considered. These two equal-length steps +form the legs of an isosceles triangle, with the base of the triangle running along the path of the +curve. The reflective property of the conic section then follows from the path of the light-beam +along the legs of an isosceles triangle that was described above. As the size of the triangle shrinks +towards zero, the reflection point converges onto the actual curve, and the base of the triangle +converges onto the actual tangent. It seems a near certainty that this proof has already appeared +somewhere in the long history of writings about conics, but I have not yet been able to find it. +Whether previously published or not, it does not appear to be widely known. It is hoped that this +simpler proof will be useful for teaching, and that it might help to make the reflective properties of +conic sections easier to understand and to explain. +Introduction: lots of proofs, but also lots of steps +Amongst the best known and most important aspects of conic sections are their reflective properties. +A parabolic mirror directs incoming parallel beams of light to its focus, as is exploited by every +satellite dish and reflecting telescope. An ellipse sends any beam emanating from one focus towards +the other focus, enabling medical devices called lithotriptors which focus ultrasound shockwaves +to shatter kidney stones, and also giving rise to entertaining illustrations with elliptical billiard +tables. A hyperbola reflects beams starting from one focus directly away from the second focus. +There exist many ways of proving these properties, but they are not quite as straightforward as +one might like. Perhaps for this reason, Algebra 2 textbooks which cover conic sections tend to +describe the reflective properties without proving them (e.g. Holliday et al., 2008; Bittinger et al., +2012). Proofs for the ellipse in particular often involve multiple steps. Perhaps the most commonly +presented proof involves first invoking Fermat’s principle that light travels along the path that +takes the least time, i.e. the shortest possible one. Then, a tangent line is drawn at some point on +the ellipse, and it is shown that the point of reflection along that tangent line that offers the shortest +possible path is the one where it touches the curve of the ellipse itself, e.g. Akopyan and Zaslavsky +1 +arXiv:2301.03695v1 [math.HO] 9 Jan 2023 + +(2007, pp.6-8). When restricted to reflection rather than also including refraction, this shortest-path +rule is also referred to as Heron’s principle (Foster and Pedersen, 1980). +Other proofs use trigonometry, e.g. the Law of Sines (Brannan et al., 2012, pp.29-30), or trigonometry +together with calculating the derivative of the velocity vector of a point moving along the path of +the curve (Glaeser et al., 2016, pp.27-30), or calculus combined with the trigonometric identity for +the tan of the difference of two angles (Stewart, 2015, p.722). +A simpler approach: bouncing light off the base of an isosceles triangle +rin +rref lect +α +θin +θref lect +β +A +B +C +Figure 1 +Here is a simpler approach. First, let us consider how light +reflects off the base of an isosceles triangle, as shown in Fig. 1. +In the triangle ABC, the legs AC and BC are of equal length. +The base angles of this isosceles triangle, namely α and β, are +therefore also equal. Let us reflect light off the thick line passing +through the vertex C, parallel to the base AB of the triangle. By +the laws of reflection, the angle of incidence θin and the angle of +reflection θreflect will be equal. Because the reflector is parallel to +AB, the angles α and θin are alternate interior angles and hence +are equal, as are β and θreflect. Thus, all four angles are equal +to each other, and the reflected beam rreflect simply follows the +path of the leg BC. +Moving along the curve of a conic by taking two equal-length steps +A +E1 +E2 +F2 +δ +δ +Line to F1 +B +F2 +F1 +δ +δ +Figure 2: Drawing an isosceles triangle in two equal-sized steps, in order to move from one point to a +neighbouring one along the curve of the ellipse. (A) A close-up view of the isosceles triangle formed by +taking two steps of size δ. (B) A broader view showing the entire ellipse. +What does this seemingly pointlessly obvious fact about isosceles triangles have to do with the +reflective properties of conics? Consider the question of how to move from one point on a conic +section curve to a nearby point on the same curve. Figure 2 shows this for the case of an ellipse. +Specifically, in Fig. 2A, we move from point E1 to point E2, both lying on the ellipse. Recall that +the ellipse is the set of all points such that the sum of the distances to the two foci is equal to some +constant. So, if we start at E1 and first move a distance δ away from the focus F1, then we can keep +this sum of distances constant and hence return to the ellipse by taking an equal-length step of δ +towards the focus F2, thereby arriving at the point E2. +2 + +The first step, moving directly away from F1 by a distance δ, added to the sum of the distances to +the foci. The second step, of equal size but this time moving directly towards F2, then subtracted +an equal but opposite amount from that sum. So, the combined result of the two steps is to leave +that sum unchanged, thus staying on the locus of the ellipse. +Crucially, these two equal-sized steps form an isosceles triangle, whose base E1E2 runs along a +small segment of the curve of the ellipse. This is exactly the sort of triangle discussed above and +shown in Figure 1. A beam of light originating from focus F1 and then bouncing off a line at the +apex of the triangle that is parallel to its base will be reflected back along the second leg of the +triangle, i.e. directly towards focus F2. +When the triangle is of finite size, as shown in the figure, this reflection takes place very slightly +outside of the ellipse, rather than directly on it. However, if we shrink the triangle ever smaller +by letting the step-size δ tend towards zero, then the reflection tends towards happening exactly +on the curve of the ellipse, and the base of the triangle tends towards being a perfect tangent. The +infinitesimally small isosceles triangle receives light directly from the first focus and reflects it back +exactly towards the second focus. +The second step returns exactly to the path of the curve, as the triangle +shrinks towards zero size +P1 +P2 +D +B: Point on ellipse? +A +Line to F1 +Line to F1 +Line to F2 +Line to F2 +Change in dist to F1 +due to step D → B +g +Chan e in dist to F2 +due to step A → D +Figure 3: The two equal steps’ orthogonal projections +onto each other are themselves of equal length. So, +the two-step journey leaves unchanged the sum of +the distances to the foci, thus returning to the ellipse. +Before showing how the same argument also +applies to parabolas and hyperbolas, it is worth +pausing a moment to check that the second step +of length δ truly does return to the locus of the +ellipse. After all, although the first step does +indeed move δ directly away from the focus F1, +it also adds some smaller but non-zero amount +to the distance from the focus F2. Similarly, the +return step towards F2 also reduces the distance +to F1. +Each such change is equal to the orthogonal pro- +jection of the δ-length step onto the other leg of +the isosceles triangle, as is shown in Figure 3. +As can be seen from that figure, the symme- +try of an isosceles triangle ensures that the two +such projections are of equal length: the two +steps A → D and D → B are both of length δ, +and they share the same projection angle ̸ ADB. +Hence, they are equal. So, the two δ-length steps and the two equal-length projections collectively +leave the sum of the distances to the two foci unchanged, thereby ensuring that the endpoint B of +the two steps does indeed lie on the ellipse, as required. +Note that the above argument requires that the lines from A to F2 and from B to F2 must be parallel +to each other, and similarly the lines from A and B to F1. This is only approximately true for an +isosceles triangle ADB of finite size, but tends towards being exactly true as the size of that triangle +shrinks towards zero compared to the distance to the foci. +3 + +Applying exactly the same argument to parabolas and hyperbolas +Focus +Directrix +δ +δ +Figure 4: A parabola. +Exactly the same argument also applies to parabolas and hyperbo- +las, with only the directions of the two steps altering. Fig. 4 shows +two equal-length steps of δ being taken in order to move along the +curve of a parabola. For points on this sort of curve, the distance +to the directrix must remain equal to the distance to the focus. So, +after taking a step of length δ straight towards the directrix, we can +get back onto the parabola by taking an equal-length step straight +towards the focus. +The first step, moving directly towards the directrix, created an +imbalance between the directrix- and focus-distances by adding +δ to the former. The second step, also of length δ but now moving +directly towards the focus, restored equality of the two distances by subtracting δ from the latter. +Here again, the resulting isosceles triangle shows why incoming light-beams that are perpendicular +to the directrix will get reflected directly towards the focus. Also, as before, this triangle is only +approximate when δ is finitely large, but it tends towards being exact as δ tends towards zero. +F2 +F1 +δ +δ +Figure 5: A hyperbola. +Figure 5 similarly shows two equal-length steps of δ being taken in +order to move along the curve of a hyperbola. In this case, the differ- +ence between the distances to the two foci must remain constant. So, +after taking a step of length δ directly away from the first focus, F1, +we can get back onto the hyperbola by taking an equal-length step +directly away from the second focus, F2. For this type of curve, light +emanating from one focus will get reflected directly away from the +second focus, rather than towards it as was the case for the ellipse. +This somewhat different reflective property of a hyperbola turns out +to be useful in combination with a parabolic reflector, in an astronom- +ical instrument called a Cassegrain telescope (Downs, 2003, ch.6), a +noteworthy example of which is the Hubble Space Telescope (NASA, +2023). As ever, the triangle shown in Fig. 5 is only approximate, but +the reflections become exact as δ tends towards zero. +Conclusion +The arguments presented here are probably not new, although I have so far been unable to find +them, either in the books on conics that referenced above (Akopyan and Zaslavsky, 2007; Downs, +2003; Glaeser et al., 2016) or anywhere else. Whether previously published or not, the approach +does not appear to be widely known. I hope that this simpler proof will be useful for teaching, and +that it might help to make the reflective properties of conic sections easier to understand and to +explain. +References +Akopyan, A. V. and Zaslavsky, A. A. (2007). Geometry of Conics. American Mathematical Society, +Providence, Rhode Island. +4 + +Bittinger, M., Ellenbogen, D., and Johnson, B. (2012). Intermediate Algebra: Graphs & Models. Pearson, +4th edition. +Brannan, D. A., Esplen, M. F., and Gray, J. J. (2012). Geometry. Cambridge University Press, 2nd +edition. +Downs, J. W. (2003). Practical Conic Sections: The Geometric Properties of Ellipses, Parabolas and +Hyperbolas. Dover Publications, Mineola, NY. +Foster, J. H. and Pedersen, J. J. (1980). +On the reflective property of ellipses. +The +American Mathematical Monthly, +87(4):294–297. +Publisher: +Taylor & Francis +eprint: +https://doi.org/10.1080/00029890.1980.11995020. +Glaeser, G., Stachel, H., and Odehnal, B. (2016). The Universe of Conics: From the ancient Greeks to +21st century developments. Springer, Berlin, 1st edition. +Holliday, B., Luchin, B., Cuevas, G. J., Carter, J. A., Marks, D., Day, R., Casey, R. M., and Hayek, +L. M. (2008). Algebra 2. Glencoe/McGraw-Hill. +NASA (2023). Hubble space telescope optics system. https://www.nasa.gov/content/goddard/hubble- +space-telescope-optics-system. Retrieved on Jan. 8, 2023. +Stewart, J. (2015). Calculus. Cengage Learning, Boston, MA, 8th edition. +5 + diff --git a/n9E2T4oBgHgl3EQfJwZ7/content/tmp_files/load_file.txt b/n9E2T4oBgHgl3EQfJwZ7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..09dce486e607a1f19a36ff81a4ef2d6dc5f8a8a8 --- /dev/null +++ b/n9E2T4oBgHgl3EQfJwZ7/content/tmp_files/load_file.txt @@ -0,0 +1,184 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf,len=183 +page_content='A simpler proof for the reflective properties of conic sections Rajeev D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Raizada January 11, 2023 Abstract Ellipses, parabolas and hyperbolas all have beautiful reflective properties, sending beams of light either towards or away from the foci or directrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' However, the proofs of these properties tend to involve multiple steps, making them less intuitive, and harder for students to grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Here, a simpler proof is presented which only requires examining the properties of an isosceles triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' First, it is noted if a light-beam runs along one of the legs of an isosceles triangle, and then reflects off a mirror that is parallel to the triangle’s base and that touches the apex, then the light will bounce back along the path of the other leg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' With that in mind, a short journey is made, starting from any point on a conic section and moving to a nearby point on the curve by taking two equal-length straight line steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Each step points either towards or away from the foci or directrix of the curve, depending on which variety of conic section is being considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' These two equal-length steps form the legs of an isosceles triangle, with the base of the triangle running along the path of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' The reflective property of the conic section then follows from the path of the light-beam along the legs of an isosceles triangle that was described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' As the size of the triangle shrinks towards zero, the reflection point converges onto the actual curve, and the base of the triangle converges onto the actual tangent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' It seems a near certainty that this proof has already appeared somewhere in the long history of writings about conics, but I have not yet been able to find it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Whether previously published or not, it does not appear to be widely known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' It is hoped that this simpler proof will be useful for teaching, and that it might help to make the reflective properties of conic sections easier to understand and to explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Introduction: lots of proofs, but also lots of steps Amongst the best known and most important aspects of conic sections are their reflective properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' A parabolic mirror directs incoming parallel beams of light to its focus, as is exploited by every satellite dish and reflecting telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' An ellipse sends any beam emanating from one focus towards the other focus, enabling medical devices called lithotriptors which focus ultrasound shockwaves to shatter kidney stones, and also giving rise to entertaining illustrations with elliptical billiard tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' A hyperbola reflects beams starting from one focus directly away from the second focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' There exist many ways of proving these properties, but they are not quite as straightforward as one might like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Perhaps for this reason, Algebra 2 textbooks which cover conic sections tend to describe the reflective properties without proving them (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Holliday et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Bittinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Proofs for the ellipse in particular often involve multiple steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Perhaps the most commonly presented proof involves first invoking Fermat’s principle that light travels along the path that takes the least time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' the shortest possible one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Then, a tangent line is drawn at some point on the ellipse, and it is shown that the point of reflection along that tangent line that offers the shortest possible path is the one where it touches the curve of the ellipse itself, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Akopyan and Zaslavsky 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content='03695v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content='HO] 9 Jan 2023 (2007, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content='6-8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' When restricted to reflection rather than also including refraction, this shortest-path rule is also referred to as Heron’s principle (Foster and Pedersen, 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Other proofs use trigonometry, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' the Law of Sines (Brannan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=', 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content='29-30), or trigonometry together with calculating the derivative of the velocity vector of a point moving along the path of the curve (Glaeser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=', 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content='27-30), or calculus combined with the trigonometric identity for the tan of the difference of two angles (Stewart, 2015, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content='722).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' A simpler approach: bouncing light off the base of an isosceles triangle rin rref lect α θin θref lect β A B C Figure 1 Here is a simpler approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' First, let us consider how light reflects off the base of an isosceles triangle, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' In the triangle ABC, the legs AC and BC are of equal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' The base angles of this isosceles triangle, namely α and β, are therefore also equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Let us reflect light off the thick line passing through the vertex C, parallel to the base AB of the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' By the laws of reflection, the angle of incidence θin and the angle of reflection θreflect will be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Because the reflector is parallel to AB, the angles α and θin are alternate interior angles and hence are equal, as are β and θreflect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Thus, all four angles are equal to each other, and the reflected beam rreflect simply follows the path of the leg BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Moving along the curve of a conic by taking two equal-length steps A E1 E2 F2 δ δ Line to F1 B F2 F1 δ δ Figure 2: Drawing an isosceles triangle in two equal-sized steps, in order to move from one point to a neighbouring one along the curve of the ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' (A) A close-up view of the isosceles triangle formed by taking two steps of size δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' (B) A broader view showing the entire ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' What does this seemingly pointlessly obvious fact about isosceles triangles have to do with the reflective properties of conics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Consider the question of how to move from one point on a conic section curve to a nearby point on the same curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Figure 2 shows this for the case of an ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Specifically, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' 2A, we move from point E1 to point E2, both lying on the ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Recall that the ellipse is the set of all points such that the sum of the distances to the two foci is equal to some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' So, if we start at E1 and first move a distance δ away from the focus F1, then we can keep this sum of distances constant and hence return to the ellipse by taking an equal-length step of δ towards the focus F2, thereby arriving at the point E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' 2 The first step, moving directly away from F1 by a distance δ, added to the sum of the distances to the foci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' The second step, of equal size but this time moving directly towards F2, then subtracted an equal but opposite amount from that sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' So, the combined result of the two steps is to leave that sum unchanged, thus staying on the locus of the ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Crucially, these two equal-sized steps form an isosceles triangle, whose base E1E2 runs along a small segment of the curve of the ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' This is exactly the sort of triangle discussed above and shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' A beam of light originating from focus F1 and then bouncing off a line at the apex of the triangle that is parallel to its base will be reflected back along the second leg of the triangle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' directly towards focus F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' When the triangle is of finite size, as shown in the figure, this reflection takes place very slightly outside of the ellipse, rather than directly on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' However, if we shrink the triangle ever smaller by letting the step-size δ tend towards zero, then the reflection tends towards happening exactly on the curve of the ellipse, and the base of the triangle tends towards being a perfect tangent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' The infinitesimally small isosceles triangle receives light directly from the first focus and reflects it back exactly towards the second focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' The second step returns exactly to the path of the curve, as the triangle shrinks towards zero size P1 P2 D B: Point on ellipse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' A Line to F1 Line to F1 Line to F2 Line to F2 Change in dist to F1 due to step D → B g Chan e in dist to F2 due to step A → D Figure 3: The two equal steps’ orthogonal projections onto each other are themselves of equal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' So, the two-step journey leaves unchanged the sum of the distances to the foci, thus returning to the ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Before showing how the same argument also applies to parabolas and hyperbolas, it is worth pausing a moment to check that the second step of length δ truly does return to the locus of the ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' After all, although the first step does indeed move δ directly away from the focus F1, it also adds some smaller but non-zero amount to the distance from the focus F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Similarly, the return step towards F2 also reduces the distance to F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Each such change is equal to the orthogonal pro- jection of the δ-length step onto the other leg of the isosceles triangle, as is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' As can be seen from that figure, the symme- try of an isosceles triangle ensures that the two such projections are of equal length: the two steps A → D and D → B are both of length δ, and they share the same projection angle ̸ ADB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Hence, they are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' So, the two δ-length steps and the two equal-length projections collectively leave the sum of the distances to the two foci unchanged, thereby ensuring that the endpoint B of the two steps does indeed lie on the ellipse, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Note that the above argument requires that the lines from A to F2 and from B to F2 must be parallel to each other, and similarly the lines from A and B to F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' This is only approximately true for an isosceles triangle ADB of finite size, but tends towards being exactly true as the size of that triangle shrinks towards zero compared to the distance to the foci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' 3 Applying exactly the same argument to parabolas and hyperbolas Focus Directrix δ δ Figure 4: A parabola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Exactly the same argument also applies to parabolas and hyperbo- las, with only the directions of the two steps altering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' 4 shows two equal-length steps of δ being taken in order to move along the curve of a parabola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' For points on this sort of curve, the distance to the directrix must remain equal to the distance to the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' So, after taking a step of length δ straight towards the directrix, we can get back onto the parabola by taking an equal-length step straight towards the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' The first step, moving directly towards the directrix, created an imbalance between the directrix- and focus-distances by adding δ to the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' The second step, also of length δ but now moving directly towards the focus, restored equality of the two distances by subtracting δ from the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Here again, the resulting isosceles triangle shows why incoming light-beams that are perpendicular to the directrix will get reflected directly towards the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Also, as before, this triangle is only approximate when δ is finitely large, but it tends towards being exact as δ tends towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' F2 F1 δ δ Figure 5: A hyperbola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Figure 5 similarly shows two equal-length steps of δ being taken in order to move along the curve of a hyperbola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' In this case, the differ- ence between the distances to the two foci must remain constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' So, after taking a step of length δ directly away from the first focus, F1, we can get back onto the hyperbola by taking an equal-length step directly away from the second focus, F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' For this type of curve, light emanating from one focus will get reflected directly away from the second focus, rather than towards it as was the case for the ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' This somewhat different reflective property of a hyperbola turns out to be useful in combination with a parabolic reflector, in an astronom- ical instrument called a Cassegrain telescope (Downs, 2003, ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content='6), a noteworthy example of which is the Hubble Space Telescope (NASA, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' As ever, the triangle shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' 5 is only approximate, but the reflections become exact as δ tends towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Conclusion The arguments presented here are probably not new, although I have so far been unable to find them, either in the books on conics that referenced above (Akopyan and Zaslavsky, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Downs, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Glaeser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=', 2016) or anywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Whether previously published or not, the approach does not appear to be widely known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' I hope that this simpler proof will be useful for teaching, and that it might help to make the reflective properties of conic sections easier to understand and to explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' References Akopyan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' and Zaslavsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Geometry of Conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' American 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=', Day, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=', Casey, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=', and Hayek, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Algebra 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Glencoe/McGraw-Hill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' NASA (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Hubble space telescope optics system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content='gov/content/goddard/hubble- space-telescope-optics-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Retrieved on Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' 8, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Stewart, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} +page_content=' Cengage Learning, Boston, MA, 8th edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfJwZ7/content/2301.03695v1.pdf'} 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Norway +{bimal.bhattarai, ole.granmo, lei.jiao, rohan.yadav, jivitesh.sharma}@uia.no +Abstract +Embedding words in vector space is a fundamen- +tal first step in state-of-the-art natural language pro- +cessing (NLP). Typical NLP solutions employ pre- +defined vector representations to improve gener- +alization by co-locating similar words in vector +space. For instance, Word2Vec is a self-supervised +predictive model that captures the context of words +using a neural network. +Similarly, GLoVe is a +popular unsupervised model incorporating corpus- +wide word co-occurrence statistics. Such word em- +bedding has significantly boosted important NLP +tasks, including sentiment analysis, document clas- +sification, and machine translation. However, the +embeddings are dense floating-point vectors, mak- +ing them expensive to compute and difficult to in- +terpret. In this paper, we instead propose to rep- +resent the semantics of words with a few defin- +ing words that are related using propositional logic. +To produce such logical embeddings, we introduce +a Tsetlin Machine-based autoencoder that learns +logical clauses self-supervised. The clauses con- +sist of contextual words like “black,” “cup,” and +“hot” to define other words like “coffee,” thus be- +ing human-understandable. We evaluate our em- +bedding approach on several intrinsic and extrinsic +benchmarks, outperforming GLoVe on six classifi- +cation tasks. Furthermore, we investigate the inter- +pretability of our embedding using the logical rep- +resentations acquired during training. We also vi- +sualize word clusters in vector space, demonstrat- +ing how our logical embedding co-locate similar +words.1 +1 +Introduction +The success of natural language processing (NLP) relies on +advances in word, sentence, and document representation. By +capturing word semantics and similarities, such representa- +tions boost the performance of downstream tasks [Borgeaud +1The Tsetlin Machine Autoencoder and logical word embedding +implementation is available here: https://github.com/cair/tmu. +et al., 2022], including clustering, topic modelling [Angelov, +2020], searching, and text mining [Huang et al., 2020]. +While straightforward, the traditional bag-of-words encod- +ing does not consider the words’ position, semantics, and +context within a document. +Distributed word representa- +tion [Bengio et al., 2000; Bojanowski et al., 2017] addresses +this lack by encoding words as low-dimensional vectors, re- +ferred to as embeddings. The purpose is to co-locate simi- +lar or contextually relevant words in vector space. There are +many algorithms for learning word embeddings. Contem- +porary self-supervised techniques like Word2Vec [Mikolov +et al., 2013], FastText [Bojanowski et al., 2017], and +GloVe [Pennington et al., 2014] have demonstrated how to +build embeddings from word co-occurrence, utilizing mas- +sive training data. +Introducing context-dependent embed- +dings, the more sophisticated language models BERT [De- +vlin et al., 2019] and ELMO [Peters et al., 2018] now per- +form remarkably well in downstream tasks [Reimers and +Gurevych, 2019]. However, they require significant compu- +tation power [Schwartz et al., 2020]. +The above approaches represent words as dense floating +point vectors. Word2Vec, for instance, typically builds a 300- +dimensional vector per word. The size and density of these +vectors make them expensive to compute and difficult to in- +terpret. Consider, for example, the word “queen.” Represent- +ing it with 300 floats seems inefficient compared to the Ox- +ford Language definition for the same word: “the female ruler +of an independent state, especially one who inherits the posi- +tion by right of birth.” From this perspective, it appears ad- +vantageous to create embeddings directly from words rather +than from arbitrary floating-point values. Such interpretable +embeddings would capture the multiple meanings of a word +using a few defining words, simplifying both computation +and interpretation. +In this paper, we propose a Tsetlin Machine (TM) +[Granmo, 2018] based autoencoder for creating interpretable +embeddings. The autoencoder builds propositional logic ex- +pressions with context words that identify each target word. +The term “coffee” can, for instance, be represented by “one,” +“hot,” “cup,” “table,” and “black.” In this manner, the TM +builds contextual representations from a vast text corpus, +which model the semantics of each word. In contrast to neu- +ral network-based embedding, the logical TM embedding is +sparse. The embedding space consists of, e.g., 500 truth val- +arXiv:2301.00709v1 [cs.CL] 2 Jan 2023 + +ues, where each truth value is a logical expression over words. +For contextual representation, each target word links to less +than ten percent of these expressions. Despite the sparsity and +crispness of this representation, it is competitive with neural +network-based embedding. +The contributions of our work are summarized below: +• We propose the first TM-based Autoencoder to learn ef- +ficient encodings in a self-supervised manner. +• We introduce TM-based word embedding that builds +human-comprehensible contextual representations from +unlabeled data. +• We compare our embedding with state-of-the-art ap- +proaches on several intrinsic and extrinsic benchmarks, +outperforming GloVe on six downstream classification +tasks. +2 +Related Work +The majority of self-supervised embedding approaches pro- +duce dense word representations based on the distributional +hypothesis [Harris, 1954], which states that words that oc- +cur in the same context are likely to have similar meaning. +Word2Vec [Mikolov et al., 2013] is one of the best-known +models. It builds embeddings from word co-occurrence using +a neural network, leveraging the hidden layer output weights. +GloVe [Pennington et al., 2014], on the other hand, em- +beds by factorizing a word co-occurrence matrix. Similarly, +canonical correlation analysis (CCA) is used in [Dhillon et +al., 2015] for embedding words to maximise context corre- +lation. In [Levy et al., 2015], it is demonstrated how pre- +cise factorization-based SVD can compete with neural em- +bedding. However, all of these methods are challenging to +train because they involve tweaking algorithms and hyperpa- +rameters toward particular applications [Lample et al., 2016], +limiting their wider applicability. +Building upon word embedding, several studies focus on +sentence embedding [Arora et al., 2017; Logeswaran and +Lee, 2018]. +Recent advances in sentence embedding in- +clude supervised data inference [Reimers and Gurevych, +2019], multitask learning [Cer et al., 2018], contrastive learn- +ing [Zhang et al., 2020], and pretrained large language mod- +els [Li et al., 2020]. +However, the majority of sentence +embedding techniques overlook intrinsic evaluations such as +similarity tasks, and instead largely focus on extrinsic eval- +uations involving downstream performance. +The most re- +cent building block for embedding originates from the trans- +former approach [Vaswani et al., 2017]. Transformers pro- +vide context awareness by utilizing stacks of self-attention +layers. BERT [Kenton and Toutanova, 2019], for instance, +employs the transformer architecture to carry out extensive +self-supervised training, making it capable of producing text +embedding. Other embedding models use a contrastive loss +function to perform supervised fine-tuning on positive and +negative text pairs [Wang et al., 2021]. +Despite the large +variety of text embedding models, they all share three main +drawbacks: i) they are computationally demanding to train; +ii) they are intrinsically complex because they are trained on +a large amount of data to tune a huge amount of parameters; +Clause Pool ++4 +-5 +Input +Output ++1 ++2 +-7 ++6 +Figure 1: Tsetlin Machine Autoencoder. In this illustration, x1 is +masked by replacing it with value 1 for inferring ˆx1. +and iii) the embeddings produced from these models are not +easily interpreted by humans. +To improve interpretability, Faruqui et al. +introduced +“Sparse Overcomplete Word Vectors” (SPOWV) which cre- +ate a sparse non-negative projection of word embedding us- +ing dictionary learning [Faruqui et al., 2015]. +Similarly, +SParse Interpretable Neural Embeddings (SPINE) employs +a k-sparse denoising autoencoder to generate sparse embed- +dings [Subramanian et al., 2018]. However, these methods +are unable to distinguish between multiple context-dependent +word meanings. To address this problem, another avenue of +research focuses on composing linear combinations of dense +vectors from Word2Vec and GloVe [Arora et al., 2018]. How- +ever, the assumption of linearity does not hold for real-world +data, yielding linear coefficients that are difficult to compre- +hend [Mu et al., 2017]. +The logical embedding approach we present here is most +closely related to Naive Bayes word sense induction and topic +modeling [Charniak and others, 2013; Lau et al., 2014]. This +approach learns word meanings from local contexts by con- +sidering each instance of the word in a document as a pseudo- +document. However, the approach is not scalable because +it requires training a single topic per target word. Our ap- +proach, on the other hand, is scalable and builds non-linear +(non-naive) logical embeddings that capture word composi- +tions. To build the logical embeddings, we propose a novel +human-interpretable algorithm based on the TM that provides +logical rules describing contexts. The TM has recently per- +formed competitively with other deep learning techniques in +many NLP tasks, including novelty detection [Bhattarai et +al., 2022a], sentiment analysis [Yadav et al., 2021], and fake +news detection [Bhattarai et al., 2022b]. Furthermore, the +local and global interpretability of TMs have been explored +through direct manipulation of the logical rules [Blakely and +Granmo, 2021]. +3 +Tsetlin Machine Autoencoder +We here detail the TM Autoencoder based on the Coalesced +TM [Glimsdal and Granmo, 2021], extended with input +masking and freezing of masked variables. For ease of ex- +planation, we use three inputs. Adding more inputs follows +trivially. + +3.1 +Architecture +Input and Output. +As seen in Figure 1, the TM Autoen- +coder digests and outputs propositional values: (x1, x2, x3) ∈ +{0, 1}3 → (�x1, �x2, �x3) ∈ {0, 1}3. For our purposes, the +propositional variables x1, x2, and x3 each represents a word, +for example, “Brilliant,” “Actor,” and “Awful.” The value 1 +means that the word occurs in the input text, while the value +0 means that it does not. I.e., we represent natural language +text as a set of words. Notice also that the input variables have +corresponding output variables �x1, �x2, and �x3. In short, �x1 +is to be predicted from x2 and x3, �x2 from x1 and x3, and +so on. Continuing our example, �x1 predicts the presence of +“Brilliant” based on knowing the occurrence of “Actor” and +“Awful.” +Clause Pool. +A pool of n conjunctive clauses, denoted +Cj, j ∈ {1, 2, . . . , n}, encodes the input in order to pre- +dict the output. A conjunctive clause Cj is simply an And- +expression over a given subset Lj ⊆ {x1, x2, x3} of the input +(our autoencoder does not use the input negations ¬x1, ¬x2, +and ¬x3): +Cj(x1, x2, x3) = +� +xk∈Lj +xk. +(1) +For example, the input subset L1 = {x1, x2} gives the clause +C1(x1, x2, x3) = x1 ∧ x2 in the figure. This clause matches +the input if x1 and x2 both are 1. In our example, the clause +accordingly encodes the concept “Brilliant Actor”. +Weights. +An integer weight matrix W connects each of the +n clauses to the three outputs �x1, �x2, and �x3: +W = +�w11 +· · · +w1n +w21 +· · · +w2n +w31 +· · · +w3n +� +∈ Z3×n. +(2) +The row index is an output while the column index is a clause. +The weight w12, for instance, connects output �x1 to clause +C2. In Figure 1, six weights connect two clauses and three +outputs: +�+4 +−5 ++1 ++2 +−7 ++6 +� +. +(3) +Consider, for example, the weights (+4, −5) of output �x1 in +the figure. The weight +4 states that clause C1(x1, x2, x3) = +x1 ∧ x2 favours �x1 being 1, while clause C2(x1, x2, x3) = +x2 ∧ x3 opposes it. For example, the concept “Awful Actor” +opposes output “Brilliant.” +3.2 +Inference +Let us consider the prediction of �x1 first. The autoencoder +predicts �x1 from the clauses and weights: +�x1 = 0 ≤ +n +� +j=1 +wj1Cj(1, x2, x3). +(4) +That is, each clause Cj is multiplied by its weight wj1 for +output �x1. The outcomes are then summed up to decide the +output. If the sum is larger than or equal to zero, the output +is �x1 = 1. Otherwise, it is �x1 = 0. Clauses with positive +Maximally +Memorized +Maximally +Forgotten +Memorized +Forgotten + +4 + +3 + +2 + +1 +Figure 2: Tsetlin Machine memory for single clause. +weight thus promote output �x1 = 1 while clauses with neg- +ative weight encourage �x1 = 0. Notice that x1 is masked by +replacing it with value 1. Accordingly, the autoencoder infers +output �x1 from the remaining inputs x2 and x3. +Correspondingly, �x2 and �x3 are calculated by respectively +masking x2 and x3: +�x2 += +0 ≤ +n +� +j=1 +wj2Cj(x1, 1, x3), +(5) +�x3 += +0 ≤ +n +� +j=1 +wj3Cj(x1, x2, 1). +(6) +Example. +Assume that the input is always either (1, 1, 0) +or (0, 1, 1). The input (1, 1, 0) could for instance represent +“Brilliant Actor” and (0, 1, 1) “Awful Actor.” Then notice +how Eq. (4) correctly determines the masked input x1 with +output �x1 in Figure 1, both for input (1, 1, 0) and (0, 1, 1). +3.3 +Learning +We next consider how to learn the variable subsets Lj for the +clauses Cj, j ∈ {1, 2, . . . , n}, as well as how to determine +the weights wji of the weight matrix W. +Clause Memory. +Each clause Cj has a graded memory that +contains the input variables, shown in Figure 2. The graded +memory enables incremental learning of the variable subsets +from data. Observe how each variable is in one of four mem- +ory positions (the number of memory positions is a user- +configurable parameter). Positions 1 − 2 means Forgotten. +Positions 3−4 means Memorized. Memorized variables take +part in the clause, while Forgotten ones do not. The memory +in Figure 2 thus gives the clause Cj(x1, x2, x3) = x1 ∧ x2. +Learning Step. +The TM Autoencoder learns incrementally +using three kinds of memory and weight updates: Type Ia, +Type Ib, and Type II. Each training example has the form +[k, (x1, x2, x3), xk], 1 ≤ k ≤ 3. The first element is an index +that identifies which input to mask and which output to pre- +dict. The second element is an input vector (x1, x2, x3) and +the third element is the target value for output �xk, which is +xk. We describe the update procedure step-by-step below for +index 1 examples (output �x1 prediction). The update proce- +dure for �x2 and �x3 follows trivially. +Clause +Update +Probability. +First, +we +calculate +the +weighted clause sum for �x1 from Eqn. +(4): +v1 += +�n +j=1 wj1Cj(1, x2, x3). The sum is then compared with a +margin T (hyper-parameter) to calculate a summation error ϵ. +The error depends on the x1-value: +ϵ = +�T − clip(v1, −T, T), +x1 = 1, +T + clip(v1, −T, T), +x1 = 0. +(7) + +Maximally +Memorized +Maximally +Forgotten +Memorized +Forgotten + +4 + +3 + +2 + +1 +Figure 3: Type Ia (Recognize) Feedback for input (1, 1, 0). The +masked variable x1 is frozen. +Maximally +Memorized +Maximally +Forgotten +Memorized +Forgotten + +4 + +3 + +2 + +1 +Figure 4: Type Ib (Erase) Feedback for input (0, 0, 1). The masked +variable x1 is frozen. +That is, for x1-value 1 the weighted clause sum should be- +come T, while for x1-value 0 the sum should become −T. +The goal of the learning is thus to reach the margin for all +inputs (x1, x2, x3), ensuring correct output from Eqn. (4). +To reach this goal, each clause Cj is updated randomly with +probability +ϵ +2T in each round. In other words, the update +probability drops with the error towards zero. +Update Types. +The kind of update depends on the values +of x1, Cj(1, x2, x3), and wj1. We first consider clauses with +positive weight, wj1 ≥ 0. According to Eqn. 4, they are to +recognize patterns for x1 = 1. Note that in all of the below +updates, the masked variable x1 is frozen, leaving it unaf- +fected by the update. +• Type Ia (Recognize) Feedback occurs when x1 = +1 and Cj(1, x2, x3) = 1. +Then one can say that +Cj(1, x2, x3) = 1 is a true positive because it correctly +predicts the masked x1-value. The Type Ia feedback re- +inforces this successful match by updating the memory +of Cj to further mimic the input (see Figure 3). That is, +1-valued variables move one step upwards in memory, +with probability 1.0.2 Conversely, 0-valued inputs move +one step downwards, however, randomly with probabil- +ity 1 +s. +Here, s is a hyperparameter called specificity, +meaning that a larger s makes the clauses more specific. +The clause overall is also reinforced by incrementing its +weight wj1 by 1. +• Type Ib (Erase) Feedback occurs when x1 = 1 and +Cj(1, x2, x3) = 0. Then we call Cj(1, x2, x3) = 0 +a false negative because it fails to promote x1 = 1. +In that case, all inputs randomly move one step down- +wards in memory (see Figure 4). Again, each downward +move happens with probability 1 +s. Here, the purpose is +to eliminate the false negative outcome by erasing vari- +ables from the clause. +2Originally, the increment probability is +s−1 +s , which can be +boosted to 1.0 to enhance learning of true positive patterns [Granmo, +2018]. +Maximally +Memorized +Maximally +Forgotten +Memorized +Forgotten + +4 + +3 + +2 + +1 +Figure 5: Type II (Reject) Feedback for input (0, 1, 0). The masked +variable x1 is frozen. +• Type II (Reject) Feedback occurs when x1 = 0 and +Cj(1, x2, x3) = 1. Then, one can say that Cj(1, x2) = 1 +is a false positive because it promotes x1 = 1 when in +fact we have x1 = 0. Then all Forgotten 0-valued in- +puts move one step upwards in memory. The purpose +is to eventually eliminate the current false positive out- +come by injecting 0-valued variables into the clause. +The clause is further diminished by decrementing its +weight wj1 by 1. +Note that the latter decrement can +switch the weight from positive to negative. In effect, +the clause then changes role, now training to recognize +x1 = 0 instead. +Clauses Cj with negative weights, wj1 < 0, are updated +the same way. However, they are to recognize patterns for +x1 = 0. To achieve this, x1 = 0 is treated as x1 = 1 and +x1 = 1 is treated as x1 = 0 when updating the memories. +Furthermore, the weight updates are reversed. Increments be- +comes decrements, and vice versa. +Algorithm 1 TM word embedding +Require: Vocabulary V; Documents D ∈ G, D ⊆ V; Accumula- +tion u; Clauses n; Margin T; Specificity s; Rounds r +1: TMCreate(n, T, s) +▷ Create TM with n clauses. +2: for r rounds do +3: +for word k ∈ V do +▷ Create one example per word. +4: +qk ← Select({0, 1}) +▷ Random target value. +5: +if qk = 1 then +6: +Gk ← {D|word k ∈ D, D ∈ G} ▷ Documents with +word k. +7: +else +8: +Gk ← {D|word k /∈ D, D ∈ G} +▷ Documents +without word k. +9: +Sk ← SelectN(Gk, u) +▷ Random subset of size u. +10: +Uk ← � +D∈Sk D +▷ Union of selected documents. +11: +xk ← (x1, x2, . . . , xm), xi = +� +1, +word i ∈ U k +0, +word i /∈ U k +12: +TMUpdate(k, xk, qk) ▷ Update TM Autoencoder for +output index k, input xk, and target value �xk = qk. +13: C, W ← TMGetState() ▷ Clauses Cj ∈ C with weights W . +14: E ← clip(W , 0, T) +▷ Elementwise clip of negative values +produces weighted logical word embeddings. +15: B ← (W > 0) +▷ Elementwise comparison with zero +produces purely logical word embeddings. +16: return C, E, B + +4 +Logical Embedding Procedure +We now use the TM Autoencoder to build logical embed- +dings. Let V = {word1, word2, . . . , wordm} be the target +vocabulary consisting of m unique words. +Pre-processing. +The first step is to pre-process the docu- +ment corpus. To this end, each document is represented by a +subset of words D ⊆ V. For example, the document “The ac- +tor was brilliant” becomes the set D = {“actor”, “brilliant”, +“the”, “was”}. The set G, in turn, contains all the documents, +D ∈ G. Finally, in propositional vector form, the word set D +becomes: +x = (x1, x2, . . . , xt), xi = +�1, +wordi ∈ D, +0, +wordi /∈ D. +(8) +Embedding. +Algorithm 1 specifies the procedure for em- +bedding the m vocabulary words from V by using n clauses, +Cj, 1 ≤ j ≤ n, forming a clause set C. +Each round of +training produces a training example [k, (x1, x2, . . . , xm), qk] +per wordk in V. First, a target value qk for the word is set +randomly to either 0 or 1. This random selection balances +the dataset. +If qk becomes 1, we randomly select u doc- +uments that contain wordk and assign them to the set Sk +(positive examples). Otherwise, we randomly select u doc- +uments that does not contain the word (negative examples). +Next, the randomly selected documents are merged by OR- +ing them together, yielding the unified document Uk. The +purpose of ORing multiple documents is to increase the fre- +quency of rare context words. Then, picking up characteristic +ones becomes easier. After that, the propositional vector form +(x1, x2, . . . , xm) of Uk is obtained. Finally, the TM Autoen- +coder is updated with [k, (x1, x2, . . . , xm), qk] following the +training procedure in Section 3. +Vector Space Representation. +The weighted logical em- +bedding of wordk ∈ V can now be obtained from row k of +matrix E (returned from Algorithm 1), while the the purely +logical embedding is found in row k of matrix B. Let ek de- +note the k’th row of E, and let el denote the l’th row. We can +then compare the similarity of two words wordk and wordl +using cosine similarity (CS) between their E-embedding: +CS(wordk, wordl) = +ek · el +||ek|| ||el||. +(9) +5 +Empirical Evaluation +We here evaluate our logical embedding scheme, comparing +it with neural network approaches. +5.1 +Datasets and Setup +We first evaluate our logical embedding intrinsically, fol- +lowed by an extrinsic evaluation using classification tasks. +Intrinsic Evaluation. +We use word similarity and catego- +rization benchmarks for intrinsic evaluation. That is, we ex- +amine to what degree our approach retains semantic word +relations. To this end, we measure how semantic relations +manifest in vector space using six datasets: SimLex-999, +WordSim-353, MEN, MTurk-287, Mturk-771, and RG-65. +Each dataset consists of human-scored word pairs, which are +compared with the corresponding vector space similarities. +The categorization tasks evaluate how well we can group +words into distinct word categories, only based on their em- +bedding. We here use three datasets: AP, BLESS, and ESS- +LLI.3 As baselines, we chose Word2Vec, GloVe, and FastText +because of their wide use. +Extrinsic Evaluation. +In our extrinsic evaluation, we in- +vestigate how well our logical embedding supports down- +stream NLP classification tasks. Using the word embeddings +as feature vectors, the performance of supervised classifica- +tion models gives insight into the embedding quality. We +employ six standard text classification datasets from SentE- +val [Conneau and Kiela, 2018]: R8, R52, TREC, SUBJ, SST- +2, and SST-5. For supervised learning, we use the standard +attention-based BiLSTM model with the Adam optimizer and +cross-entropy loss function. In this manner, we directly con- +trast GloVe embedding against the logical TM approach. +Embedding Datasets. +For extrinsic evaluation with BiL- +STM, we use standard 300-dimensional GloVe embeddings, +pre-trained on the Wikipedia 2014 + Gigaword 5 datasets (6B +tokens).4 The purpose is to compare the TM embedding per- +formance against widely used and successful GloVe embed- +dings on downstream tasks. To directly compare the intrinsic +properties of Word2Vec, GloVe, FastText, and TM embed- +ding, we also train them from scratch using the One Billion +Word dataset [Chelba et al., 2014]. For training the TM, we +use Algorithm 1 with r = 2000 training rounds, produc- +ing 2000 examples per word by accumulating u = 25 con- +texts per example. We use the following hyperparameters: a +pool of n = 600 clauses, margin T = 1200, and specificity +s = 5.0.5 Word2Vec Skip-Gram is trained with 10 passes +over the data, using separated embeddings for the input and +output contexts. The window size is 5 and we use five nega- +tive samples per example. Similarly, GloVe is trained for 30 +epochs with a window size of 10 and a learning rate of 0.05.6 +5.2 +Results and Discussion +As presented in Section 5.1, we employ two kinds of evalu- +ation: intrinsic and extrinsic. Table 1 contains the intrinsic +evaluation results from the six word similarity tasks. We here +compute the Spearman correlation, the Kendall coefficient, +and the cosine similarity between the human-set similarity +scores and the predicted similarity scores per dataset. Con- +sidering Spearman and Kendall score, Word2Vec and GloVe +are marginally better than the comparable FastText and TM +embedding. However, as reported in [Rastogi et al., 2015], +3To obtain the categorization accuracy, +we use KMeans +clustering from sklearn on the word embeddings and exam- +ine the cluster quality by calculating the purity score from +(https://github.com/purity). +4The pre-trained GloVe embeddings can be found here: +https://nlp.stanford.edu/projects/glove/ +5The TM Autoencoder and logical word embedding implemen- +tation can be found here: https://github.com/cair/tmu. +6Word2Vec +and +FastText +have +been +trained +using +the +standard +gensim +library +(https://github.com/RaRe- +Technologies/gensim/tree/develop/gensi). GloVe has been trained +using https://github.com/maciejkula/glove-python. + +Dataset +W2V +FastText +TM +GloVe +Spearman +Kendall +Cosine +Spearman +Kendall +Cosine +Spearman +Kendall +Cosine +Spearman +Kendall +Cosine +WS-353 +0.53 +0.37 +0.87 +0.46 +0.32 +0.79 +0.45 +0.31 +0.90 +0.41 +0.28 +0.90 +SIM999 +0.26 +0.18 +0.79 +0.23 +0.16 +0.79 +0.14 +0.10 +0.76 +0.25 +0.17 +0.80 +MEN +0.71 +0.50 +0.91 +0.71 +0.51 +0.94 +0.64 +0.45 +0.94 +0.73 +0.53 +0.95 +MTURK287 +0.66 +0.47 +0.77 +0.63 +0.44 +0.93 +0.63 +0.44 +0.92 +0.66 +0.47 +0.86 +MTURK717 +0.57 +0.39 +0.86 +0.52 +0.36 +0.93 +0.48 +0.32 +0.91 +0.58 +0.40 +0.94 +RG65 +0.72 +0.58 +0.89 +0.67 +0.49 +0.88 +0.75 +0.63 +0.92 +0.78 +0.62 +0.93 +Average +0.58 +0.42 +0.85 +0.54 +0.38 +0.88 +0.52 +0.38 +0.89 +0.57 +0.42 +0.90 +Table 1: Performance comparison of TM embedding with baseline algorithms on the similarity task. +Target words +Clauses +Clauses + +student +surgery +heart +baseball +football +queen +princess +coffee +tea + +Figure 6: Interpretability of clauses capturing distinct meanings of target words in the TM embedding. +Dataset +W2V +FastText +TM +GloVe +AP +0.50 +0.35 +0.41 +0.41 +BLESS +0.64 +0.66 +0.62 +0.66 +ESSLI +0.63 +0.60 +0.57 +0.56 +Average +0.59 +0.54 +0.53 +0.54 +Table 2: Performance comparison of TM embedding with baseline +embeddings on the categorization task. +Dataset +GloVe +TM +TMhybrid +Acc. +F1 +Acc. +F1 +Acc. +F1 +R8 +96.31 +0.88 +96.10 +0.88 +97.80 +0.94 +TREC +95.20 +0.95 +96.40 +0.96 +96.80 +0.96 +R52 +90.34 +0.58 +91.23 +0.62 +94.23 +0.68 +SUBJ +86.20 +0.86 +85.80 +0.85 +86.70 +0.87 +SST-2 +76.38 +0.75 +75.61 +0.74 +79.30 +0.78 +SST-5 +47.47 +0.46 +47.80 +0.43 +49.75 +0.44 +Table 3: Performance comparison of our embedding with standard +GloVe embedding on the classification task. +small differences in correlation-based measures are not nec- +essarily significant for smaller datasets. +To more robustly +assess performance, we therefore also use cosine similarity +to compare predicted word similarities with the human-set +similarities. In terms of cosine score, our model outperforms +Word2Vec and FastText on the majority of the datasets, while +performing competitively with GloVe. This means that the +angles between the human-set similarities and the GloVe/TM- +predicted similarities are quite similar. Finally, Table 2 shows +the outcome for the word categorization tasks. As seen, the +performance of the selected embedding techniques are com- +parable, with Word2Vec being slightly ahead. +Previous research indicates that intrinsic word similarity +performance is minimally or even negatively correlated with +downstream NLP performance [Wang et al., 2021]. There- +fore, we also include an extrinsic evaluation with six down- +stream classification tasks. To avoid overfitting and robustly +assess downstream properties, we keep our experimental +setup from above. Table 3 reports the outcome of the eval- +uation, where the embeddings have been fed to an attention- +based BiLSTM model. The first configuration (GloVe) uses +the pre-trained GloVe embeddings from the Wikipedia 2014 ++ Gigaword 5 datasets. The second configuration consists +of our purely logical TM embedding from One Billion Word +(embedding B from Algorithm 1). Being five times smaller, +the One Billion Word dataset only provides about 80 percent +of the vocabulary required for the classification tasks. We em- +bed the remaining 20 percent of the words randomly. Hence, +the TM approach can potentially have a disadvantage in the +evaluation. In the third configuration (TMhybrid), we replace +the 20 percent random embeddings with the corresponding +GloVe embeddings (approximately 80% TM + 20% GloVe). +We note that the downstream accuracy of BiLSTM is simi- +lar for both TM and GloVe. Specifically, the TM embedding + +exceeds GloVe by a small margin on TREC, R52, and SST- +5. The hybrid embedding, on the other hand, clearly outper- +forms the other two. In particular, for R52, SST-2, and SST-5, +the hybrid embedding is able to surpass GloVe by a substan- +tial margin of roughly 2 − 4%. Given that the datasets are +not completely balanced, we also compute F1 macro scores. +We again observe that the TM embedding either outperforms +or is competive with GloVe. For R8 and R52, the hybrid em- +bedding surpasses GloVe by a large margin, respectively by +around 6% and 10%. Based on these results, we conjecture +that logical TM embedding can successfully replace neural +network embedding. Even with 20% of the vocabulary miss- +ing, trained on five times smaller data, the logical embedding +perform competitively with GloVe. Interestingly, the hybrid +approach performed even better. One possible explanation of +this higher performance can be the extra information added +by the larger vocabulary. Additionally, there may be synergy +between the neural and logical representations that manifest +in the hybrid approach. +5.3 +Interpretability and Visualization +In this section, we investigate the nature of the TM embed- +dings in more detail, focusing on interpretability. Our em- +bedding consists of the positive clause weights E, or, alter- +natively, the propositional version B, explained by the set +of clauses C. As demonstrated in Figure 6, each clause in C +captures a facet of a context. The dotted lines in the figure +showcase the connection between the target words and their +clauses from matrix B (and, accordingly, E). Each target +word gets its own color to more easily discern the connec- +tions. In the figure, we provide an excerpt of 18 connections +from B, involving 8 target words and the 11 most triggered +clauses for these words. +Consider for example the target +words surgery and heart. These two target words share two +clauses: [went ∧ hospital] and [old ∧ disease ∧ patient]. +The two clauses capture two joint contexts, both related to +health. The clauses thus represent commonality between the +target words, providing information on one particular mean- +ing of the words. +The two target words are also semantically different. The +differences are captured by the clauses they do not share. The +target word heart, for example, also relates to the meaning +[woman ∧ love], which surgery does not. Surgery, on the +other hand, connects with [injury ∧ game ∧ racing]. In this +manner, the unique meanings and relations between words +are represented through sharing of logical expressions. Ac- +cordingly, it is feasible to capture a wide range of possible +contextual representations with concise logical expressions. +As such, the logical embedding provides a sparse representa- +tion of words and their relations. Indeed, at most 10% of the +clauses connect to each word in our experiments. As shown in +the intrinsic evaluations from the previous subsection, these +contextual representations are effective for measuring word +similarity and categorizing words. Similarly, we observed +that the logical embedding is boosting downstream NLP clas- +sification tasks. +To cast further light on the TM embedding approach, we +visualize the embedding of 400 words from the SimLex-999 +dataset in Figure 7, plotted using t-SNE. The figure indicates +Figure 7: TM embedding visualization plotted using t-SNE. +that we are able to cluster contextually similar words in vec- +tor space. +To scrutinize the clusters, we zoom in on two +of them. Consider the upper right cluster first. Notice how +the words in the cluster relate to hospital, such as heart and +diseases. As seen, the word embeddings are closely located +in vector space. Similarly, we can observe that terminology +connected to weather and geography are grouped together in +the bottom cluster. From these two examples, it seems clear +that the TM embedding incorporates semantic relationships +among words. +6 +Conclusion and Future Work +In this work, we first discussed the challenge and necessity +of finding computationally simpler and more interpretable +word embedding approaches. We then motivated an efficient +self-supervised approach, namely, a TM-based autoencoder, +for producing sparse and interpretable logical word embed- +dings. We evaluated our approach on a wide range of intrin- +sic and extrinsic tasks, demonstrating that it is competitive +with dense neural network-based embedding schemes such +as Word2Vec, GloVe, and FastText. Further, we investigated +the interpretability our embedding through visualization and +a case study. Our conclusion from the study is that the log- +ical embedding is able to represent words with logical ex- +pressions. This structure makes the representation sparse, en- +abling a clear-cut decomposition of each word into sets of +semantic concepts. +Future work includes scaling up our implementation using +GPUs to support building of large scale vocabularies from +more massive datasets. Also, we intend to investigate how +sentence-level and document-level embedding can be created +using clauses, for instance applicable for downstream sen- +tence similarity tasks. + +500 +600 +condition +hospital +surgery +450 +heart +physician +infe&tion +doctor +disease +400 +illness +Tiver +pain +organ +blood +alcohol +400 +350 +bone +sick +nerMuscle +300 +400 +350 +-300 +-250 +-200 +200 +0 +（ +250 +bath +-200 +cliff +south +inn +valley +hill +north +foot + gate +area +tower +detk +beach +highway +canyon +boundary +roofbrick +tear +island +150 +tree +coast +wobd +corridor +dirtmud +hurreaggitude +ocean +.400 +grass +ice +10Q: +storm +sea +airport +tin +narrow +stream +rwigather +dawn +andient +water +plane +50 - +salt +parade +heavy +sky sun +iet +250 +350 +400: +450 +558 +3Q0 +500 +600 +400 +269 +400 +600References +[Angelov, 2020] Dimo Angelov. 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Tsdae: Using transformer-based sequential de- +noising auto-encoder for unsupervised sentence embed- +ding learning. arXiv preprint arXiv:2104.06979, 2021. +[Yadav et al., 2021] Rohan Yadav, Lei Jiao, Ole-Christoffer +Granmo, and Morten Goodwin. +Human-Level Inter- +pretable Learning for Aspect-Based Sentiment Analysis. +In Proceedings of AAAI, 2021. +[Zhang et al., 2020] Yan Zhang, Ruidan He, Zuozhu Liu, +Kwan Hui Lim, and Lidong Bing. An Unsupervised Sen- +tence Embedding Method by Mutual Information Max- +imization. +In Proceedings of the 2020 Conference +on Empirical Methods in Natural Language Processing +(EMNLP), pages 1601–1610, 2020. + diff --git a/rdAyT4oBgHgl3EQfz_kx/content/tmp_files/load_file.txt b/rdAyT4oBgHgl3EQfz_kx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..378eac3f8357eb8476d1bd9704baaed86e32d7c3 --- /dev/null +++ b/rdAyT4oBgHgl3EQfz_kx/content/tmp_files/load_file.txt @@ -0,0 +1,700 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf,len=699 +page_content='Tsetlin Machine Embedding: Representing Words Using Logical Expressions Bimal Bhattarai , Ole-Christoffer Granmo , Lei Jiao , Rohan Yadav and Jivitesh Sharma Centre for AI Research (CAIR), University of Agder, Norway {bimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='bhattarai, ole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='granmo, lei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='jiao, rohan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='yadav, jivitesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='sharma}@uia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='no Abstract Embedding words in vector space is a fundamen- tal first step in state-of-the-art natural language pro- cessing (NLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Typical NLP solutions employ pre- defined vector representations to improve gener- alization by co-locating similar words in vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' For instance, Word2Vec is a self-supervised predictive model that captures the context of words using a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Similarly, GLoVe is a popular unsupervised model incorporating corpus- wide word co-occurrence statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Such word em- bedding has significantly boosted important NLP tasks, including sentiment analysis, document clas- sification, and machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' However, the embeddings are dense floating-point vectors, mak- ing them expensive to compute and difficult to in- terpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In this paper, we instead propose to rep- resent the semantics of words with a few defin- ing words that are related using propositional logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' To produce such logical embeddings, we introduce a Tsetlin Machine-based autoencoder that learns logical clauses self-supervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The clauses con- sist of contextual words like “black,” “cup,” and “hot” to define other words like “coffee,” thus be- ing human-understandable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We evaluate our em- bedding approach on several intrinsic and extrinsic benchmarks, outperforming GLoVe on six classifi- cation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Furthermore, we investigate the inter- pretability of our embedding using the logical rep- resentations acquired during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We also vi- sualize word clusters in vector space, demonstrat- ing how our logical embedding co-locate similar words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='1 1 Introduction The success of natural language processing (NLP) relies on advances in word, sentence, and document representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' By capturing word semantics and similarities, such representa- tions boost the performance of downstream tasks [Borgeaud 1The Tsetlin Machine Autoencoder and logical word embedding implementation is available here: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='com/cair/tmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2022], including clustering, topic modelling [Angelov, 2020], searching, and text mining [Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' While straightforward, the traditional bag-of-words encod- ing does not consider the words’ position, semantics, and context within a document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Distributed word representa- tion [Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Bojanowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2017] addresses this lack by encoding words as low-dimensional vectors, re- ferred to as embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The purpose is to co-locate simi- lar or contextually relevant words in vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' There are many algorithms for learning word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Contem- porary self-supervised techniques like Word2Vec [Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2013], FastText [Bojanowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2017], and GloVe [Pennington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2014] have demonstrated how to build embeddings from word co-occurrence, utilizing mas- sive training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Introducing context-dependent embed- dings, the more sophisticated language models BERT [De- vlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2019] and ELMO [Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2018] now per- form remarkably well in downstream tasks [Reimers and Gurevych, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' However, they require significant compu- tation power [Schwartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The above approaches represent words as dense floating point vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Word2Vec, for instance, typically builds a 300- dimensional vector per word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The size and density of these vectors make them expensive to compute and difficult to in- terpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Consider, for example, the word “queen.” Represent- ing it with 300 floats seems inefficient compared to the Ox- ford Language definition for the same word: “the female ruler of an independent state, especially one who inherits the posi- tion by right of birth.” From this perspective, it appears ad- vantageous to create embeddings directly from words rather than from arbitrary floating-point values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Such interpretable embeddings would capture the multiple meanings of a word using a few defining words, simplifying both computation and interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In this paper, we propose a Tsetlin Machine (TM) [Granmo, 2018] based autoencoder for creating interpretable embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The autoencoder builds propositional logic ex- pressions with context words that identify each target word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The term “coffee” can, for instance, be represented by “one,” “hot,” “cup,” “table,” and “black.” In this manner, the TM builds contextual representations from a vast text corpus, which model the semantics of each word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In contrast to neu- ral network-based embedding, the logical TM embedding is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The embedding space consists of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 500 truth val- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='00709v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='CL] 2 Jan 2023 ues, where each truth value is a logical expression over words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' For contextual representation, each target word links to less than ten percent of these expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Despite the sparsity and crispness of this representation, it is competitive with neural network-based embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The contributions of our work are summarized below: We propose the first TM-based Autoencoder to learn ef- ficient encodings in a self-supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We introduce TM-based word embedding that builds human-comprehensible contextual representations from unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We compare our embedding with state-of-the-art ap- proaches on several intrinsic and extrinsic benchmarks, outperforming GloVe on six downstream classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 2 Related Work The majority of self-supervised embedding approaches pro- duce dense word representations based on the distributional hypothesis [Harris, 1954], which states that words that oc- cur in the same context are likely to have similar meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Word2Vec [Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2013] is one of the best-known models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' It builds embeddings from word co-occurrence using a neural network, leveraging the hidden layer output weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' GloVe [Pennington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2014], on the other hand, em- beds by factorizing a word co-occurrence matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Similarly, canonical correlation analysis (CCA) is used in [Dhillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2015] for embedding words to maximise context corre- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In [Levy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2015], it is demonstrated how pre- cise factorization-based SVD can compete with neural em- bedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' However, all of these methods are challenging to train because they involve tweaking algorithms and hyperpa- rameters toward particular applications [Lample et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2016], limiting their wider applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Building upon word embedding, several studies focus on sentence embedding [Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Logeswaran and Lee, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Recent advances in sentence embedding in- clude supervised data inference [Reimers and Gurevych, 2019], multitask learning [Cer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2018], contrastive learn- ing [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2020], and pretrained large language mod- els [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' However, the majority of sentence embedding techniques overlook intrinsic evaluations such as similarity tasks, and instead largely focus on extrinsic eval- uations involving downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The most re- cent building block for embedding originates from the trans- former approach [Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Transformers pro- vide context awareness by utilizing stacks of self-attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' BERT [Kenton and Toutanova, 2019], for instance, employs the transformer architecture to carry out extensive self-supervised training, making it capable of producing text embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Other embedding models use a contrastive loss function to perform supervised fine-tuning on positive and negative text pairs [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Despite the large variety of text embedding models, they all share three main drawbacks: i) they are computationally demanding to train;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' ii) they are intrinsically complex because they are trained on a large amount of data to tune a huge amount of parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Clause Pool +4 5 Input Output +1 +2 7 +6 Figure 1: Tsetlin Machine Autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In this illustration, x1 is masked by replacing it with value 1 for inferring ˆx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' and iii) the embeddings produced from these models are not easily interpreted by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' To improve interpretability, Faruqui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' introduced “Sparse Overcomplete Word Vectors” (SPOWV) which cre- ate a sparse non-negative projection of word embedding us- ing dictionary learning [Faruqui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Similarly, SParse Interpretable Neural Embeddings (SPINE) employs a k-sparse denoising autoencoder to generate sparse embed- dings [Subramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' However, these methods are unable to distinguish between multiple context-dependent word meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' To address this problem, another avenue of research focuses on composing linear combinations of dense vectors from Word2Vec and GloVe [Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' How- ever, the assumption of linearity does not hold for real-world data, yielding linear coefficients that are difficult to compre- hend [Mu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The logical embedding approach we present here is most closely related to Naive Bayes word sense induction and topic modeling [Charniak and others, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Lau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' This approach learns word meanings from local contexts by con- sidering each instance of the word in a document as a pseudo- document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' However, the approach is not scalable because it requires training a single topic per target word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Our ap- proach, on the other hand, is scalable and builds non-linear (non-naive) logical embeddings that capture word composi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' To build the logical embeddings, we propose a novel human-interpretable algorithm based on the TM that provides logical rules describing contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The TM has recently per- formed competitively with other deep learning techniques in many NLP tasks, including novelty detection [Bhattarai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2022a], sentiment analysis [Yadav et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2021], and fake news detection [Bhattarai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2022b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Furthermore, the local and global interpretability of TMs have been explored through direct manipulation of the logical rules [Blakely and Granmo, 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 3 Tsetlin Machine Autoencoder We here detail the TM Autoencoder based on the Coalesced TM [Glimsdal and Granmo, 2021], extended with input masking and freezing of masked variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' For ease of ex- planation, we use three inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Adding more inputs follows trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='1 Architecture Input and Output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' As seen in Figure 1, the TM Autoen- coder digests and outputs propositional values: (x1, x2, x3) ∈ {0, 1}3 → (�x1, �x2, �x3) ∈ {0, 1}3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' For our purposes, the propositional variables x1, x2, and x3 each represents a word, for example, “Brilliant,” “Actor,” and “Awful.” The value 1 means that the word occurs in the input text, while the value 0 means that it does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', we represent natural language text as a set of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Notice also that the input variables have corresponding output variables �x1, �x2, and �x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In short, �x1 is to be predicted from x2 and x3, �x2 from x1 and x3, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Continuing our example, �x1 predicts the presence of “Brilliant” based on knowing the occurrence of “Actor” and “Awful.” Clause Pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' A pool of n conjunctive clauses, denoted Cj, j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' , n}, encodes the input in order to pre- dict the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' A conjunctive clause Cj is simply an And- expression over a given subset Lj ⊆ {x1, x2, x3} of the input (our autoencoder does not use the input negations ¬x1, ¬x2, and ¬x3): Cj(x1, x2, x3) = � xk∈Lj xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' (1) For example, the input subset L1 = {x1, x2} gives the clause C1(x1, x2, x3) = x1 ∧ x2 in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' This clause matches the input if x1 and x2 both are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In our example, the clause accordingly encodes the concept “Brilliant Actor”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' An integer weight matrix W connects each of the n clauses to the three outputs �x1, �x2, and �x3: W = �w11 · · w1n w21 · · w2n w31 · · w3n � ∈ Z3×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' (2) The row index is an output while the column index is a clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The weight w12, for instance, connects output �x1 to clause C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In Figure 1, six weights connect two clauses and three outputs: �+4 −5 +1 +2 −7 +6 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' (3) Consider, for example, the weights (+4, −5) of output �x1 in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The weight +4 states that clause C1(x1, x2, x3) = x1 ∧ x2 favours �x1 being 1, while clause C2(x1, x2, x3) = x2 ∧ x3 opposes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' For example, the concept “Awful Actor” opposes output “Brilliant.” 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='2 Inference Let us consider the prediction of �x1 first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The autoencoder predicts �x1 from the clauses and weights: �x1 = 0 ≤ n � j=1 wj1Cj(1, x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' (4) That is, each clause Cj is multiplied by its weight wj1 for output �x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The outcomes are then summed up to decide the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' If the sum is larger than or equal to zero, the output is �x1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Otherwise, it is �x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Clauses with positive Maximally Memorized Maximally Forgotten Memorized Forgotten 4 3 2 1 Figure 2: Tsetlin Machine memory for single clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' weight thus promote output �x1 = 1 while clauses with neg- ative weight encourage �x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Notice that x1 is masked by replacing it with value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Accordingly, the autoencoder infers output �x1 from the remaining inputs x2 and x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Correspondingly, �x2 and �x3 are calculated by respectively masking x2 and x3: �x2 = 0 ≤ n � j=1 wj2Cj(x1, 1, x3), (5) �x3 = 0 ≤ n � j=1 wj3Cj(x1, x2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' (6) Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Assume that the input is always either (1, 1, 0) or (0, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The input (1, 1, 0) could for instance represent “Brilliant Actor” and (0, 1, 1) “Awful Actor.” Then notice how Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' (4) correctly determines the masked input x1 with output �x1 in Figure 1, both for input (1, 1, 0) and (0, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='3 Learning We next consider how to learn the variable subsets Lj for the clauses Cj, j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' , n}, as well as how to determine the weights wji of the weight matrix W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Clause Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Each clause Cj has a graded memory that contains the input variables, shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The graded memory enables incremental learning of the variable subsets from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Observe how each variable is in one of four mem- ory positions (the number of memory positions is a user- configurable parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Positions 1 − 2 means Forgotten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Positions 3−4 means Memorized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Memorized variables take part in the clause, while Forgotten ones do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The memory in Figure 2 thus gives the clause Cj(x1, x2, x3) = x1 ∧ x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Learning Step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The TM Autoencoder learns incrementally using three kinds of memory and weight updates: Type Ia, Type Ib, and Type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Each training example has the form [k, (x1, x2, x3), xk], 1 ≤ k ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The first element is an index that identifies which input to mask and which output to pre- dict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The second element is an input vector (x1, x2, x3) and the third element is the target value for output �xk, which is xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We describe the update procedure step-by-step below for index 1 examples (output �x1 prediction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The update proce- dure for �x2 and �x3 follows trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Clause Update Probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' First, we calculate the weighted clause sum for �x1 from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' (4): v1 = �n j=1 wj1Cj(1, x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The sum is then compared with a margin T (hyper-parameter) to calculate a summation error ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The error depends on the x1-value: ϵ = �T − clip(v1, −T, T), x1 = 1, T + clip(v1, −T, T), x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' (7) Maximally Memorized Maximally Forgotten Memorized Forgotten 4 3 2 1 Figure 3: Type Ia (Recognize) Feedback for input (1, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The masked variable x1 is frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Maximally Memorized Maximally Forgotten Memorized Forgotten 4 3 2 1 Figure 4: Type Ib (Erase) Feedback for input (0, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The masked variable x1 is frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' That is, for x1-value 1 the weighted clause sum should be- come T, while for x1-value 0 the sum should become −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The goal of the learning is thus to reach the margin for all inputs (x1, x2, x3), ensuring correct output from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' To reach this goal, each clause Cj is updated randomly with probability ϵ 2T in each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In other words, the update probability drops with the error towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Update Types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The kind of update depends on the values of x1, Cj(1, x2, x3), and wj1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We first consider clauses with positive weight, wj1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' According to Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 4, they are to recognize patterns for x1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Note that in all of the below updates, the masked variable x1 is frozen, leaving it unaf- fected by the update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Type Ia (Recognize) Feedback occurs when x1 = 1 and Cj(1, x2, x3) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Then one can say that Cj(1, x2, x3) = 1 is a true positive because it correctly predicts the masked x1-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The Type Ia feedback re- inforces this successful match by updating the memory of Cj to further mimic the input (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' That is, 1-valued variables move one step upwards in memory, with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='2 Conversely, 0-valued inputs move one step downwards, however, randomly with probabil- ity 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Here, s is a hyperparameter called specificity, meaning that a larger s makes the clauses more specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The clause overall is also reinforced by incrementing its weight wj1 by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Type Ib (Erase) Feedback occurs when x1 = 1 and Cj(1, x2, x3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Then we call Cj(1, x2, x3) = 0 a false negative because it fails to promote x1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In that case, all inputs randomly move one step down- wards in memory (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Again, each downward move happens with probability 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Here, the purpose is to eliminate the false negative outcome by erasing vari- ables from the clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 2Originally, the increment probability is s−1 s , which can be boosted to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='0 to enhance learning of true positive patterns [Granmo, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Maximally Memorized Maximally Forgotten Memorized Forgotten 4 3 2 1 Figure 5: Type II (Reject) Feedback for input (0, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The masked variable x1 is frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Type II (Reject) Feedback occurs when x1 = 0 and Cj(1, x2, x3) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Then, one can say that Cj(1, x2) = 1 is a false positive because it promotes x1 = 1 when in fact we have x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Then all Forgotten 0-valued in- puts move one step upwards in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The purpose is to eventually eliminate the current false positive out- come by injecting 0-valued variables into the clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The clause is further diminished by decrementing its weight wj1 by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Note that the latter decrement can switch the weight from positive to negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In effect, the clause then changes role, now training to recognize x1 = 0 instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Clauses Cj with negative weights, wj1 < 0, are updated the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' However, they are to recognize patterns for x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' To achieve this, x1 = 0 is treated as x1 = 1 and x1 = 1 is treated as x1 = 0 when updating the memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Furthermore, the weight updates are reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Increments be- comes decrements, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Algorithm 1 TM word embedding Require: Vocabulary V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Documents D ∈ G, D ⊆ V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Accumula- tion u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Clauses n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Margin T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Specificity s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Rounds r 1: TMCreate(n, T, s) ▷ Create TM with n clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 2: for r rounds do 3: for word k ∈ V do ▷ Create one example per word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 4: qk ← Select({0, 1}) ▷ Random target value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 5: if qk = 1 then 6: Gk ← {D|word k ∈ D, D ∈ G} ▷ Documents with word k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 7: else 8: Gk ← {D|word k /∈ D, D ∈ G} ▷ Documents without word k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 9: Sk ← SelectN(Gk, u) ▷ Random subset of size u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 10: Uk ← � D∈Sk D ▷ Union of selected documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 11: xk ← (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' , xm), xi = � 1, word i ∈ U k 0, word i /∈ U k 12: TMUpdate(k, xk, qk) ▷ Update TM Autoencoder for output index k, input xk, and target value �xk = qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 13: C, W ← TMGetState() ▷ Clauses Cj ∈ C with weights W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 14: E ← clip(W , 0, T) ▷ Elementwise clip of negative values produces weighted logical word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 15: B ← (W > 0) ▷ Elementwise comparison with zero produces purely logical word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 16: return C, E, B 4 Logical Embedding Procedure We now use the TM Autoencoder to build logical embed- dings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Let V = {word1, word2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' , wordm} be the target vocabulary consisting of m unique words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The first step is to pre-process the docu- ment corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' To this end, each document is represented by a subset of words D ⊆ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' For example, the document “The ac- tor was brilliant” becomes the set D = {“actor”, “brilliant”, “the”, “was”}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The set G, in turn, contains all the documents, D ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Finally, in propositional vector form, the word set D becomes: x = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' , xt), xi = �1, wordi ∈ D, 0, wordi /∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' (8) Embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Algorithm 1 specifies the procedure for em- bedding the m vocabulary words from V by using n clauses, Cj, 1 ≤ j ≤ n, forming a clause set C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Each round of training produces a training example [k, (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' , xm), qk] per wordk in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' First, a target value qk for the word is set randomly to either 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' This random selection balances the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' If qk becomes 1, we randomly select u doc- uments that contain wordk and assign them to the set Sk (positive examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Otherwise, we randomly select u doc- uments that does not contain the word (negative examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Next, the randomly selected documents are merged by OR- ing them together, yielding the unified document Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The purpose of ORing multiple documents is to increase the fre- quency of rare context words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Then, picking up characteristic ones becomes easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' After that, the propositional vector form (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' , xm) of Uk is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Finally, the TM Autoen- coder is updated with [k, (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' , xm), qk] following the training procedure in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Vector Space Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The weighted logical em- bedding of wordk ∈ V can now be obtained from row k of matrix E (returned from Algorithm 1), while the the purely logical embedding is found in row k of matrix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Let ek de- note the k’th row of E, and let el denote the l’th row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We can then compare the similarity of two words wordk and wordl using cosine similarity (CS) between their E-embedding: CS(wordk, wordl) = ek · el ||ek|| ||el||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' (9) 5 Empirical Evaluation We here evaluate our logical embedding scheme, comparing it with neural network approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='1 Datasets and Setup We first evaluate our logical embedding intrinsically, fol- lowed by an extrinsic evaluation using classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Intrinsic Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We use word similarity and catego- rization benchmarks for intrinsic evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' That is, we ex- amine to what degree our approach retains semantic word relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' To this end, we measure how semantic relations manifest in vector space using six datasets: SimLex-999, WordSim-353, MEN, MTurk-287, Mturk-771, and RG-65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Each dataset consists of human-scored word pairs, which are compared with the corresponding vector space similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The categorization tasks evaluate how well we can group words into distinct word categories, only based on their em- bedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We here use three datasets: AP, BLESS, and ESS- LLI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='3 As baselines, we chose Word2Vec, GloVe, and FastText because of their wide use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Extrinsic Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In our extrinsic evaluation, we in- vestigate how well our logical embedding supports down- stream NLP classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Using the word embeddings as feature vectors, the performance of supervised classifica- tion models gives insight into the embedding quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We employ six standard text classification datasets from SentE- val [Conneau and Kiela, 2018]: R8, R52, TREC, SUBJ, SST- 2, and SST-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' For supervised learning, we use the standard attention-based BiLSTM model with the Adam optimizer and cross-entropy loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In this manner, we directly con- trast GloVe embedding against the logical TM approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Embedding Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' For extrinsic evaluation with BiL- STM, we use standard 300-dimensional GloVe embeddings, pre-trained on the Wikipedia 2014 + Gigaword 5 datasets (6B tokens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='4 The purpose is to compare the TM embedding per- formance against widely used and successful GloVe embed- dings on downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' To directly compare the intrinsic properties of Word2Vec, GloVe, FastText, and TM embed- ding, we also train them from scratch using the One Billion Word dataset [Chelba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' For training the TM, we use Algorithm 1 with r = 2000 training rounds, produc- ing 2000 examples per word by accumulating u = 25 con- texts per example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We use the following hyperparameters: a pool of n = 600 clauses, margin T = 1200, and specificity s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='5 Word2Vec Skip-Gram is trained with 10 passes over the data, using separated embeddings for the input and output contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The window size is 5 and we use five nega- tive samples per example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Similarly, GloVe is trained for 30 epochs with a window size of 10 and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='2 Results and Discussion As presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='1, we employ two kinds of evalu- ation: intrinsic and extrinsic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Table 1 contains the intrinsic evaluation results from the six word similarity tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We here compute the Spearman correlation, the Kendall coefficient, and the cosine similarity between the human-set similarity scores and the predicted similarity scores per dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Con- sidering Spearman and Kendall score, Word2Vec and GloVe are marginally better than the comparable FastText and TM embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' However, as reported in [Rastogi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2015], 3To obtain the categorization accuracy, we use KMeans clustering from sklearn on the word embeddings and exam- ine the cluster quality by calculating the purity score from (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='com/purity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 4The pre-trained GloVe embeddings can be found here: https://nlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='edu/projects/glove/ 5The TM Autoencoder and logical word embedding implemen- tation can be found here: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='com/cair/tmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 6Word2Vec and FastText have been trained using the standard gensim library (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='com/RaRe- Technologies/gensim/tree/develop/gensi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' GloVe has been trained using https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='com/maciejkula/glove-python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Dataset W2V FastText TM GloVe Spearman Kendall Cosine Spearman Kendall Cosine Spearman Kendall Cosine Spearman Kendall Cosine WS-353 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='53 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='90 Table 1: Performance comparison of TM embedding with baseline algorithms on the similarity task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Target words Clauses Clauses student surgery heart baseball football queen princess coffee tea Figure 6: Interpretability of clauses capturing distinct meanings of target words in the TM embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Dataset W2V FastText TM GloVe AP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='41 BLESS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='66 ESSLI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='56 Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='54 Table 2: Performance comparison of TM embedding with baseline embeddings on the categorization task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Dataset GloVe TM TMhybrid Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' F1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' F1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' F1 R8 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='88 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='88 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='94 TREC 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='95 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='96 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='96 R52 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='58 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='62 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='68 SUBJ 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='86 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='85 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='87 SST-2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='75 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='74 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='78 SST-5 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='46 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='43 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='44 Table 3: Performance comparison of our embedding with standard GloVe embedding on the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' small differences in correlation-based measures are not nec- essarily significant for smaller datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' To more robustly assess performance, we therefore also use cosine similarity to compare predicted word similarities with the human-set similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In terms of cosine score, our model outperforms Word2Vec and FastText on the majority of the datasets, while performing competitively with GloVe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' This means that the angles between the human-set similarities and the GloVe/TM- predicted similarities are quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Finally, Table 2 shows the outcome for the word categorization tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' As seen, the performance of the selected embedding techniques are com- parable, with Word2Vec being slightly ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Previous research indicates that intrinsic word similarity performance is minimally or even negatively correlated with downstream NLP performance [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' There- fore, we also include an extrinsic evaluation with six down- stream classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' To avoid overfitting and robustly assess downstream properties, we keep our experimental setup from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Table 3 reports the outcome of the eval- uation, where the embeddings have been fed to an attention- based BiLSTM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The first configuration (GloVe) uses the pre-trained GloVe embeddings from the Wikipedia 2014 + Gigaword 5 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The second configuration consists of our purely logical TM embedding from One Billion Word (embedding B from Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Being five times smaller, the One Billion Word dataset only provides about 80 percent of the vocabulary required for the classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We em- bed the remaining 20 percent of the words randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Hence, the TM approach can potentially have a disadvantage in the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In the third configuration (TMhybrid), we replace the 20 percent random embeddings with the corresponding GloVe embeddings (approximately 80% TM + 20% GloVe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We note that the downstream accuracy of BiLSTM is simi- lar for both TM and GloVe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Specifically, the TM embedding exceeds GloVe by a small margin on TREC, R52, and SST- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The hybrid embedding, on the other hand, clearly outper- forms the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In particular, for R52, SST-2, and SST-5, the hybrid embedding is able to surpass GloVe by a substan- tial margin of roughly 2 − 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Given that the datasets are not completely balanced, we also compute F1 macro scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We again observe that the TM embedding either outperforms or is competive with GloVe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' For R8 and R52, the hybrid em- bedding surpasses GloVe by a large margin, respectively by around 6% and 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Based on these results, we conjecture that logical TM embedding can successfully replace neural network embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Even with 20% of the vocabulary miss- ing, trained on five times smaller data, the logical embedding perform competitively with GloVe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Interestingly, the hybrid approach performed even better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' One possible explanation of this higher performance can be the extra information added by the larger vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Additionally, there may be synergy between the neural and logical representations that manifest in the hybrid approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='3 Interpretability and Visualization In this section, we investigate the nature of the TM embed- dings in more detail, focusing on interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Our em- bedding consists of the positive clause weights E, or, alter- natively, the propositional version B, explained by the set of clauses C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' As demonstrated in Figure 6, each clause in C captures a facet of a context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The dotted lines in the figure showcase the connection between the target words and their clauses from matrix B (and, accordingly, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Each target word gets its own color to more easily discern the connec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In the figure, we provide an excerpt of 18 connections from B, involving 8 target words and the 11 most triggered clauses for these words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Consider for example the target words surgery and heart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' These two target words share two clauses: [went ∧ hospital] and [old ∧ disease ∧ patient].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The two clauses capture two joint contexts, both related to health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The clauses thus represent commonality between the target words, providing information on one particular mean- ing of the words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The two target words are also semantically different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The differences are captured by the clauses they do not share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The target word heart, for example, also relates to the meaning [woman ∧ love], which surgery does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Surgery, on the other hand, connects with [injury ∧ game ∧ racing].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In this manner, the unique meanings and relations between words are represented through sharing of logical expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Ac- cordingly, it is feasible to capture a wide range of possible contextual representations with concise logical expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' As such, the logical embedding provides a sparse representa- tion of words and their relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Indeed, at most 10% of the clauses connect to each word in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' As shown in the intrinsic evaluations from the previous subsection, these contextual representations are effective for measuring word similarity and categorizing words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Similarly, we observed that the logical embedding is boosting downstream NLP clas- sification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' To cast further light on the TM embedding approach, we visualize the embedding of 400 words from the SimLex-999 dataset in Figure 7, plotted using t-SNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' The figure indicates Figure 7: TM embedding visualization plotted using t-SNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' that we are able to cluster contextually similar words in vec- tor space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' To scrutinize the clusters, we zoom in on two of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Consider the upper right cluster first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Notice how the words in the cluster relate to hospital, such as heart and diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' As seen, the word embeddings are closely located in vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Similarly, we can observe that terminology connected to weather and geography are grouped together in the bottom cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' From these two examples, it seems clear that the TM embedding incorporates semantic relationships among words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 6 Conclusion and Future Work In this work, we first discussed the challenge and necessity of finding computationally simpler and more interpretable word embedding approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We then motivated an efficient self-supervised approach, namely, a TM-based autoencoder, for producing sparse and interpretable logical word embed- dings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' We evaluated our approach on a wide range of intrin- sic and extrinsic tasks, demonstrating that it is competitive with dense neural network-based embedding schemes such as Word2Vec, GloVe, and FastText.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Further, we investigated the interpretability our embedding through visualization and a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Our conclusion from the study is that the log- ical embedding is able to represent words with logical ex- pressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' This structure makes the representation sparse, en- abling a clear-cut decomposition of each word into sets of semantic concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Future work includes scaling up our implementation using GPUs to support building of large scale vocabularies from more massive datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' Also, we intend to investigate how sentence-level and document-level embedding can be created using clauses, for instance applicable for downstream sen- tence similarity tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' 500 600 condition hospital surgery 450 heart physician infe&tion doctor disease 400 illness Tiver pain organ blood alcohol 400 350 bone sick nerMuscle 300 400 350 300 250 200 200 0 （ 250 bath 200 cliff south inn valley hill north foot gate area tower detk beach highway canyon boundary roofbrick tear island 150 tree coast wobd corridor dirtmud hurreaggitude ocean .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content='400 grass ice 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 1601–1610, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAyT4oBgHgl3EQfz_kx/content/2301.00709v1.pdf'} diff --git a/s9AzT4oBgHgl3EQfrv3p/vector_store/index.pkl b/s9AzT4oBgHgl3EQfrv3p/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..768a328f38aeea64c32544bcb1062c518768b841 --- /dev/null +++ b/s9AzT4oBgHgl3EQfrv3p/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a2372d8b322c0f147e9aacaf194130b783caf6f24c85dc8bbd007d1a9f983524 +size 224004 diff --git a/sdE2T4oBgHgl3EQfLgYA/vector_store/index.faiss b/sdE2T4oBgHgl3EQfLgYA/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..397fa985303c4f9f7d055e98971c554c219ab48e --- 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sha256:3a3dc23bc8b5cac3b5c463b780ca39b9b56e95bbbcfe8d255ee1335339ee3a3f +size 2949165 diff --git a/v9E0T4oBgHgl3EQf-AJM/content/tmp_files/load_file.txt b/v9E0T4oBgHgl3EQf-AJM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..96fce29c8911905de47cfed4d3b249f8a21c36cd --- /dev/null +++ b/v9E0T4oBgHgl3EQf-AJM/content/tmp_files/load_file.txt @@ -0,0 +1,481 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf,len=480 +page_content='High-efficiency entanglement of two microwave fields in cavity opto-magnomechanical systems Ke Di1, Shuai Tan1, Liyong Wang2,*, Anyu Cheng1, Xi Wang1, Yu Liu1, Jiajia Du1,* 1 Chongqing University of Post and Telecommunications, Chongqing 400065, China 2 Department of Applied Physics, Wuhan University of Science and Technology, Wuhan 430081, China Authors to whom any correspondence should be addressed E-mail: wangliyong@wust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='cn and dujj@cqupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='cn January 2023 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' We demonstrate a scheme to realize high-efficiency entanglement of two microwave fields in a dual opto-magnomechanical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The magnon mode simultaneously couples with the microwave cavity mode and phonon mode via magnetic dipole interaction and magnetostrictive interaction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Meanwhile, the phonon mode couples with the optical cavity mode via radiation pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Each magnon mode and optical cavity mode adopts a strong red detuning driving field to activate the beam splitter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Therefore, the entangled state generated by the injected two-mode squeezed light in optical cavities can be eventually transferred into two microwave cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' A stationary entanglement Ea1a2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='54 is obtained when the input two-mode squeezed optical field has a squeezing parameter r=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Meanwhile, the entanglement Ea1a2 increases as the squeezing parameter r increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' This demonstrates the flexible tunability of our scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The entanglement survives up to an environmental temperature about 385 mK, which shows high robustness of the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Our result is useful for applications which require high entanglement of microwave fields like quantum radar, quantum navigation, quantum teleportation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Introduction Entangled states have been widely investigated for decades since it is an essential resource in fields of quantum computation [1], quantum key distribution[2, 3], quantum teleportation [4, 5, 6], quantum network construction [7, 8] and fundamental physics [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Since the wave- particle duality, the entangled state of light field exists in two forms: one is the entangled photon pair in discrete variables (DV), and the other is the two-mode squeezed state in continuous variable (CV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' In particular, the light field of CV has the characteristics of superior broadband, excellent stabilization and high operating efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Generally, entangled optical fields in CV are prepared with second-order or third-order nonlinear crystal by parametric down-conversion process [11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Compared to optical fields, there are few methods to prepare entangled microwave fields of CV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The nonlinear Josephson circuit parametric amplifier [14, 15, 16], circuit quantum electrodynamics (QED) [17, 18] and optomechanics [19, 20] systems are traditionally used to generate entangled states of high quality microwave fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' However, these schemes require complex equipment and adjustment operations, which make the integration and elaborate control difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' It is also demonstrated that entanglement of two microwave fields can be generated deterministically in an optomechanical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' However, it is difficult to obtain resolved sideband [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Recently, preparing steady-state entanglement of two microwave fields based on nonlinear magnetostrictive effects in a cavity magnomechanical system is reported [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' It is easy to satisfy the resolved-sideband condition for cooling the mechanical oscillator to ground-state since the linewidth of the magnon is much smaller than the frequency of the phonon [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' However, the entanglement strength of two microwave fields in cavity magnomechanical systems is much lower compared with the traditional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' How to further enhance the stationary entanglement EN in cavity magnomechanical system is an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Cavity magnomechanical system is useful for studying quantum states at macroscopic scales [23, 24, 25, 26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' As a ferromagnetic material, Yttrium iron garnet (YIG) is a key component of the cavity magnomechanical system [23, 24], and it has high spin density and low dissipation rate [28, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The magnon is an embodiment of the collective excitation for spin waves inside the YIG, and it can strongly couple with the microwave cavity mode via magnetic dipole interaction [23, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' In addition, the magnon mode can also couple with the phonon mode by magnetostriction-induced deformation of YIG crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Recently, it has been demonstrated that the phonon mode can be successfully coupled to the optical cavity mode via radiation pressure [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' It provides a route for communication of entangled state between microwave field and optical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Due to the strong coupling characteristics of magnon [32, 33, 34], bipartite or tripartite entanglement in a cavity magnomechanical system is feasible [29, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' For example, Magnons can be used to generate magnomechanically induced transparency (MMIT) phenomena, which have been observed experimentally [28, 36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Lately, multi- channel MMIT phenomena have been realized by coupling magnon modes with multiple different physical subsystems [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' In 2020, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' proposed a scheme for entangling two microwave fields in a cavity magnomechanical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The entanglement of two microwave fields reaches up to EN =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='18, and the entanglement survives at an environmental temperature about 140 mK [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Since then, lots of approaches have been tried to improve the microwave entanglement EN and the robustness to temperature, but few results can perfectly satisfy the application expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Inspired by this, here we propose a scheme to generate high-efficiency stationary entanglement of two microwave fields in a dual opto-magnomechanical system which consists of two optical cavities and two microwave cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Two YIG crystals are embedded in each microwave cavity under an uniform bias magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' A microwave driving field is applied in the perpendicular direction to activate the phonon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Meanwhile, the magnons couple with the microwave fields via magnetic dipole interaction, and the phonons couple with the optical cavity modes via radiation pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' A two-mode squeezed optical field is injected into two optical cavities which makes two optical cavities quantum correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Each optical cavity is driven by a red-detuned laser field to activate the optomechanical beam splitter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Therefore, the entanglement between two optical cavities is transferred to two phonon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Simultaneously, two magnon modes are driven by strong red detuning microwave fields to activate magnomechanical beam splitter interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Then the entanglement of two phonon modes is further transferred to two magnon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Furthermore, two microwave cavities are entangled due to the magnon-microwave beam splitter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' As a consequence, quantum state transfer is realized from photons to phonons, then to magnons, and finally to microwaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The proposed scheme for preparing stationary entanglement of two microwave fields is high efficiency (Ea1a2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='54) and robustness to the environmental temperature (T=385 mK), and it will be useful for a variety of applications like quantum radar [40, 41], quantum teleportation [4, 5, 6], quantum network construction [7, 8], fundamental physics [9, 10], quantum wireless fidelity (Wi-Fi) network, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Theoretical Model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='1 shows the dual-cavity opto-magnomechanical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Each cavity opto- magnomechanical subsystem consists of a microwave cavity mode, a magnon mode, a phonon mode and an optical cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The magnon and phonon modes are constructed by YIG crystal (a 5 × 2 × 100 µm3 YIG cuboid) with a micro-bridge structure [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The YIG crystal is placed in the microwave cavity and in a bias magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The optical cavity consists of two highly reflective mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The right cavity mirror attached to the surface of a YIG micro-bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' An uniform bias magnetic field and a microwave field interact with the YIG crystal to activate the magnon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The microwave field is injected into the microwave cavity from the right side of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The magnon mode couples with the microwave cavity mode and the phonon mode through magnetic dipole interaction and nonlinear magnetostrictive interaction [34, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Furthermore, the phonon mode couples with the optical cavity mode through radiation pressure [31, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The strong red detuning microwave field and optical field are used to drive the magnon mode and optical cavity mode, respectively [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The red detuning field here can cool the mechanical motion and increase the magnomechanical (optomechanical) coupling strength, thus the magnon-phonon-photon state-swap interaction is activated [23, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The Hamiltonian for the dual-cavity opto-magnomechanical system can be written as [23, 42]: H/h= {wa,a, aj + Wm,m, mj + wc,cfcj + Wb,bfbj j=1,2 (1) + ga, (a mj +ajm) + gm,m,m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='(b +bj)+ gc,ccj(b + bj) CjeiwLit))Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Scheme diagram of dual-cavity opto-magnomechanical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' (a) Model diagram of the system to realize stationary entanglement of two microwave fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Two YIG micro-bridges are embedded in two microwave cavities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' High- reflection mirrors are attached to the left side of the YIG crystals, which are used to construct the optical cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' (b) Scheme diagram of entangled state transfer process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Red arrows denote state-swap interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' (c) The frequency diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' j = 1, 2 denote the first subsystem and the second subsystem, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The resonant frequency of the microwave field (optical cavity) is ωaj (ωcj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The magnon mode with frequency ωmj is driven by a microwave field with frequency ω0j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The optical cavity is driven by a two-mode squeezed optical field with frequency ωLj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Both the optical cavity modes and the magnon modes are driven by strong red detuning fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The mechanical motion triggered by two driving fields generates the Skotos sideband (ω0(L)j−ωbj) and the anti-Skotos sideband (ω0(L)j + ωbj) excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Stationary entanglement of two microwave fields can be realized when the optical cavity mode, the magnon mode and the microwave cavity modes in each opto-magnomechanical subsystems resonate with the anti-Skotos sidebands simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The first four terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' (1) describe the energies of four different cavity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' ja , j m , jc and jb ( ja\uf02b , j m\uf02b , jc\uf02b and jb\uf02b ) are the annihilation (creation) operators for the microwave cavity mode, the magnon mode, the optical cavity mode and the phonon mode, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' It satisfies [ j O , j O\uf02b ] = 1(O = a, m, c, b) and j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Oj \uf077 to different resonant frequencies of four cavity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The frequency of the magnon is mj j H \uf077 \uf067 \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' / 2 \uf067 \uf070 = 28GHz/T is the gyromagnetic ratio and Hj is the external bias magnetic field intensity [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The fifth, sixth and seventh terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' (1) denote the interaction terms of the magnon and microwave cavity modes, the magnon and Microwave cavity Opticalcavity (a) a1 m1 b1+ Microwave cavity Optical cavity a2 m2 (b) c WL1 Gbet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='0 ga mj CI b1 mi a1 WL2 W02 Gbea Gmb2 b m2 ga2 a2 0phonon modes, as well as the phonon and optical cavity modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' gaj is the magnon- microwave coupling strength, which is larger than the decay rate of the microwave cavity and magnon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=', gaj > κaj , κmj [33, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' κ aj (κmj ) is decay rate of the microwave cavity (magnon) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' gmj and gcj are the magnon-phonon coupling rate and phonon-optical cavity coupling rate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The eighth and ninth terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' (1) denote the driving fields for the magnon mode and optical cavity mode, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The Rabi frequency 5 4 j sj dj N H \uf067 \uf057 \uf03d denotes the coupling strength of the magnon mode, where sj j N V \uf072 \uf03d denotes the total spin number of the ferrimagnet and Hdj is the amplitude of the drive magnetic field [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' ρ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='22 × 1027 m−3 and Vj = 5 × 2 × 100 µm3 are the density and volume of the YIG cuboid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' \uf028 \uf029 2 / j cj Lj Lj E P \uf06b \uf077 \uf03d \uf068 is the coupling strength of the optical cavity, where κcj is the decay rate of the optical cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' PLj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='64 mW and ωLj are the power and frequency of the laser with wavelength of 1550 nm [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The quantum Langevin equation for different modes can be obtained by solving equation 1 , j j O O H i \uf0e9 \uf0f9 \uf03d \uf0eb \uf0fb \uf067 \uf068 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Introduce the input noise terms [44, 23], we get: where n n oj \uf077 \uf077 \uf044 \uf03d \uf02d ( , j j n a m \uf03d ), cj cj Lj \uf077 \uf077 \uf044 \uf03d \uf02d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' bj \uf067 is the the mechanical damping rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' in ja , in j m , in jb and in jc are the input noise operators corresponding to different modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The mean value of the input noise in j K (K = a, m, b) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The correlation functions of the input noises can be described by [45]: where 1 exp( 1) j Kj B K N T \uf077 \uf06b \uf02d \uf0e9 \uf0f9 \uf03d \uf02d \uf0ea \uf0fa \uf0eb \uf0fb \uf068 is mean thermal excitation number of each cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' kB is the Boltzmann constant and T is the bath temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' 1, the two-mode vacuum squeezed light field is injected into different optical cavities to make the noise operators of two optical cavities quantum correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The input noise correlation function for the two-mode squeezed light field can be described as [46, 47]: mj = -im,mj - Km,mi - iga,aj - igm,mi(b + bj) + 2j + 2hm,m, (2) bj = -iwb, - jbj - igm,m,mj +ige,c, cj + V2jbm Cj = -i△ce,Cj - Ke,Cj + igc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='Cj(b +b,) + E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' + V2ke,cinKin(t)Kin+(t)) = (Nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' + 1)s(t -t) (3) Km+(t)Kn(t)) = (Nk,)s(t-t)where N=sinh2r, M=sinhrcoshr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' r denotes the squeezing parameter of two-mode squeezed light field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Since the driving frequency and the resonant frequency of optical cavity are not equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=', , 0 cj ck \uf044 \uf044 \uf0b9 , phase factors related to cj \uf044 and ck \uf044 are not zero in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The strong microwave (optical) driving field can significantly strengthen the effective coupling rate of the magnon- phonon (phonon-optical cavity) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The magnon mode and optical mode have large amplitude 1 j m \uf03f and 1 jc \uf03f [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Considering the linearly coupling relation of the microwave cavity mode and the magnon mode, the large amplitudes 1 ja \uf03f is also obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Therefore, the dynamics of the system can be linearized and each operator can be described in the form of a fluctuation around a large average value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=', Oj = (Oj ) + δOj (O = a, m, b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Neglect the high- order fluctuation terms in the linearization process, the steady-state values of each mode can be obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' (2) as follows: Here \uf0b0 2 Re(b ) mj mj mj j g \uf044 \uf03d \uf044 \uf02b and \uf0b0 2 Re(b ) cj cj cj j g \uf044 \uf03d \uf044 \uf02b , and both of them contain a detuning term and a frequency shift term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The frequency shift terms are caused by mechanical displacements due to magnetostrictive interaction and radiation pressure interaction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' The frequency shift terms are usually small, \uf0b0 mj mj \uf044 \uf044 \uf03b , \uf0b0 cj cj \uf044 \uf044 \uf03b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' \uf0b0 \uf0b0 , , , , mj cj aj bj mj cj j \uf077 \uf06b \uf06b \uf067 \uf044 \uf044 \uf044 \uf03b \uf03f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content=' Therefore, the average value of each mode can be safely approximated as / j aj j aj a ig m i \uf02d \uf044 \uf03b , \uf0b0 \uf028 \uf029 2 / mj j j aj aj m i g \uf02d \uf057 \uf044 \uf02d \uf044 \uf044 \uf03b , 2 2 / j mj j cj j bj b ig m ig c i\uf077 \uf02d \uf02b \uf03b , / j j cj c E i\uf044 \uf03b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E0T4oBgHgl3EQf-AJM/content/2301.02808v1.pdf'} +page_content='The fluctuation terms of the quantum Langevin equations for the system can be further described as: ( - )o(I + ) = <(0)+u(0)u) (cin+(t)cgn(t)) =Ns(t -t), (j = 1,2) (4) (+ - 0)9(a%+0)*W = ((0) +0(0)+u0) ，(≠k=1,2)iga, 0 i.e for every bounded stopping time τ, +E (Xτ) ≤ E (Aτ) . Then, for every k ∈ (0, 1), +E +�� +sup +t≥0 +Xt +�k� +≤ 2 − k +1 − k E +� +Ak +∞ +� +Theorem 1.1. There exists two constants cp,g and Cp,g such that forall g-martingale Y +vanishing at zero; +cp,gE[⟨Y ⟩∞] ≤ E[Y ∗ +∞] ≤ Cp,gE[⟨Y ⟩∞] +Proof. By stopping it is enough to prove the result for bounded M. Let q ≥ 2. From +It¯o’s formula we have +d |Yt|q = q |Yt|q−1 sgn (Yt) dYt + 1 +2q(q − 1) |Yt|q−2 d⟨Y ⟩t += q sgn (Yt) |Yt|q−1 (−g(t, Yt, Zt)dt + ZtdBt) + 1 +2q(q − 1) |Yt|q−2 Z2 +t dt += −q sgn(Yt) |Yt|q−1 g(t, Yt, Zt)dt + 1 +2q(q − 1) |Yt|q−2 Z2 +t dt + q sgn (Yt) |Yt|q−1 ZtdBt +|Yt|q = +� t +0 +−q sgn(Ys) |Ys|q−1 g(t, Ys, Zs)dt + 1 +2q(q − 1) |Ys|q−2 Z2 +sds + +� t +0 +q sgn (Ys) |Ys|q−1 ZsdBs +E (|Yt|q | F0) ≤ E +�� t +0 +−q sgn(Ys) |Ys|q−1 g(t, Ys, Zs)ds + 1 +2q(q − 1) |Ys|q−2 Z2 +sds | F0 +� +E +�� t +0 +qµ |Ys|q−1 |Zs|ds + 1 +2q(q − 1) |Ys|q−2 Z2 +sds | F0 +� +2 + +From the Lenglart’s domination inequality, we deduce then that for every k ∈ (0, 1), +E + + +� +sup +0≤t≤T +|Yt|q +�k + ≤ 2 − k +1 − kE +�� T +0 +qµ |Ys|q−1 |Zs|ds + 1 +2q(q − 1) |Ys|q−2 Z2 +sds +�k +≤ 2 − k +1 − kE +�� T +0 +qµ |Ys|q−2 (δ2|Zs|2 + 1 +δ2 |Ys|2) + 1 +2q(q − 1) |Ys|q−2 Z2 +sds +�k += 2 − k +1 − kE +�� T +0 +qµ +δ2 |Ys|q + (1 +2q(q − 1) + qµδ2) |Ys|q−2 Z2 +sds +�k +≤ 2 − k +1 − kE +�� T +0 +qµ +δ2 |Ys|q ds +�k ++ 2 − k +1 − kE +�� T +0 +(1 +2q(q − 1) + qµδ2) |Ys|q−2 Z2 +sds +�k +≤ 2 − k +1 − k(qµ +δ2 T)kE + + +� +sup +0≤t≤T +|Yt|q +�k + ++ 2 − k +1 − k(1 +2q(q − 1) + qµδ2)kE +�� +sup +0≤t≤T +|Yt|k(q−2) +� �� T +0 +Z2 +sds +�k� +Therefore +(1 − 2 − k +1 − k(qµ +δ2 T)k)E + + +� +sup +0≤t≤T +|Yt|q +�k + ≤ 2 − k +1 − k(1 +2q(q − 1) + qµδ2)kE +�� +sup +0≤t≤T +|Yt|k(q−2) +� �� T +0 +Z2 +sds +�k� +By Holder inequality we obtain +(1 − 2 − k +1 − k(qµ +δ2 T)k)E +� +( sup +0≤t≤T +|Yt|)qk +� +≤ 2 − k +1 − k(1 +2q(q − 1) + qµδ2)k +� +E( sup +0≤t≤T +|Yt|)kq +�1− 2 +q + +E +�� T +0 +Z2 +sds +� kq +2 + + +q +2 +By by choosing δ large enough such that κ = 1 − 2−k +1−k(qµ +δ2 T)k > 0 and taking p = qk, we +obtain +E +� +( sup +0≤t≤T +|Yt|)p +� +≤ +2 − k +κ(1 − k)(1 +2q(q − 1) + qµδ2)k +� +E +� T +0 +Z2 +sds +� p +2 +We proceed now to the proof of the left hand side inequality. +For each integer n ⩾ 1, let us introduce the stopping time +τn = inf +� +t ∈ [0, T], +� t +0 +|Zr|2 dr ⩾ n +� +∧ T +Itˆo’s formula gives us +� τn +0 +|Zs|2 ds = |Yτn|2 + 2 +� τn +0 +Ysg (s, Ys, Zs) ds − 2 +� τn +0 +YsZs dBs +3 + +But, from the assumption on g, we have g (s, Y, z) ⩽ µ|z|, and so +2yg(s, y, z) ⩽ 2µ|y|2 + 1 +2|z|2 +Thus, since τn ⩽ T, we deduce that +1 +2 +� τn +0 +|Zs|2 ds ⩽ Y 2 +∗ + 2µTY 2 +∗ + 2 +���� +� τn +0 +YsZs dBs +���� . +It follows that +� τn +0 +|Zs|2 ds ⩽ (2 + 4µT)Y 2 +∗ + 4 +���� +� τn +0 +YsZs dBs +���� . +and thus that +�� τn +0 +|Zs|2 ds +�p/2 +⩽ kp +� +Y p +∗ + +���� +� τn +0 +sYsZs dBss +���� +p/2� +But by the BDG inequality, we get +cpE +����� +� τn +0 +⟨Yr, Zr dBr⟩ +���� +p/2� +⩽ dpE +��� τn +0 +|Yr|2 |Zr|2 dr +�p/4� +⩽ dpE +� +Y p/2 +∗ +�� τn +0 +|Zr|2 dr +�p/4� +cpE +����� +� τn +0 +⟨Yr, Zr dBr⟩ +���� +p/2� +⩽ d2 +p +2 E [Y p +∗ ] + 1 +2E +��� τn +0 +|Zr|2 dr +�p/2� +Coming back to estimate (5), we get, for each n ⩾ 1 +E +��� τn +0 +|Zr|2 dr +�p/2� +⩽ CpE +� +Y p +∗ + +�� T +0 +fr dr +�p� +and, Fatou’s lemma implies that +E +��� T +0 +|Zr|2 dr +�p/2� +⩽ CpE +� +Y p +∗ + +�� T +0 +fr dr +�p� +The result follows. +References +[1] Peng, S. 1997. Backward SDE and related g-expectations. In Backward Stochastic +Differential Equations; El Karoui, N., andMazliak, L., Eds. Pitman Research Notes +inMathematics Series, 364. London: Longman Scientific & Technical, 141–159. +[2] Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation, +Systems and Control Letters 14(1), 55–61 (1990) +[3] E. Lenglart, Relation de domination entre deux processus, Ann. Inst. H. Poincar´e +Sect. B (N.S.) 13 (1977), no. 2, 171–179. MR 0471069 (57 #10810) +4 + diff --git a/wNFQT4oBgHgl3EQfvjYz/content/tmp_files/load_file.txt b/wNFQT4oBgHgl3EQfvjYz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..13d065236bb5f0a53b19d36456bac29d714629b1 --- /dev/null +++ b/wNFQT4oBgHgl3EQfvjYz/content/tmp_files/load_file.txt @@ -0,0 +1,228 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf,len=227 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='13398v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='PR] 31 Jan 2023 Burkholder-Davis-Gundy Inequality for g- martingale Wahid Faidi ∗ e-mail: faidiwahid@su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='sa February 1, 2023 Abstract In this work we establish an BDG inequality type for certain nonlinear martingale arizing from BSDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' 1 The linear case We first recall the notion of g -expectations, defined in [1] from which most basic material of this section is taken .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' We are given a function g : g(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' z) : Ω × [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' T] × R × Rd �−→ R satisfying H: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 (i) g(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' z) ∈ L2 F(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' +∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' for each y ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' z ∈ Rd (ii) g(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' 0) ≡ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' for each y ∈ R (iii) there exist two positive non-random functionsvandu such that ∀y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' y2 ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' z2 ∈ Rd |g (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' z1) − g (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' z2)| ≤ u(t) |y1 − y2| + v(t) |z1 − z2| (iv) � +∞ 0 u(t)dt < ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' � +∞ 0 v2(t)dt < ∞ For each given X ∈ L2 (Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' P) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' let � yX(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' zX(·) � ∈ L2 F � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' R1 × Rd� be the unique solution of the following backward stochastic differential equation (BSDE): −dyX(t) = g � t, yX(t), zX(t) � dt − zX(t)dBt yX(T) = X (We refer to [2] for definitions and basic results about BSDEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' it will be enough here to remember that, provided that g satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1), there is a unique pair � yX(·), zX(·) � of adapted processes solving the equation above ) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Condition (ii) and (iii) implies ∀(t, y, z) ∈ R × R × Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' |g (t, y, z)| ≤ v(t) |z| indeed |g (t, y, z)| = |g (t, y, z) − g (t, 0, z)| u(t) |y − y| + v(t) |z − 0| = v(t) |z| ∗University Tunis El Manar 1 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' (g -expectation) The g -expectation Eg[·] : L2(Ω, F, P) �−→ R is defined by Eg[X] = yX(0) Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' (conditional g -expectation) The conditional g -expectation of X with respect to Ft is defined by Eg [X | Ft] = yX(t) If τ ≤ T is a stopping time, we define similarly Eg [X | Fτ] = yX(τ) g -expectations and conditional g -expectations are in general not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' However, they meet the following basic properties of usual expectations (see [1] for proofs): Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' A process (Yt)0≤t≤T such that E � Y 2 t � < ∞ for all t is a g-martingale (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' g -supermartingale, g -submartingale) iff Eg [Yt | Fs] = Ys, ( resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' ≤ Ys, ≥ Ys) , ∀s ≤ t ≤ T Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' (Lenglart) Let (Xt)t≥0 be a positive adapted right-continuous process dom- inated by a predictable increasing process (At)t>0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='e for every bounded stopping time τ, E (Xτ) ≤ E (Aτ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Then, for every k ∈ (0, 1), E �� sup t≥0 Xt �k� ≤ 2 − k 1 − k E � Ak ∞ � Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' There exists two constants cp,g and Cp,g such that forall g-martingale Y vanishing at zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' cp,gE[⟨Y ⟩∞] ≤ E[Y ∗ ∞] ≤ Cp,gE[⟨Y ⟩∞] Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' By stopping it is enough to prove the result for bounded M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Let q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' From It¯o’s formula we have d |Yt|q = q |Yt|q−1 sgn (Yt) dYt + 1 2q(q − 1) |Yt|q−2 d⟨Y ⟩t = q sgn (Yt) |Yt|q−1 (−g(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Yt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Zt)dt + ZtdBt) + 1 2q(q − 1) |Yt|q−2 Z2 t dt = −q sgn(Yt) |Yt|q−1 g(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Yt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Zt)dt + 1 2q(q − 1) |Yt|q−2 Z2 t dt + q sgn (Yt) |Yt|q−1 ZtdBt |Yt|q = � t 0 −q sgn(Ys) |Ys|q−1 g(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Ys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Zs)dt + 1 2q(q − 1) |Ys|q−2 Z2 sds + � t 0 q sgn (Ys) |Ys|q−1 ZsdBs E (|Yt|q | F0) ≤ E �� t 0 −q sgn(Ys) |Ys|q−1 g(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Ys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Zs)ds + 1 2q(q − 1) |Ys|q−2 Z2 sds | F0 � E �� t 0 qµ |Ys|q−1 |Zs|ds + 1 2q(q − 1) |Ys|q−2 Z2 sds | F0 � 2 From the Lenglart’s domination inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' we deduce then that for every k ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0≤t≤T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='|Yt|q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�k\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='\uf8f8 ≤ 2 − k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1 − kE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='qµ |Ys|q−1 |Zs|ds + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='2q(q − 1) |Ys|q−2 Z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='sds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='≤ 2 − k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1 − kE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='qµ |Ys|q−2 (δ2|Zs|2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='δ2 |Ys|2) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='2q(q − 1) |Ys|q−2 Z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='sds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='= 2 − k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1 − kE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='qµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='δ2 |Ys|q + (1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='2q(q − 1) + qµδ2) |Ys|q−2 Z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='sds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='≤ 2 − k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1 − kE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='qµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='δ2 |Ys|q ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='+ 2 − k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1 − kE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='(1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='2q(q − 1) + qµδ2) |Ys|q−2 Z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='sds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='≤ 2 − k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1 − k(qµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='δ2 T)kE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0≤t≤T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='|Yt|q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�k\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='\uf8f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='+ 2 − k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1 − k(1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='2q(q − 1) + qµδ2)kE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0≤t≤T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='|Yt|k(q−2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='� �� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='Z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='sds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�k� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='Therefore ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='(1 − 2 − k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1 − k(qµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='δ2 T)k)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0≤t≤T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='|Yt|q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�k\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='\uf8f8 ≤ 2 − k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1 − k(1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='2q(q − 1) + qµδ2)kE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0≤t≤T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='|Yt|k(q−2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='� �� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='Z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='sds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�k� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='By Holder inequality we obtain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='(1 − 2 − k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1 − k(qµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='δ2 T)k)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='( sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0≤t≤T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='|Yt|)qk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='≤ 2 − k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1 − k(1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='2q(q − 1) + qµδ2)k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='E( sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0≤t≤T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='|Yt|)kq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�1− 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='\uf8edE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='�� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='Z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='sds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='� kq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='\uf8f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='By by choosing δ large enough such that κ = 1 − 2−k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='1−k(qµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='δ2 T)k > 0 and taking p = qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' we obtain E � ( sup 0≤t≤T |Yt|)p � ≤ 2 − k κ(1 − k)(1 2q(q − 1) + qµδ2)k � E � T 0 Z2 sds � p 2 We proceed now to the proof of the left hand side inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' For each integer n ⩾ 1, let us introduce the stopping time τn = inf � t ∈ [0, T], � t 0 |Zr|2 dr ⩾ n � ∧ T Itˆo’s formula gives us � τn 0 |Zs|2 ds = |Yτn|2 + 2 � τn 0 Ysg (s, Ys, Zs) ds − 2 � τn 0 YsZs dBs 3 But, from the assumption on g, we have g (s, Y, z) ⩽ µ|z|, and so 2yg(s, y, z) ⩽ 2µ|y|2 + 1 2|z|2 Thus, since τn ⩽ T, we deduce that 1 2 � τn 0 |Zs|2 ds ⩽ Y 2 ∗ + 2µTY 2 ∗ + 2 ���� � τn 0 YsZs dBs ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' It follows that � τn 0 |Zs|2 ds ⩽ (2 + 4µT)Y 2 ∗ + 4 ���� � τn 0 YsZs dBs ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' and thus that �� τn 0 |Zs|2 ds �p/2 ⩽ kp � Y p ∗ + ���� � τn 0 sYsZs dBss ���� p/2� But by the BDG inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' we get cpE ����� � τn 0 ⟨Yr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Zr dBr⟩ ���� p/2� ⩽ dpE ��� τn 0 |Yr|2 |Zr|2 dr �p/4� ⩽ dpE � Y p/2 ∗ �� τn 0 |Zr|2 dr �p/4� cpE ����� � τn 0 ⟨Yr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Zr dBr⟩ ���� p/2� ⩽ d2 p 2 E [Y p ∗ ] + 1 2E ��� τn 0 |Zr|2 dr �p/2� Coming back to estimate (5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' we get,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' for each n ⩾ 1 E ��� τn 0 |Zr|2 dr �p/2� ⩽ CpE � Y p ∗ + �� T 0 fr dr �p� and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Fatou’s lemma implies that E ��� T 0 |Zr|2 dr �p/2� ⩽ CpE � Y p ∗ + �� T 0 fr dr �p� The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' References [1] Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Backward SDE and related g-expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' In Backward Stochastic Differential Equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' El Karoui, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=', andMazliak, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=', Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Pitman Research Notes inMathematics Series, 364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' London: Longman Scientific & Technical, 141–159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' [2] Pardoux, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=', Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=': Adapted solution of a backward stochastic differential equation, Systems and Control Letters 14(1), 55–61 (1990) [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Lenglart, Relation de domination entre deux processus, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' Poincar´e Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' B (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=') 13 (1977), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' 2, 171–179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} +page_content=' MR 0471069 (57 #10810) 4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFQT4oBgHgl3EQfvjYz/content/2301.13398v1.pdf'} diff --git a/wdFJT4oBgHgl3EQfgCxB/content/tmp_files/2301.11559v1.pdf.txt b/wdFJT4oBgHgl3EQfgCxB/content/tmp_files/2301.11559v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c281efbeecd4caef0bce1f59909a716a357bd74e --- /dev/null +++ b/wdFJT4oBgHgl3EQfgCxB/content/tmp_files/2301.11559v1.pdf.txt @@ -0,0 +1,1057 @@ +Enabling Multi-threading in Heterogeneous +Quantum-Classical Programming Models +Akihiro Hayashi∗ Austin Adams∗ Jeffrey Young∗ Alexander McCaskey† Eugene Dumitrescu‡ +Vivek Sarkar∗ Thomas M. Conte∗ +∗Georgia Institute of Technology †NVIDIA Corporation ‡Oak Ridge National Laboratory +Email: {ahayashi,aja,jyoung9,vsarkar,conte}@gatech.edu, amccaskey@nvidia.com, dumitrescuef@ornl.gov +Abstract—While quantum computers enable significant perfor- +mance improvements for certain classes of applications, building +a well-defined programming model has been a pressing issue. In +this paper, we address some of the key limitations to realizing a +generic heterogeneous parallel programming model for quantum- +classical heterogeneous platforms. We discuss our experience +in enabling user-level multi-threading in QCOR [1] as well as +challenges that need to be addressed for programming future +quantum-classical systems. +Specifically, we discuss our design and implementation of +introducing C++-based parallel constructs to enable 1) parallel +execution of a quantum kernel with std::thread and 2) +asynchronous execution with std::async. To do so, we provide +a detailed overview of the current implementation of the QCOR +programming model and runtime, and discuss how we add 1) +thread-safety to some of its user-facing API routines, and 2) +increase parallelism in QCOR by removing data races that +inhibit multi-threading so as to better utilize available computing +resources. +We also present preliminary performance results with the +Quantum++ [2] back end on a single-node Ryzen9 3900X machine +that has 12 physical cores (24 hardware threads) with 128GB of +RAM. The results show that running two Bell kernels with 12 +threads per kernel in parallel outperforms running the kernels +one after the other each with 24 threads (1.63× improvement). In +addition, we observe the same trend when running two Shor’s +algorthm kernels in parallel (1.22× faster than executing the +kernels one after the other). It is worth noting that the trends +remain the same even when we only use physical cores instead +of threads. +We believe that our design, implementation, and results will +open up an opportunity not only for 1) enabling quicker prototyp- +ing of parallel/asynchrony-aware quantum-classical algorithms +on quantum circuit simulators in the short-term, but also for 2) +realizing a generic heterogeneous parallel programming model +for quantum-classical heterogeneous platforms in the long-term. +Index Terms—Quantum-Classical Programming Models, Par- +allel Programming Models, QCOR, Heterogeneous Computing +I. INTRODUCTION +Quantum computing is a rapidly evolving field that lever- +ages the laws of quantum mechanics for computation. Since +near-term quantum computers are susceptible to significant +levels of noise, a hybrid combination of classical computers +and quantum computers, namely quantum-classical comput- +ers, is explored to mitigate noise while achieving orders-of- +magnitude performance improvements for certain classes of +applications. Such a hybrid combination can be viewed as one +Fig. 1: QCOR Machine Model [3] +realization of heterogeneous computing where different types +of processing elements, including special purpose accelerators, +simultaneously and asynchronously work together. +QCOR [1] is a programming system to realize such a +heterogeneous quantum-classical model. It is based on the +C++-based programming language and a compiler that is +built on top of XACC [4] As shown in Figure 1, QCOR’s +target machine is a heterogeneous system where multiple +CPUs (cores) are connected with quantum devices and other +accelerators such as GPUs and FPGAs. +To program quantum devices in QCOR, the user writes +a quantum kernel (i.e., a function that will be executed on +a quantum device) in quantum computing domain-specific +languages (DSLs), such as XACC’s XASM or IBM’s Open- +QASM [5]. Similar to other GPU-based heterogeneous pro- +gramming models such as CUDA [6], SYCL [7], and +OpenCL [8], QCOR allows the user to write quantum kernels +and CPU control code in the same program. This single-source +programming model greatly facilitates quantum-classical pro- +gramming. +However, one open research question for QCOR and +other quantum DSLs is how to provide well-defined, user- +level multi-threading support. Specifically, as the machine +model in Figure 1 implies, it is possible that multiple CPU +cores might simultaneously utilize one or more quantum de- +vices. Currently, there is no user-facing API-level support for +multi-threading in quantum-classical programming models like +QCOR and DSLs like OpenQASM, although it is typical to +internally use multi-threading for accelerating quantum circuit +arXiv:2301.11559v1 [quant-ph] 27 Jan 2023 + +GPU +FPGA +Instruction +Channel +APU +Quantum +Device +Quantum State +Measurement +Quantum +Output Buffer +Memory SystemAlgorithm 1 Shor’s Algorithm (Pseudocode) +Input: N: A natural number to be factorized. +Output: A non-trivial divisor(s) of N. +1: procedure MAIN(N) +2: +repeat +3: +a ← random(1, N); +▷ 1 < a < N +4: +K ← gcd(a, N); +5: +if K == 1 then +6: +SHOR(N, a); +7: +else +8: +return K +9: +until a divisor(s) is found or explored all +10: procedure SHOR(N, a) +11: +for s = 1, ..., nShots do +12: +rs ← SHORKERNEL(N, a) +13: +r ← r1, ..., rs +▷ Estimate r from the measurements +14: +if r mod 2 ≡ 1 or ar mod N ≡ −1 then +15: +return φ; +16: +else +17: +return gcd(ar/2 ± 1, N); +simulations [2], [9]–[11]. +In this paper, we explore the possibility of enabling user- +level multi-threading in QCOR, which enables coarser grain +parallelism in quantum-classical programming models. We be- +lieve this is an important step towards realizing an end-to-end +heterogeneous programming system that can work on general +heterogeneous platforms that include quantum computers. This +work makes the following key contributions: +• Design and implementation of multi-threading support for +a heterogeneous quantum-classical programming model. +• Discussion of scenarios and use cases where user-level +multi-threading is beneficial for near-term quantum sys- +tems. +• A demonstration which shows that running two quantum +kernels in parallel using N/2-threads for each kernel +outperforms running the kernel one-by-one using N- +threads, by factors of 1.22× to 1.63× for the evaluated +kernels. +II. MOTIVATION +This section highlights our motivation for enabling user- +level multi-threading in quantum-classical computing by dis- +cussing potential parallelism in quantum-classical programs. +Let us use Shor’s algorithm as a motivating example. In +Algorithm 1, SHOR is a quantum-classical task that invokes +the period-finding quantum kernel (SHORKERNEL) to estimate +exponent r. Notice that SHOR can be called multiple times +until one or more (non-)trivial divisors are found or the entire +search space is explored. +From the perspective of parallel processing, one possibility +of parallelizing this algorithm is to run multiple instances of +SHOR in parallel. Furthermore, since it can require multiple +shots to find r, it would be also possible to further parallelize +Algorithm 2 Parallel Shor’s Algorithm (Pseudocode) +Input: N: A natural number to be factorized. +Output: A non-trivial divisor(s) of N. +1: procedure MAIN(N) +2: +repeat +3: +a ← random(1, N); +▷ 1 < a < N +4: +K ← gcd(a, N); +5: +if K == 1 then +6: +async SHOR(N, a); +7: +else +8: +return K +9: +until a divisor(s) is found or explored all +10: procedure SHOR(N, a) +11: +foreach s = 1, ..., nShots do +12: +rs ← SHORKERNEL(N, a) +13: +r ← r1, ..., rs +▷ Estimate r from the measurements +14: +if r mod 2 ≡ 1 or ar mod N ≡ −1 then +15: +return φ; +16: +else +17: +return gcd(ar/2 ± 1, N); +the shot loop in SHOR (Line 11). Finally, if the SHORKERNEL +is executed on a simulator, there is a massive amount of +parallelism as in [2], [9]–[11]. Algorithm 2 is a pseudo-parallel +version of Algorithm 1. As in the X10 language [12], async +represents parallel task creation and execution and foreach +represents parallel loop creation and execution. +Figure 2 graphically illustrates the potential parallelism in +Shor’s algorithm across these three levels. Based on what +we discussed for Algorithm 2 and observe in Figure 2, +we identify the following multiple levels of parallelism in +quantum-classical programs: +Task level parallelism: multiple independent classical tasks +that can include quantum kernels are executed in parallel. +Shot level parallelism: multiple independent shots are exe- +cuted in parallel. +Inner simulator level parallelism: quantum simulators, in- +cluding state vector and tensor network simulators such as [2], +[9]–[11], are typically parallelized using OpenMP, CUDA, and +the Eigen library to utilize a massive amount of parallelism +on CPUs and/or GPUs. +It is worth noting that the actual amount of available +parallelism depends not only on algorithms but also on the +simulated or physical quantum back ends that are targeted. +One example would be when a user executes their program +on a current-day single QPU system in which there would +be limited parallelism due to the lack of additional physical +hardware. However, in most cases, we believe that allowing the +user to specify all available parallelism for a quantum-classical +task will greatly enhance the performance and expressiveness +of quantum-classical programs because there are plenty of +computing resources (CPUs, GPUs, and FPGAs) that can +accelerate the development of quantum-classical algorithms +even on conventional systems. + +Task1: +Shor (N = 15, a = 2); +7! ≡ 1 𝑚𝑜𝑑 15 +gcd 7" − 1, 15 = 3 +gcd 7" + 1, 15 = 5 +Found! +Task level parallelism +Shot level parallelism +Attempt 1: +r = x? +Quantum +Device +Attempt 2: +r = y? +Quantum +Device +Task2: +Shor (N = 15, a = 4); +Attempt 1: +r = z? +Quantum +Device +Attempt 2: +r = w? +Quantum +Device +Task3: +Shor (N = 15, a = 7); +Attempt 1: +r = 4? +Quantum +Device +Inner simulator level parallelism +(when a simulator is used) +Fig. 2: Multi-level parallelism in a quantum-classical program (Shor’s algorithm). +Listing 1: A 2-qubit Bell kernel implementation in QCOR +1 using namespace std; +2 // the Bell kernel +3 __qpu__ void bell(qreg q) { +4 +using qcor::xasm; +5 +H(q[0]); +6 +CX(q[0], q[1]); +7 +for (int i = 0; i < q.size(); i++) { +8 +Measure(q[i]); +9 +} +10 } +11 int main(int argc, char **argv) { +12 +// Create two qubit registers, each size 2 +13 +auto q = qalloc(2); +14 +// Run the quantum kernel +15 +bell(q); +16 +// dump the results +17 +q.print(); +18 } +Listing 2: An example output of the Bell kernel (1024 shots) +1 "AcceleratorBuffer": { +2 +"name": "qrg_bmQBh", +3 +"size": 2, +4 +"Information": {}, +5 +"Measurements": { +6 +"00": 513, +7 +"11": 511 +8 +} +9 } +Thus, we believe that enabling user-level multi-threading +in quantum-classical programming models will 1) accelerate +the development of a quantum-classical algorithm, and 2) +facilitate porting an existing heterogeneous algorithm to a +quantum-classical one. It is also worth noting that the goal of +this work is not optimizing and fine-tuning quantum-classical +parallel programs for a specific target system. Instead, we +look to motivate and introduce concrete parallel programming +constructs (std::thread and std::async) for quantum- +classical programming models. +III. QCOR +QCOR is a C++-based high-level quantum-classical pro- +gramming model. One of the key features of QCOR is that +Listing 3: A VQE implementation in QCOR +1 __qpu__ void ansatz(qreg q, double theta) { +2 +X(q[0]); +3 +Ry(q[1], theta); +4 +CX(q[1], q[0]); +5 } +6 +7 int main(int argc, char **argv) { +8 +// Allocate 2 qubits +9 +auto q = qalloc(2); +10 +11 +// Programmer needs to set +12 +// the number of variational params +13 +auto n_variational_params = 1; +14 +15 +// Create the Deuteron Hamiltonian +16 +auto H = 5.907 - 2.1433 * X(0) * X(1) - +17 +2.1433 * Y(0) * Y(1) + .21829 * Z(0) - +18 +6.125 * Z(1); +19 +20 +// Create the ObjectiveFunction +21 +auto obj = createObjectiveFunction(ansatz, H, q, +22 +n_variational_params, +23 +{{"gradient-strategy", +24 +"central"}, +25 +{"step", 1e-3}}); +26 +27 +// Create the Optimizer. +28 +auto opt = createOptimizer("nlopt", +29 +{{"nlopt-optimizer", +30 +"l-bfgs"}}); +31 +// Optimize +32 +auto [opt_val, opt_params] = opt->optimize(objective); +33 +std::cout << opt_val << std::endl; +34 } +the user can write both quantum and classical kernels and +functions in the same code. This feature is not only anal- +ogous to existing heterogeneous programming models such +as CUDA, OpenCL, and SYCL, but it also also provides a +new programming model for heterogeneous quantum-classical +computing programs that achieve hybrid quantum-classical +workflows. As shown in the machine model in Figure 1, in +theory, the user is free to leverage different kinds of processors +(e.g., CPUs, GPUs, FPGAs, Quantum Devices) that could all +be enabled through a QCOR-style programming model. +Listing 1 shows an example of QCOR program that executes +the Bell kernel. First, on Line 13, the qalloc API is called to +allocate 2-qubits. Then, the kernel written in XASM is invoked + +Listing +4: +Simultaneously +Launching +two +Bell +kernels +(std::thread) +1 using namespace std; +2 // the bell kernel +3 __qpu__ void bell(qreg q) { +4 +using qcor::xasm; +5 +H(q[0]); +6 +CX(q[0], q[1]); +7 +for (int i = 0; i < q.size(); i++) { +8 +Measure(q[i]); +9 +} +10 } +11 void foo() { +12 +// Create two qubit registers, each size 2 +13 +auto q = qalloc(2); +14 +// Run the quantum kernel +15 +bell(q); +16 +// dump the results +17 +q.print(); +18 } +19 int main(int argc, char **argv) { +20 +thread t0(foo); thread t1(foo); +21 +// Other classical/quantum work +22 +... +23 +t0.join(); t1.join(); +24 } +on Line 15. Notice that the kernel is defined on Line 3 - 10. +After the kernel is invoked, the measurement results can be +inspected by printing the content of the quantum register as +shown on Line 17. An example output of the QCOR program +can be found in Listing 2. +In addition to the simple quantum circuit simulation above, +for completeness, we would like to emphasize that QCOR +is expressive enough to write a wide variety of quantum- +classical algorithms such as the variational quantum eigen- +solver (VQE) and the Quantum Approximate Optimization +Algorithm (QAOA). Listing 3 shows a VQE implementation +in QCOR. Note that createObjectiveFunction and +createOptimizer are built-in QCOR helper functions that +facilitate creating and invoking a classical optimizer with a +user-defined objective function with the Deuteron Hamiltonian +and the ansatz kernel. More details can be found in [1], [3]. +IV. DESIGN +A. Multi-threading Design Overview +Since QCOR is primarily written in C++, we look to +enable user-level multi-threading in QCOR in a way that +is acceptable to both QCOR and C++ programmers. For +QCOR programmers, our goal is to minimize modifications +to the code required for enabling multi-threading. For C++ +programmers, our goal is to provide a threading interface that +is natural to use. To that end, we leverage C++’s standard +threading constructs (std::thread and std::async). +However, in terms of general applicability, our discussions +should apply to other parallel programming systems for C++, +such as OpenMP [13], Kokkos [14], and RAJA [15]. +Our current focus is on enabling coarse-grain parallelism +to exploit the full capability of a CPU-QPU system. In one +scenario, the user would like a one-to-one relation between +a CPU and a QPU to simultaneously perform N independent +Listing +5: +Asynchronously +Launching +the +Bell +kernel +(std::async) +1 using namespace std; +2 int main(int argc, char **argv) { +3 +std::future f = async(launch::async, +4 +[=]() -> int { foo(); return 1; }); +5 +// Other classical/quantum work +6 +... +7 +// +8 +f.get(); +9 } +tasks, where N is the number of CPU-QPU pairs. Another sce- +nario might be a one-to-many/many-to-one relation between +CPU(s) and QPU(s). It is worth noting that the QPU part is +not necessarily a hardware QPU device. Since QCOR offers +different backends, the QPU part can be a quantum circuit +simulation on either a local machine or a cloud service and +can also incorporate coarser tasks such as VQE. +B. User-Facing API +1) std::thread: Listing 4 shows an example where two +threads simultaneously run the Bell kernel using thread. +The main function creates two threads (t0 and t1), each +of which executes the foo function. In the foo function, it +first allocates 2-qubits using qalloc, then invokes the kernel +written in XASM in Line 3 - 10, and finally gets the results. +This approach enables the user to overlap other work on the +main thread with the two threads. Also, the main function can +wait on each thread by calling join(). +2) std::async: Another example (Listing 5) is asynchronous +execution where the main function asynchronously launches +the foo() function with async. Similar to the thread +example, the user may want to overlap other work with the +launched task. However, one interesting difference is that +async returns a future object, which helps the user to +check the status of the asynchronously launched task and +take further action depending on the return value of the task +(get()). +C. Enabling Thread Safety +Thread safety is usually attributed to a function/routine that +can be safely invoked by multiple threads simultaneously. It is +very common that thread safety is guaranteed in conventional +heterogeneous programming models such as CUDA, OpenCL, +and SYCL. For example, the SYCL specification [16] de- +scribes this in the following manner: “SYCL guarantees that +all the member functions and special member functions of the +SYCL classes described are thread safe.” +It is worth noting that enabling thread safety does not +necessarily mean improving performance because it essen- +tially prevents multiple threads from simultaneously accessing +shared data. In this work, our first priority is to enable thread +safety for QCOR’s user-facing API. For portions where paral- +lelization is important, we explore the possibility of increasing +parallelism in Section V. + +Listing 6: Making qalloc() thread-safe with Mutex Lock +1 mutex m; +2 qbit qalloc(const int n) { +3 +lock_guard lock(m); +4 +... +5 +allocated_buffers.insert({...}); +6 +... +7 } +Listing 7: How a QPU instance is declared and created +1 +namespace xacc { +2 +namespace internal_compiler { +3 +// global variable +4 +std::shared_ptr qpu = nullptr; +5 +... +6 }} +7 // Getting an instance of qpp +8 qpu = xacc::getAccelerator("qpp"); +V. IMPLEMENTATION +This section discusses how we enable user-level multi- +threading in QCOR and XACC. +Since the QCOR and XACC systems include over 200K +lines of code written in modern C++, we focus on discussing +a few common cases that can possibly inhibit user-level multi- +threading. Essentially, these cases are focused on identifying +potential sources of data races when multi-threading is added. +A. Identifying sources of data races +1) Global Variables: Global variables are the most common +source of data races because these variables can be accessed +simultaneously by multiple threads. The following is a global +std::map object that is used to implement qalloc(). +// global variable +map> +allocated_buffers{}; +Because qalloc() internally invokes map’s insert(), +which is not thread-safe, concurrent invocations of qalloc() +can be problematic. +2) Services: QCOR depends on different software com- +ponents provided by QCOR itself and XACC. Typically, +xacc::getService(...) is used to obtain a shared +pointer to a specific service, namely T in this example. +For services that do not derive xacc::Cloneable, the +xacc::getService(...) always returns a pointer +to the same instance, which can be another source of a data +race. The following is an example where a pointer to the +qpp accelerator, a software simulator in QCOR/XACC (i.e., +Quamtum++ [2]), which is used to run the Bell kernel in +Listing 4 and Listing 5, is stored into acc. +shared_ptr acc; // a local variable +acc = xacc::getService("qpp", ...); +Because +Accelerator +is +not +Cloneable, +getService(...) +always +returns +the same qpp instance. This can cause a data collision since +multiple threads can simultaneously register their gates to the +Listing 8: QPU Manager Implementation (Simplified) +1 using namespace std; +2 class QPUManager { +3 +public: +4 +static QPUManager& getInstance() { +5 +static QPUManager instance; return instance; +6 +} +7 +private: +8 +QPUManager() {} +9 +map> qpu_map; +10 +public: +11 +shared_ptr getQPU(); +12 +void setQPU(std::shared_ptr _qpu); +13 }; +same accelerator and can thus end up simulating an erroneous +circuit. +B. Implementation Details +In general, we pursue the following two approaches to +remove data races that inhibit multi-threading in QCOR and +XACC: 1) enabling thread safety and 2) increasing parallelism. +The former goal is achieved by adding safety to multi-threaded +execution with mutex locks. The latter approach explores the +possibility of leveraging multi-threading to accelerate user +programs. +1) Enabling thread-safety: For enabling thread-safety, we +leverage std::mutex or std::recursive_mutex to +enable mutual exclusions. For example, Listing 6 shows +qalloc(), which has a non-thread-safe call in Line 5. +We first create a mutex object in the global scope, and +then the object is used to create a critical section with +std::lock_guard. +2) Increasing Parallelism: For increasing parallelism, we +use a quantum accelerator object (qpu) as a motivating +example. In the original implementation, as shown in Listing 7, +the qpu object is declared as a global variable and is initialized +by calling xacc::getAccelerator(), which internally +calls xacc::getService(). Thus, this +example includes the two data race scenarios discussed above +in Section V-A. +We remove the data races by i) making Accelerator +cloneable +to +create +different +instances +every +time +xacc::getAccelerator() is called, and ii) providing +a map that maps a current thread ID to the corresponding +accelerator object, the latter of which is called QPUManager. +Listing 8 shows a brief overview of QPUManager. +QPUManager is implemented by using the singleton pattern +and contains the setter and getter functions. The setter function +takes the return variable of xacc::getAccelerator() +and registers the accelerator instance along with a current +thread id to the map. Similarly, the getter function returns +a qpu instance that corresponds to a current thread. +C. Current Implementation Status +We have implemented these changes to enable thread- +safety for QCOR and have created a pull request against the +QCOR [17], QCOR SPEC [18], and XACC [19] repositories. + +1.00 +0.96 +1.30 +1.63 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +12 threads +24 threads +2 x (6 +threads/task) +2 x (12 +threads/task) +One-by-One (Conventional) +Parallel (Our approach) +Speed up over 12 threads +2 Bell kernels +Fig. 3: Bell Kernel +For increasing multi-threaded parallelism, we have confirmed +that the examples (Listing 4 and Listing 5) and Shor’s kernel +work in a parallel fashion, and we plan to create another pull +request to share that functionality. +One small limitation of our implementation is that the user +needs to manually call quantum::initialize() API at +the beginning of each thread so the runtime can register its +thread ID to the QPUManager. In the future, we plan to +create a compiler pass that automatically inserts this API +call. Alternatively, we could provide qcor::thread and +qcor::async wrappers for the original C++ constructs that +internally call this initialization function. +VI. PRELIMINARY PERFORMANCE EVALUATION +This section presents the results of an empirical evaluation +of our extended QCOR programming model and runtime +implementation on a single-node platform to demonstrate its +performance benefits. +Purposes: The goal of our evaluation is two-fold: +1) to demonstrate that our extended QCOR programming +model and runtime system with C++ threading model +enables parallel quantum kernel execution. +2) to demonstrate that enabling parallel quantum kernel +execution is beneficial in terms of performance. +Platform: We present the performance results on a single-node +AMD server, which consists of a 12-core, 24-thread Ryzen9 +3900X CPu running at 3.8GHz with 128GB of DRAM. +Quantum Kernels: +We use the following quantum kernels written in XASM: +1) Bell Kernel: The 2-qubit Bell kernel shown in Listing 4. +The number of shots is 1024. +2) Shor’s Kernel: The period-finding quantum kernel, +which is based on [20]. The number of shots is 10 and +we run SHOR(N=15, a=2) and SHOR(N=15, a=7) as in +Algorithm 1. +Experimental variants: For kernel simulations, we use the +QppAccelerator back end in QCOR, which uses the +Quantum++ library [2]. +We compare the following two variants in terms of perfor- +mance: +1.00 +1.02 +1.20 +1.22 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +12 threads +24 threads +2 x (6 +threads/task) +2 x (12 +threads/task) +One-by-One (Conventional) +Parallel (Our approach) +Speed up over 12 threads +2 Shor's kernels +Fig. 4: Shor’s Kernel +1) One-by-One (baseline, conventional): Run the first +kernel with N-threads and then run the second kernel +with N-threads. +2) Parallel: Run the two kernels in parallel, each of which +uses N/2-threads. +Note that each kernel is executed on multiple physical +cores/threads even in the baseline version because Quantum++ +uses OpenMP [13]. For both variants, we appropriately set +the OMP_NUM_THREADS parameter to specify the number of +threads per kernel. However, tuning this parameter for the best +performance is beyond the scope of this paper. Instead, our +goal is to study scenarios where running multiple quantum +kernels simultaneously could lead to performance benefits. +Finally, note that shot-level parallelism is not exploited in these +versions. +A. Bell kernel +Figure 3 shows relative performance improvements over +the baseline execution (one-by-one execution with 12-threads). +In one-by-one execution, increasing the number of threads +degrades performance because the kernel is too small to fully +utilize all the threads/cores. In contrast, parallel execution of +the two kernels enables a further performance improvement of +up to 1.63×. +B. Shor’s kernel +Figure 4 shows relative performance improvements over +the baseline execution (one-by-one execution with 12-threads). +Similar to the Bell kernel results, parallel execution enables +a further performance improvement of up to 1.22× over the +baseline execution. +VII. DISCUSSION +As shown in Section VI, we demonstrated a scenario where +running multiple kernels simultaneously is beneficial. The +goal of this section is to summarize difference application +scenarios that we believe are good candidates for user-level +multi-threading: +Shor’s algorithm: As we discussed in Section II, suppose we +factorize N using Shor’s algorithm, we can create p parallel + +tasks with a random number ap s.t. 1 < ap < N and +gcd(ap, N) = 1, each of which invokes Shor’s kernel to +estimate rp and checks if rp is even and arp mod N ≡ 1 in +parallel. Algorithm 2 summarizes the parallel algorithm and +Figure 4 shows that running two Shor’s kernels in parallel +outperforms one-by-one execution. We anticipate that the +performance improvement will be more significant if CPUs +with more cores and GPUs are used for simulating Shor’s +circuit. +VQE: VQE [21] optimizes a (Hamiltonian H) cost function +over a parameterized manifold of quantum states |ψ(⃗θ)⟩ = +U(⃗θ)|ψ0⟩ as min +⃗θ +⟨ψ(⃗θ)|H|ψ(⃗θ)⟩. For QMA-hard Hamiltoni- +ans, dim(⃗θ) is large but for many interesting models in +physical sciences dim(⃗θ) may scale (sub-)polynomially, in +which case the optimization problem at hand may still be quite +challenging. The pleasantly parallel nature of the optimization +process can be utilized with multiple asynchronous quantum +kernel instances minimizing over ⃗θ-space. +Asynchronous Quantum JIT Compilation: Shi et al. [22] +discusses a scenario where a GPU is used to compile and +optimize quantum circuits, which can take several hours. +With user-level multi-threading enabled, it is possible to avoid +blocking computing resources by asynchronously offloading +a compilation task onto a GPU and launching the compiled +kernel on a QPU only when it is ready. +Parallel Quantum-Classical Workflow: As generalizations +of different parallel execution scenarios discussed above, one +can write an entire workflow in which different tasks run on +different processing units including CPUs, QPUs, GPUs, and +FPGAs. +VIII. RELATED WORK +While domain-specific languages (DSLs) for quantum com- +puting significantly facilitate the development of quantum al- +gorithms, many DSLs only focus on the kernel part and do not +provide a system-wide programming model. We believe that +such a system-wide programming model will become more +important in quantum-classical computing because exploiting +classical parallelism such as thread-level parallelism can im- +prove end-to-end performance as discussed in Section VI. +Here, we briefly discuss existing programming models from +the viewpoint of classical parallelism on non-quantum devices. +Qiskit [23] has been one of the most popular programming +frameworks for quantum computing. However, it is not appro- +priate to directly map Qiskit programs to quantum-classical +systems unless there is an AOT/JIT-level smart compiler that +is aware of the underlying parallel hardware because the +Global Interpreter Lock (GIL) may hinder Python-level multi- +threaded execution. +Q# is a programming language designed to express hybrid +quantum-classical algorithms [24]. Currently, there is no way +to express the concept of threads in the Q# language itself +[25], nor in the Q# standard library [26]. Additionally, QIR +(Quantum Intermediate Representation), a hybrid quantum- +classical IR based on LLVM IR that is generated by the Q# +compiler front-end, does not explicitly guarantee thread-safety +for any runtime functions [27]. Indeed, the reference QIR +runtime [28] may exhibit data races if used in multi-threaded +code. It is worth noting that QParallel [29] allows the user to +explicitly express parallelism in the quantum kernel part, not +in the classical part. +Other newer platforms for hybrid quantum-classical com- +puting have been proposed like NVIDIA’s QODA [30], which +is designed for the simulation of quantum circuits with GPUs +and QPUs. It is unclear what multi-threaded support model +QODA uses as it is a proprietary product. +IX. CONCLUSIONS AND FUTURE WORK +This paper explores the possibility of enabling user-level +multi-threading in QCOR. We made enhancements to QCOR +to support C++-based parallel and asynchronous execution of +quantum kernels by 1) adding thread safety to QCOR API +routines, and 2) increase parallelism by removing data races +that inhibit multi-threading. +Our preliminary results with the Bell and Shor’s algorithm +kernels show that enabling user-level multi-threading gives +us performance improvements over the conventional baseline +version in which each kernel is still executed by multiple +threads, but is executed one-by-one. +We believe this multi-threading design for heterogeneous +quantum-classical programming models will open up an op- +portunity for rapidly prototyping and developing quantum- +classical programs on conventional systems in the short- +term. 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Available: https://arxiv.org/abs/2210.03680 +[30] NVIDIA, “NVIDIA QODA: The Platform for Hybrid Quantum- +Classical Computing,” https://developer.nvidia.com/qoda, 2022. + diff --git a/wdFJT4oBgHgl3EQfgCxB/content/tmp_files/load_file.txt b/wdFJT4oBgHgl3EQfgCxB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6aef4f9316e7f66e9793f0e94ce175a3d8abd22 --- /dev/null +++ b/wdFJT4oBgHgl3EQfgCxB/content/tmp_files/load_file.txt @@ -0,0 +1,650 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf,len=649 +page_content='Enabling Multi-threading in Heterogeneous Quantum-Classical Programming Models Akihiro Hayashi∗ Austin Adams∗ Jeffrey Young∗ Alexander McCaskey† Eugene Dumitrescu‡ Vivek Sarkar∗ Thomas M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Conte∗ ∗Georgia Institute of Technology †NVIDIA Corporation ‡Oak Ridge National Laboratory Email: {ahayashi,aja,jyoung9,vsarkar,conte}@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='edu, amccaskey@nvidia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='com, dumitrescuef@ornl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='gov Abstract—While quantum computers enable significant perfor- mance improvements for certain classes of applications, building a well-defined programming model has been a pressing issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' In this paper, we address some of the key limitations to realizing a generic heterogeneous parallel programming model for quantum- classical heterogeneous platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' We discuss our experience in enabling user-level multi-threading in QCOR [1] as well as challenges that need to be addressed for programming future quantum-classical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Specifically, we discuss our design and implementation of introducing C++-based parallel constructs to enable 1) parallel execution of a quantum kernel with std::thread and 2) asynchronous execution with std::async.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' To do so, we provide a detailed overview of the current implementation of the QCOR programming model and runtime, and discuss how we add 1) thread-safety to some of its user-facing API routines, and 2) increase parallelism in QCOR by removing data races that inhibit multi-threading so as to better utilize available computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' We also present preliminary performance results with the Quantum++ [2] back end on a single-node Ryzen9 3900X machine that has 12 physical cores (24 hardware threads) with 128GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' The results show that running two Bell kernels with 12 threads per kernel in parallel outperforms running the kernels one after the other each with 24 threads (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='63× improvement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' In addition, we observe the same trend when running two Shor’s algorthm kernels in parallel (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='22× faster than executing the kernels one after the other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' It is worth noting that the trends remain the same even when we only use physical cores instead of threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' We believe that our design, implementation, and results will open up an opportunity not only for 1) enabling quicker prototyp- ing of parallel/asynchrony-aware quantum-classical algorithms on quantum circuit simulators in the short-term, but also for 2) realizing a generic heterogeneous parallel programming model for quantum-classical heterogeneous platforms in the long-term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Index Terms—Quantum-Classical Programming Models, Par- allel Programming Models, QCOR, Heterogeneous Computing I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' INTRODUCTION Quantum computing is a rapidly evolving field that lever- ages the laws of quantum mechanics for computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Since near-term quantum computers are susceptible to significant levels of noise, a hybrid combination of classical computers and quantum computers, namely quantum-classical comput- ers, is explored to mitigate noise while achieving orders-of- magnitude performance improvements for certain classes of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Such a hybrid combination can be viewed as one Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 1: QCOR Machine Model [3] realization of heterogeneous computing where different types of processing elements, including special purpose accelerators, simultaneously and asynchronously work together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' QCOR [1] is a programming system to realize such a heterogeneous quantum-classical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' It is based on the C++-based programming language and a compiler that is built on top of XACC [4] As shown in Figure 1, QCOR’s target machine is a heterogeneous system where multiple CPUs (cores) are connected with quantum devices and other accelerators such as GPUs and FPGAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' To program quantum devices in QCOR, the user writes a quantum kernel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=', a function that will be executed on a quantum device) in quantum computing domain-specific languages (DSLs), such as XACC’s XASM or IBM’s Open- QASM [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Similar to other GPU-based heterogeneous pro- gramming models such as CUDA [6], SYCL [7], and OpenCL [8], QCOR allows the user to write quantum kernels and CPU control code in the same program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' This single-source programming model greatly facilitates quantum-classical pro- gramming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' However, one open research question for QCOR and other quantum DSLs is how to provide well-defined, user- level multi-threading support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Specifically, as the machine model in Figure 1 implies, it is possible that multiple CPU cores might simultaneously utilize one or more quantum de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Currently, there is no user-facing API-level support for multi-threading in quantum-classical programming models like QCOR and DSLs like OpenQASM, although it is typical to internally use multi-threading for accelerating quantum circuit arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='11559v1 [quant-ph] 27 Jan 2023 GPU FPGA Instruction Channel APU Quantum Device Quantum State Measurement Quantum Output Buffer Memory SystemAlgorithm 1 Shor’s Algorithm (Pseudocode) Input: N: A natural number to be factorized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Output: A non-trivial divisor(s) of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 1: procedure MAIN(N) 2: repeat 3: a ← random(1, N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' ▷ 1 < a < N 4: K ← gcd(a, N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 5: if K == 1 then 6: SHOR(N, a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 7: else 8: return K 9: until a divisor(s) is found or explored all 10: procedure SHOR(N, a) 11: for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=', nShots do 12: rs ← SHORKERNEL(N, a) 13: r ← r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=', rs ▷ Estimate r from the measurements 14: if r mod 2 ≡ 1 or ar mod N ≡ −1 then 15: return φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 16: else 17: return gcd(ar/2 ± 1, N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' simulations [2], [9]–[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' In this paper, we explore the possibility of enabling user- level multi-threading in QCOR, which enables coarser grain parallelism in quantum-classical programming models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' We be- lieve this is an important step towards realizing an end-to-end heterogeneous programming system that can work on general heterogeneous platforms that include quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' This work makes the following key contributions: Design and implementation of multi-threading support for a heterogeneous quantum-classical programming model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Discussion of scenarios and use cases where user-level multi-threading is beneficial for near-term quantum sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' A demonstration which shows that running two quantum kernels in parallel using N/2-threads for each kernel outperforms running the kernel one-by-one using N- threads, by factors of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='22× to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='63× for the evaluated kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' MOTIVATION This section highlights our motivation for enabling user- level multi-threading in quantum-classical computing by dis- cussing potential parallelism in quantum-classical programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Let us use Shor’s algorithm as a motivating example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' In Algorithm 1, SHOR is a quantum-classical task that invokes the period-finding quantum kernel (SHORKERNEL) to estimate exponent r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Notice that SHOR can be called multiple times until one or more (non-)trivial divisors are found or the entire search space is explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' From the perspective of parallel processing, one possibility of parallelizing this algorithm is to run multiple instances of SHOR in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Furthermore, since it can require multiple shots to find r, it would be also possible to further parallelize Algorithm 2 Parallel Shor’s Algorithm (Pseudocode) Input: N: A natural number to be factorized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Output: A non-trivial divisor(s) of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 1: procedure MAIN(N) 2: repeat 3: a ← random(1, N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' ▷ 1 < a < N 4: K ← gcd(a, N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 5: if K == 1 then 6: async SHOR(N, a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 7: else 8: return K 9: until a divisor(s) is found or explored all 10: procedure SHOR(N, a) 11: foreach s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=', nShots do 12: rs ← SHORKERNEL(N, a) 13: r ← r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=', rs ▷ Estimate r from the measurements 14: if r mod 2 ≡ 1 or ar mod N ≡ −1 then 15: return φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 16: else 17: return gcd(ar/2 ± 1, N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' the shot loop in SHOR (Line 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Finally, if the SHORKERNEL is executed on a simulator, there is a massive amount of parallelism as in [2], [9]–[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Algorithm 2 is a pseudo-parallel version of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' As in the X10 language [12], async represents parallel task creation and execution and foreach represents parallel loop creation and execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Figure 2 graphically illustrates the potential parallelism in Shor’s algorithm across these three levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Based on what we discussed for Algorithm 2 and observe in Figure 2, we identify the following multiple levels of parallelism in quantum-classical programs: Task level parallelism: multiple independent classical tasks that can include quantum kernels are executed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Shot level parallelism: multiple independent shots are exe- cuted in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Inner simulator level parallelism: quantum simulators, in- cluding state vector and tensor network simulators such as [2], [9]–[11], are typically parallelized using OpenMP, CUDA, and the Eigen library to utilize a massive amount of parallelism on CPUs and/or GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' It is worth noting that the actual amount of available parallelism depends not only on algorithms but also on the simulated or physical quantum back ends that are targeted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' One example would be when a user executes their program on a current-day single QPU system in which there would be limited parallelism due to the lack of additional physical hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' However, in most cases, we believe that allowing the user to specify all available parallelism for a quantum-classical task will greatly enhance the performance and expressiveness of quantum-classical programs because there are plenty of computing resources (CPUs, GPUs, and FPGAs) that can accelerate the development of quantum-classical algorithms even on conventional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Task1: Shor (N = 15, a = 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 7!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' ≡ 1 𝑚𝑜𝑑 15 gcd 7" − 1, 15 = 3 gcd 7" + 1, 15 = 5 Found!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Task level parallelism Shot level parallelism Attempt 1: r = x?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Quantum Device Attempt 2: r = y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Quantum Device Task2: Shor (N = 15, a = 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Attempt 1: r = z?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Quantum Device Attempt 2: r = w?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Quantum Device Task3: Shor (N = 15, a = 7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Attempt 1: r = 4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Quantum Device Inner simulator level parallelism (when a simulator is used) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 2: Multi-level parallelism in a quantum-classical program (Shor’s algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Listing 1: A 2-qubit Bell kernel implementation in QCOR 1 using namespace std;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 2 // the Bell kernel 3 __qpu__ void bell(qreg q) { 4 using qcor::xasm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 5 H(q[0]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 6 CX(q[0], q[1]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 7 for (int i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' i < q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='size();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' i++) { 8 Measure(q[i]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 9 } 10 } 11 int main(int argc, char **argv) { 12 // Create two qubit registers, each size 2 13 auto q = qalloc(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 14 // Run the quantum kernel 15 bell(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 16 // dump the results 17 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='print();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 18 } Listing 2: An example output of the Bell kernel (1024 shots) 1 "AcceleratorBuffer": { 2 "name": "qrg_bmQBh", 3 "size": 2, 4 "Information": {}, 5 "Measurements": { 6 "00": 513, 7 "11": 511 8 } 9 } Thus, we believe that enabling user-level multi-threading in quantum-classical programming models will 1) accelerate the development of a quantum-classical algorithm, and 2) facilitate porting an existing heterogeneous algorithm to a quantum-classical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' It is also worth noting that the goal of this work is not optimizing and fine-tuning quantum-classical parallel programs for a specific target system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Instead, we look to motivate and introduce concrete parallel programming constructs (std::thread and std::async) for quantum- classical programming models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' QCOR QCOR is a C++-based high-level quantum-classical pro- gramming model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' One of the key features of QCOR is that Listing 3: A VQE implementation in QCOR 1 __qpu__ void ansatz(qreg q, double theta) { 2 X(q[0]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 3 Ry(q[1], theta);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 4 CX(q[1], q[0]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 5 } 6 7 int main(int argc, char **argv) { 8 // Allocate 2 qubits 9 auto q = qalloc(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 10 11 // Programmer needs to set 12 // the number of variational params 13 auto n_variational_params = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 14 15 // Create the Deuteron Hamiltonian 16 auto H = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='907 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='1433 * X(0) * X(1) - 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='1433 * Y(0) * Y(1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='21829 * Z(0) - 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='125 * Z(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 19 20 // Create the ObjectiveFunction 21 auto obj = createObjectiveFunction(ansatz, H, q, 22 n_variational_params, 23 {{"gradient-strategy", 24 "central"}, 25 {"step", 1e-3}});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 26 27 // Create the Optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 28 auto opt = createOptimizer("nlopt", 29 {{"nlopt-optimizer", 30 "l-bfgs"}});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 31 // Optimize 32 auto [opt_val, opt_params] = opt->optimize(objective);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 33 std::cout << opt_val << std::endl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 34 } the user can write both quantum and classical kernels and functions in the same code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' This feature is not only anal- ogous to existing heterogeneous programming models such as CUDA, OpenCL, and SYCL, but it also also provides a new programming model for heterogeneous quantum-classical computing programs that achieve hybrid quantum-classical workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' As shown in the machine model in Figure 1, in theory, the user is free to leverage different kinds of processors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=', CPUs, GPUs, FPGAs, Quantum Devices) that could all be enabled through a QCOR-style programming model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Listing 1 shows an example of QCOR program that executes the Bell kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' First, on Line 13, the qalloc API is called to allocate 2-qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Then, the kernel written in XASM is invoked Listing 4: Simultaneously Launching two Bell kernels (std::thread) 1 using namespace std;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 2 // the bell kernel 3 __qpu__ void bell(qreg q) { 4 using qcor::xasm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 5 H(q[0]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 6 CX(q[0], q[1]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 7 for (int i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' i < q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='size();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' i++) { 8 Measure(q[i]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 9 } 10 } 11 void foo() { 12 // Create two qubit registers, each size 2 13 auto q = qalloc(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 14 // Run the quantum kernel 15 bell(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 16 // dump the results 17 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='print();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 18 } 19 int main(int argc, char **argv) { 20 thread t0(foo);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' thread t1(foo);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 21 // Other classical/quantum work 22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 23 t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='join();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='join();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 24 } on Line 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Notice that the kernel is defined on Line 3 - 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' After the kernel is invoked, the measurement results can be inspected by printing the content of the quantum register as shown on Line 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' An example output of the QCOR program can be found in Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' In addition to the simple quantum circuit simulation above, for completeness, we would like to emphasize that QCOR is expressive enough to write a wide variety of quantum- classical algorithms such as the variational quantum eigen- solver (VQE) and the Quantum Approximate Optimization Algorithm (QAOA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Listing 3 shows a VQE implementation in QCOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Note that createObjectiveFunction and createOptimizer are built-in QCOR helper functions that facilitate creating and invoking a classical optimizer with a user-defined objective function with the Deuteron Hamiltonian and the ansatz kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' More details can be found in [1], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' DESIGN A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Multi-threading Design Overview Since QCOR is primarily written in C++, we look to enable user-level multi-threading in QCOR in a way that is acceptable to both QCOR and C++ programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' For QCOR programmers, our goal is to minimize modifications to the code required for enabling multi-threading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' For C++ programmers, our goal is to provide a threading interface that is natural to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' To that end, we leverage C++’s standard threading constructs (std::thread and std::async).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' However, in terms of general applicability, our discussions should apply to other parallel programming systems for C++, such as OpenMP [13], Kokkos [14], and RAJA [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Our current focus is on enabling coarse-grain parallelism to exploit the full capability of a CPU-QPU system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' In one scenario, the user would like a one-to-one relation between a CPU and a QPU to simultaneously perform N independent Listing 5: Asynchronously Launching the Bell kernel (std::async) 1 using namespace std;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 2 int main(int argc, char **argv) { 3 std::future f = async(launch::async, 4 [=]() -> int { foo();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' return 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' });' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 5 // Other classical/quantum work 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 7 // 8 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='get();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 9 } tasks, where N is the number of CPU-QPU pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Another sce- nario might be a one-to-many/many-to-one relation between CPU(s) and QPU(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' It is worth noting that the QPU part is not necessarily a hardware QPU device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Since QCOR offers different backends, the QPU part can be a quantum circuit simulation on either a local machine or a cloud service and can also incorporate coarser tasks such as VQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' User-Facing API 1) std::thread: Listing 4 shows an example where two threads simultaneously run the Bell kernel using thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' The main function creates two threads (t0 and t1), each of which executes the foo function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' In the foo function, it first allocates 2-qubits using qalloc, then invokes the kernel written in XASM in Line 3 - 10, and finally gets the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' This approach enables the user to overlap other work on the main thread with the two threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Also, the main function can wait on each thread by calling join().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 2) std::async: Another example (Listing 5) is asynchronous execution where the main function asynchronously launches the foo() function with async.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Similar to the thread example, the user may want to overlap other work with the launched task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' However, one interesting difference is that async returns a future object, which helps the user to check the status of the asynchronously launched task and take further action depending on the return value of the task (get()).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Enabling Thread Safety Thread safety is usually attributed to a function/routine that can be safely invoked by multiple threads simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' It is very common that thread safety is guaranteed in conventional heterogeneous programming models such as CUDA, OpenCL, and SYCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' For example, the SYCL specification [16] de- scribes this in the following manner: “SYCL guarantees that all the member functions and special member functions of the SYCL classes described are thread safe.” It is worth noting that enabling thread safety does not necessarily mean improving performance because it essen- tially prevents multiple threads from simultaneously accessing shared data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' In this work, our first priority is to enable thread safety for QCOR’s user-facing API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' For portions where paral- lelization is important, we explore the possibility of increasing parallelism in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Listing 6: Making qalloc() thread-safe with Mutex Lock 1 mutex m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 2 qbit qalloc(const int n) { 3 lock_guard lock(m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 5 allocated_buffers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='insert({.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 7 } Listing 7: How a QPU instance is declared and created 1 namespace xacc { 2 namespace internal_compiler { 3 // global variable 4 std::shared_ptr qpu = nullptr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 6 }} 7 // Getting an instance of qpp 8 qpu = xacc::getAccelerator("qpp");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' IMPLEMENTATION This section discusses how we enable user-level multi- threading in QCOR and XACC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Since the QCOR and XACC systems include over 200K lines of code written in modern C++, we focus on discussing a few common cases that can possibly inhibit user-level multi- threading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Essentially, these cases are focused on identifying potential sources of data races when multi-threading is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Identifying sources of data races 1) Global Variables: Global variables are the most common source of data races because these variables can be accessed simultaneously by multiple threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' The following is a global std::map object that is used to implement qalloc().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' // global variable map> allocated_buffers{};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Because qalloc() internally invokes map’s insert(), which is not thread-safe, concurrent invocations of qalloc() can be problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 2) Services: QCOR depends on different software com- ponents provided by QCOR itself and XACC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Typically, xacc::getService(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=') is used to obtain a shared pointer to a specific service, namely T in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' For services that do not derive xacc::Cloneable, the xacc::getService(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=') always returns a pointer to the same instance, which can be another source of a data race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' The following is an example where a pointer to the qpp accelerator, a software simulator in QCOR/XACC (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=', Quamtum++ [2]), which is used to run the Bell kernel in Listing 4 and Listing 5, is stored into acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' shared_ptr acc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' // a local variable acc = xacc::getService("qpp", .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=');' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Because Accelerator is not Cloneable, getService(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=') always returns the same qpp instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' This can cause a data collision since multiple threads can simultaneously register their gates to the Listing 8: QPU Manager Implementation (Simplified) 1 using namespace std;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 2 class QPUManager { 3 public: 4 static QPUManager& getInstance() { 5 static QPUManager instance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' return instance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 6 } 7 private: 8 QPUManager() {} 9 map> qpu_map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 10 public: 11 shared_ptr getQPU();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 12 void setQPU(std::shared_ptr _qpu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 13 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' same accelerator and can thus end up simulating an erroneous circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Implementation Details In general, we pursue the following two approaches to remove data races that inhibit multi-threading in QCOR and XACC: 1) enabling thread safety and 2) increasing parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' The former goal is achieved by adding safety to multi-threaded execution with mutex locks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' The latter approach explores the possibility of leveraging multi-threading to accelerate user programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 1) Enabling thread-safety: For enabling thread-safety, we leverage std::mutex or std::recursive_mutex to enable mutual exclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' For example, Listing 6 shows qalloc(), which has a non-thread-safe call in Line 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' We first create a mutex object in the global scope, and then the object is used to create a critical section with std::lock_guard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 2) Increasing Parallelism: For increasing parallelism, we use a quantum accelerator object (qpu) as a motivating example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' In the original implementation, as shown in Listing 7, the qpu object is declared as a global variable and is initialized by calling xacc::getAccelerator(), which internally calls xacc::getService().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Thus, this example includes the two data race scenarios discussed above in Section V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' We remove the data races by i) making Accelerator cloneable to create different instances every time xacc::getAccelerator() is called, and ii) providing a map that maps a current thread ID to the corresponding accelerator object, the latter of which is called QPUManager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Listing 8 shows a brief overview of QPUManager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' QPUManager is implemented by using the singleton pattern and contains the setter and getter functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' The setter function takes the return variable of xacc::getAccelerator() and registers the accelerator instance along with a current thread id to the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Similarly, the getter function returns a qpu instance that corresponds to a current thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Current Implementation Status We have implemented these changes to enable thread- safety for QCOR and have created a pull request against the QCOR [17], QCOR SPEC [18], and XACC [19] repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='63 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='8 12 threads 24 threads 2 x (6 threads/task) 2 x (12 threads/task) One-by-One (Conventional) Parallel (Our approach) Speed up over 12 threads 2 Bell kernels Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 3: Bell Kernel For increasing multi-threaded parallelism, we have confirmed that the examples (Listing 4 and Listing 5) and Shor’s kernel work in a parallel fashion, and we plan to create another pull request to share that functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' One small limitation of our implementation is that the user needs to manually call quantum::initialize() API at the beginning of each thread so the runtime can register its thread ID to the QPUManager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' In the future, we plan to create a compiler pass that automatically inserts this API call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Alternatively, we could provide qcor::thread and qcor::async wrappers for the original C++ constructs that internally call this initialization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' PRELIMINARY PERFORMANCE EVALUATION This section presents the results of an empirical evaluation of our extended QCOR programming model and runtime implementation on a single-node platform to demonstrate its performance benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Purposes: The goal of our evaluation is two-fold: 1) to demonstrate that our extended QCOR programming model and runtime system with C++ threading model enables parallel quantum kernel execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 2) to demonstrate that enabling parallel quantum kernel execution is beneficial in terms of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Platform: We present the performance results on a single-node AMD server, which consists of a 12-core, 24-thread Ryzen9 3900X CPu running at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='8GHz with 128GB of DRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Quantum Kernels: We use the following quantum kernels written in XASM: 1) Bell Kernel: The 2-qubit Bell kernel shown in Listing 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' The number of shots is 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 2) Shor’s Kernel: The period-finding quantum kernel, which is based on [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' The number of shots is 10 and we run SHOR(N=15, a=2) and SHOR(N=15, a=7) as in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Experimental variants: For kernel simulations, we use the QppAccelerator back end in QCOR, which uses the Quantum++ library [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' We compare the following two variants in terms of perfor- mance: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='22 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content="4 12 threads 24 threads 2 x (6 threads/task) 2 x (12 threads/task) One-by-One (Conventional) Parallel (Our approach) Speed up over 12 threads 2 Shor's kernels Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 4: Shor’s Kernel 1) One-by-One (baseline, conventional): Run the first kernel with N-threads and then run the second kernel with N-threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 2) Parallel: Run the two kernels in parallel, each of which uses N/2-threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Note that each kernel is executed on multiple physical cores/threads even in the baseline version because Quantum++ uses OpenMP [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' For both variants, we appropriately set the OMP_NUM_THREADS parameter to specify the number of threads per kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' However, tuning this parameter for the best performance is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Instead, our goal is to study scenarios where running multiple quantum kernels simultaneously could lead to performance benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Finally, note that shot-level parallelism is not exploited in these versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Bell kernel Figure 3 shows relative performance improvements over the baseline execution (one-by-one execution with 12-threads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' In one-by-one execution, increasing the number of threads degrades performance because the kernel is too small to fully utilize all the threads/cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' In contrast, parallel execution of the two kernels enables a further performance improvement of up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='63×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Shor’s kernel Figure 4 shows relative performance improvements over the baseline execution (one-by-one execution with 12-threads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Similar to the Bell kernel results, parallel execution enables a further performance improvement of up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='22× over the baseline execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' DISCUSSION As shown in Section VI, we demonstrated a scenario where running multiple kernels simultaneously is beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' The goal of this section is to summarize difference application scenarios that we believe are good candidates for user-level multi-threading: Shor’s algorithm: As we discussed in Section II, suppose we factorize N using Shor’s algorithm, we can create p parallel tasks with a random number ap s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' 1 < ap < N and gcd(ap, N) = 1, each of which invokes Shor’s kernel to estimate rp and checks if rp is even and arp mod N ≡ 1 in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Algorithm 2 summarizes the parallel algorithm and Figure 4 shows that running two Shor’s kernels in parallel outperforms one-by-one execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' We anticipate that the performance improvement will be more significant if CPUs with more cores and GPUs are used for simulating Shor’s circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' VQE: VQE [21] optimizes a (Hamiltonian H) cost function over a parameterized manifold of quantum states |ψ(⃗θ)⟩ = U(⃗θ)|ψ0⟩ as min ⃗θ ⟨ψ(⃗θ)|H|ψ(⃗θ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' For QMA-hard Hamiltoni- ans, dim(⃗θ) is large but for many interesting models in physical sciences dim(⃗θ) may scale (sub-)polynomially, in which case the optimization problem at hand may still be quite challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' The pleasantly parallel nature of the optimization process can be utilized with multiple asynchronous quantum kernel instances minimizing over ⃗θ-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Asynchronous Quantum JIT Compilation: Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' [22] discusses a scenario where a GPU is used to compile and optimize quantum circuits, which can take several hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' With user-level multi-threading enabled, it is possible to avoid blocking computing resources by asynchronously offloading a compilation task onto a GPU and launching the compiled kernel on a QPU only when it is ready.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Parallel Quantum-Classical Workflow: As generalizations of different parallel execution scenarios discussed above, one can write an entire workflow in which different tasks run on different processing units including CPUs, QPUs, GPUs, and FPGAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' RELATED WORK While domain-specific languages (DSLs) for quantum com- puting significantly facilitate the development of quantum al- gorithms, many DSLs only focus on the kernel part and do not provide a system-wide programming model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' We believe that such a system-wide programming model will become more important in quantum-classical computing because exploiting classical parallelism such as thread-level parallelism can im- prove end-to-end performance as discussed in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Here, we briefly discuss existing programming models from the viewpoint of classical parallelism on non-quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Qiskit [23] has been one of the most popular programming frameworks for quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' However, it is not appro- priate to directly map Qiskit programs to quantum-classical systems unless there is an AOT/JIT-level smart compiler that is aware of the underlying parallel hardware because the Global Interpreter Lock (GIL) may hinder Python-level multi- threaded execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Q# is a programming language designed to express hybrid quantum-classical algorithms [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Currently, there is no way to express the concept of threads in the Q# language itself [25], nor in the Q# standard library [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Additionally, QIR (Quantum Intermediate Representation), a hybrid quantum- classical IR based on LLVM IR that is generated by the Q# compiler front-end, does not explicitly guarantee thread-safety for any runtime functions [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Indeed, the reference QIR runtime [28] may exhibit data races if used in multi-threaded code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' It is worth noting that QParallel [29] allows the user to explicitly express parallelism in the quantum kernel part, not in the classical part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Other newer platforms for hybrid quantum-classical com- puting have been proposed like NVIDIA’s QODA [30], which is designed for the simulation of quantum circuits with GPUs and QPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' It is unclear what multi-threaded support model QODA uses as it is a proprietary product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' CONCLUSIONS AND FUTURE WORK This paper explores the possibility of enabling user-level multi-threading in QCOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' We made enhancements to QCOR to support C++-based parallel and asynchronous execution of quantum kernels by 1) adding thread safety to QCOR API routines, and 2) increase parallelism by removing data races that inhibit multi-threading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' Our preliminary results with the Bell and Shor’s algorithm kernels show that enabling user-level multi-threading gives us performance improvements over the conventional baseline version in which each kernel is still executed by multiple threads, but is executed one-by-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' We believe this multi-threading design for heterogeneous quantum-classical programming models will open up an op- portunity for rapidly prototyping and developing quantum- classical programs on conventional systems in the short- term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' At the same time, we envision that this initial design would be a good starting point for longer-term explorations of heterogeneous programming systems for future quantum- classical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' In future work, we plan to run other quantum-classical tasks, such as VQE, with additional quantum simulation and physical back ends and also use different back ends to demonstrate where user-level multi-threading is most beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' ACKNOWLEDGEMENT We acknowledge DOE ASCR funding under the Quan- tum Computing Application Teams program, FWP number ERKJ347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' We also acknowledge support for this work from NSF planning grant #2016666, “Enabling Quantum Computer Science and Engineering”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' REFERENCES [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdFJT4oBgHgl3EQfgCxB/content/2301.11559v1.pdf'} +page_content=' M.' metadata={'source': 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Entanglement-enhanced matter-wave +interferometry in a high-finesse cavity +Liam P. McGuinness +Laser Physics Centre, Research School of Physics, +Australian National University, Acton, +Australian Capital Territory 2601, Australia +Email: +liam@grtoet.com +In their paper “Entanglement-enhanced matter-wave interferometry in a high- +finesse cavity” Nature (2022) [1], Greve et. +al. +claim to use entanglement in a +matter-wave interferometer to achieve a sensitivity beyond that achievable with the +same number of independent particles – a limit known as the standard quantum limit +(SQL). In particular, using squeezed momentum states of 700 atoms, the authors +claim to directly observe a sensitivity 3.4 dB (a factor of 1.5) below the SQL. This +claim is incorrect. The authors do not measure anything beyond the SQL, nor do +they achieve a sensitivity beyond what one could obtain with a single atom. The +achieved sensitivity is at least a factor of 39 worse than the claimed value. +In “Entanglement-enhanced matter-wave interferometry in a high-finesse cavity” Nature +(2022) [1], Greve et. al. describe measuring the phase φ between two quantum states |a⟩ , |b⟩, +given by +1 +√ +2 +� +|a⟩ + eiφ |b⟩ +� +. With no prior information on φ, i.e. 0 < φ ≤ 2π, one cannot +estimate this phase with a single measurement to an angular uncertainty better than ∆φ = +1 radians. Allowing for N trials, which can be implemented by encoding φ into the state of N +independent atoms, the uncertainty must be greater than ∆φSQL = 1/ +√ +N rad, a limit known +as the standard quantum limit (SQL). As Greve et. al. note, this is a fundamental limit +which cannot be improved upon, even with entanglement. In fact, if one entangles N atoms +to obtain the state +1 +√ +2 +� +|aN⟩ + eiφ |bN⟩ +� +, where |aN⟩ , |bN⟩ are states in an N-dimensonal +Hilbert space, measurement of this phase has an uncertainty is restricted to ∆φ > 1 rad, +much worse than the SQL. The reason being that we have lost the ability to perform N +independent trials and are thus limited to the uncertainty of a single measurement. This is +clear to see, since the entangled state is identical to the single atom state up to a relabelling. + +While the above is well-known for estimating an unknown quantum phase, it is widely +accepted that despite this entanglement can lead to improved estimation of some other phase +θ. Why is this? The argument is that entangled states accumulate a bigger quantum phase +in response to some physical Hamiltonian to be measured (N-fold greater than a single +atom). With particular reference to Mach-Zehnder interferometry, discussed by Greve et. +al., this is the phase that particles in one arm of the interferometer accumulate with respect +to particles in the other arm. If we also allow somewhat sneakily that we now have more +prior information on φ – it is known to within a much narrow range, then it is expected that, +for the same measurement time, entangled states achieve greater sensitivity to small phase +shifts than unentangled states [2]. As a result, although the uncertainty in estimating φ is +√ +N worse with entanglement than with independent atoms, the value of φ in an entangled +state is N-fold greater than any of the independent atomic states. Again, it is important +to be clear here. With entanglement we have not improved the uncertainty in estimating φ, +it has gotten worse. Whenever measurement of a quantum phase is performed as described +above, one can immediately rule out surpassing the SQL, it is only when the quantum phase +is used to infer some other parameter θ that the uncertainty in θ can be reduced. +Let’s assume an element with phase θ is in one arm of the interferometer, and passing a +single atom through the interferometer performs a one-to-one mapping of θ to the atomic +phase φ. Then with no entanglement we have φ = θ, whereas with entanglement φ = Nθ. In +the former case (assuming no additional errors) ∆θ = ∆φ and estimation of θ with N atoms +is limited by the SQL to: ∆φ > 1/ +√ +N =⇒ ∆θ > 1/ +√ +N. With entanglement ∆θ = ∆ (φ/N) +again assuming no additional errors, so we obtain ∆φ > 1 =⇒ ∆θ > 1/N. If the second +inequality can be saturated with no overheads, then entanglement outperforms unentangled +sensors. Contrary to what is often stated, this superiority of entanglement in sensing is not +proven and only holds under strong experimental assumptions. For that reason, experimental +evidence with correct analysis and complete details is critical in validating the theory [3]. +So what is the phase shown in Fig. 1b that Greve et. +al. +measure with a precision +beyond the SQL? Described in section Entangled matter-wave interferometry: as “A +relative phase accumulates between the wave packets during a free evolution time Tevol”, at +first reading one might be surprised to find that the cause of the relative phase shift is not +explicitly defined in the main text. One reason for this oversight could be that Fig. 1b and +the description of the matter-wave interferometer is somewhat misleading. Greve et. al. do +2 + +not really perform Mach-Zehnder interferometry because they do not measure a phase shift +between different arms of the interferometer. So what are the authors measuring? Greve +et. al. allow the atoms to fall through a gravitational potential and measure the energy +shift of the atoms – manifesting as a Doppler shift of atomic resonance with respect to the +Raman laser (see Methods – Raman transitions and velocity selection). Notably this shift is +the same for both arms of the interferometer and can be measured without a Mach-Zehnder +interferometer. Most importantly, entangled atoms do not experience a greater phase shift +(see Fig. 4c), since their velocity is the same as unentangled atoms. +Put simply, the authors measure the difference between the atomic phase φ and the laser +phase θ at the end of the experiment. As entanglement produces no enhanced phase shift, +the uncertainty in measuring either phase when using entanglement is strictly worse than +the SQL, and we have shown that Greve et. al.’s claim in beating the SQL is incorrect. In +fact, assuming the ensemble is fully entangled, the obtained sensitivity must be worse than +the single atom precision limit. Even assuming the ensemble is not fully entangled, there +are many experimental imperfections that prevent Greve et. al. reaching the single atom +limit. With N = 660 atoms, this means that the achieved sensitivity is at least a factor +of 31 worse than the claimed 1.7 dB enhancement. Similar analysis can show that all other +claims made by the authors are similarly incorrect. So how can Greve et. al. claim to have +done otherwise? Maybe it is better to reframe this question with a focus on the audience. +These are the questions one should ask anybody claiming to beat the SQL. +1. Precisely what parameter do you measure? +2. What sensitivity/uncertainty for this parameter do you explicitly achieve in your ex- +periment? This should have the correct units, including the total measurement time +and the number of particles used. +3. Is it really impossible to measure this parameter with better sensitivity using the same +measurement time and the same number of independent particles? How about just a +single particle. Is it impossible to measure this parameter with better sensitivity using +the same measurement time and a single particle? +If 1) and 2) are properly defined, then the answer for all experiments to date, including +the work by Greve et. al. is a resounding – ‘No!’. +3 + +It is important that the scientific community is made aware of the current state of the art +in quantum metrology. I am sure that many people would be extremely surprised to learn +that entanglement has never been used to improve any experiment beyond what one could +achieve without entanglement. Even more surprising is that fully entangled ensembles have +never demonstrated a precision beyond the single particle limit. The community should be +made aware of this for a variety of reasons. First, if the experimental data conflicts with +the message being presented then we should demand better scientific rigour in published +papers. In quantum metrology broadly, the standards have a long way to improve. Secondly, +misrepresentation of data is hindering progress since people are currently unaware of a +massive discrepancy between quantum mechanics as interpreted and experiment; even going +so far as to prevent plausible explanations from being investigated. Thirdly, there is currently +huge investment in technologies dependent on quantum entanglement (in both quantum +sensing and quantum computing), if entangled ensembles cannot provide fundamentally +more information than a single atom then these technologies will never reach their goals. +[1] Greve, G. P., Luo, C., Wu, B. & Thompson, J. K. +Entanglement-enhanced matter- +wave interferometry in a high-finesse cavity. +Nature +610, +472–477 +(2022). +URL +https://doi.org/10.1038/s41586-022-05197-9. +[2] I am presenting the view of the general quantum metrology community, however I must make +my stance clear. I do not share the same views. Indeed I expect that entanglement cannot +improve the amount of information that can be obtained from the same number of atoms in +the same time. Specifically the information gain from fully entangled states cannot exceed that +obtained with a single spin in the same time. +[3] One specific overhead is the total measurement time T. For example by increasing the inter- +action time of a single atom with the phase element we can also achieve a mapping φ = Nθ +with a single atom. This is why ∆θ > 1/ +√ +N is not the SQL for estimating θ. The SQL for +estimating θ is ∆θ > 1/(ΩT +√ +N), where Ω is the interaction strength between the atom and +the phase element. +4 + diff --git a/xNE3T4oBgHgl3EQfOwlv/content/tmp_files/load_file.txt b/xNE3T4oBgHgl3EQfOwlv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a5a344b8a92dfa7d49b50bf0fae9cc1b7d5cd0b0 --- /dev/null +++ b/xNE3T4oBgHgl3EQfOwlv/content/tmp_files/load_file.txt @@ -0,0 +1,118 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf,len=117 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content='04396v1 [quant-ph] 11 Jan 2023 Matters Arising: Entanglement-enhanced matter-wave interferometry in a high-finesse cavity Liam P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' McGuinness Laser Physics Centre, Research School of Physics, Australian National University, Acton, Australian Capital Territory 2601, Australia Email: liam@grtoet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content='com In their paper “Entanglement-enhanced matter-wave interferometry in a high- finesse cavity” Nature (2022) [1], Greve et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' claim to use entanglement in a matter-wave interferometer to achieve a sensitivity beyond that achievable with the same number of independent particles – a limit known as the standard quantum limit (SQL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' In particular, using squeezed momentum states of 700 atoms, the authors claim to directly observe a sensitivity 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content='4 dB (a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content='5) below the SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' This claim is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' The authors do not measure anything beyond the SQL, nor do they achieve a sensitivity beyond what one could obtain with a single atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' The achieved sensitivity is at least a factor of 39 worse than the claimed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' In “Entanglement-enhanced matter-wave interferometry in a high-finesse cavity” Nature (2022) [1], Greve et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' describe measuring the phase φ between two quantum states |a⟩ , |b⟩, given by 1 √ 2 � |a⟩ + eiφ |b⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' With no prior information on φ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' 0 < φ ≤ 2π, one cannot estimate this phase with a single measurement to an angular uncertainty better than ∆φ = 1 radians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Allowing for N trials, which can be implemented by encoding φ into the state of N independent atoms, the uncertainty must be greater than ∆φSQL = 1/ √ N rad, a limit known as the standard quantum limit (SQL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' As Greve et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' note, this is a fundamental limit which cannot be improved upon, even with entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' In fact, if one entangles N atoms to obtain the state 1 √ 2 � |aN⟩ + eiφ |bN⟩ � , where |aN⟩ , |bN⟩ are states in an N-dimensonal Hilbert space, measurement of this phase has an uncertainty is restricted to ∆φ > 1 rad, much worse than the SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' The reason being that we have lost the ability to perform N independent trials and are thus limited to the uncertainty of a single measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' This is clear to see, since the entangled state is identical to the single atom state up to a relabelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' While the above is well-known for estimating an unknown quantum phase, it is widely accepted that despite this entanglement can lead to improved estimation of some other phase θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Why is this?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' The argument is that entangled states accumulate a bigger quantum phase in response to some physical Hamiltonian to be measured (N-fold greater than a single atom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' With particular reference to Mach-Zehnder interferometry, discussed by Greve et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=', this is the phase that particles in one arm of the interferometer accumulate with respect to particles in the other arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' If we also allow somewhat sneakily that we now have more prior information on φ – it is known to within a much narrow range, then it is expected that, for the same measurement time, entangled states achieve greater sensitivity to small phase shifts than unentangled states [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' As a result, although the uncertainty in estimating φ is √ N worse with entanglement than with independent atoms, the value of φ in an entangled state is N-fold greater than any of the independent atomic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Again, it is important to be clear here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' With entanglement we have not improved the uncertainty in estimating φ, it has gotten worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Whenever measurement of a quantum phase is performed as described above, one can immediately rule out surpassing the SQL, it is only when the quantum phase is used to infer some other parameter θ that the uncertainty in θ can be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Let’s assume an element with phase θ is in one arm of the interferometer, and passing a single atom through the interferometer performs a one-to-one mapping of θ to the atomic phase φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Then with no entanglement we have φ = θ, whereas with entanglement φ = Nθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' In the former case (assuming no additional errors) ∆θ = ∆φ and estimation of θ with N atoms is limited by the SQL to: ∆φ > 1/ √ N =⇒ ∆θ > 1/ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' With entanglement ∆θ = ∆ (φ/N) again assuming no additional errors, so we obtain ∆φ > 1 =⇒ ∆θ > 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' If the second inequality can be saturated with no overheads, then entanglement outperforms unentangled sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Contrary to what is often stated, this superiority of entanglement in sensing is not proven and only holds under strong experimental assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' For that reason, experimental evidence with correct analysis and complete details is critical in validating the theory [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' So what is the phase shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' 1b that Greve et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' measure with a precision beyond the SQL?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Described in section Entangled matter-wave interferometry: as “A relative phase accumulates between the wave packets during a free evolution time Tevol”, at first reading one might be surprised to find that the cause of the relative phase shift is not explicitly defined in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' One reason for this oversight could be that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' 1b and the description of the matter-wave interferometer is somewhat misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Greve et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' do 2 not really perform Mach-Zehnder interferometry because they do not measure a phase shift between different arms of the interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' So what are the authors measuring?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Greve et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' allow the atoms to fall through a gravitational potential and measure the energy shift of the atoms – manifesting as a Doppler shift of atomic resonance with respect to the Raman laser (see Methods – Raman transitions and velocity selection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Notably this shift is the same for both arms of the interferometer and can be measured without a Mach-Zehnder interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Most importantly, entangled atoms do not experience a greater phase shift (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' 4c), since their velocity is the same as unentangled atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Put simply, the authors measure the difference between the atomic phase φ and the laser phase θ at the end of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' As entanglement produces no enhanced phase shift, the uncertainty in measuring either phase when using entanglement is strictly worse than the SQL, and we have shown that Greve et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' al.’s claim in beating the SQL is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' In fact, assuming the ensemble is fully entangled, the obtained sensitivity must be worse than the single atom precision limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Even assuming the ensemble is not fully entangled, there are many experimental imperfections that prevent Greve et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' reaching the single atom limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' With N = 660 atoms, this means that the achieved sensitivity is at least a factor of 31 worse than the claimed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content='7 dB enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Similar analysis can show that all other claims made by the authors are similarly incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' So how can Greve et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' claim to have done otherwise?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Maybe it is better to reframe this question with a focus on the audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' These are the questions one should ask anybody claiming to beat the SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Precisely what parameter do you measure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' What sensitivity/uncertainty for this parameter do you explicitly achieve in your ex- periment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' This should have the correct units, including the total measurement time and the number of particles used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Is it really impossible to measure this parameter with better sensitivity using the same measurement time and the same number of independent particles?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' How about just a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Is it impossible to measure this parameter with better sensitivity using the same measurement time and a single particle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' If 1) and 2) are properly defined, then the answer for all experiments to date, including the work by Greve et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' is a resounding – ‘No!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' 3 It is important that the scientific community is made aware of the current state of the art in quantum metrology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' I am sure that many people would be extremely surprised to learn that entanglement has never been used to improve any experiment beyond what one could achieve without entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Even more surprising is that fully entangled ensembles have never demonstrated a precision beyond the single particle limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' The community should be made aware of this for a variety of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' First, if the experimental data conflicts with the message being presented then we should demand better scientific rigour in published papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' In quantum metrology broadly, the standards have a long way to improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Secondly, misrepresentation of data is hindering progress since people are currently unaware of a massive discrepancy between quantum mechanics as interpreted and experiment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' even going so far as to prevent plausible explanations from being investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Thirdly, there is currently huge investment in technologies dependent on quantum entanglement (in both quantum sensing and quantum computing), if entangled ensembles cannot provide fundamentally more information than a single atom then these technologies will never reach their goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' [1] Greve, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=', Luo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=', Wu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' & Thompson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Entanglement-enhanced matter- wave interferometry in a high-finesse cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Nature 610, 472–477 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content='1038/s41586-022-05197-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' [2] I am presenting the view of the general quantum metrology community, however I must make my stance clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' I do not share the same views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Indeed I expect that entanglement cannot improve the amount of information that can be obtained from the same number of atoms in the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' Specifically the information gain from fully entangled states cannot exceed that obtained with a single spin in the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' [3] One specific overhead is the total measurement time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' For example by increasing the inter- action time of a single atom with the phase element we can also achieve a mapping φ = Nθ with a single atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' This is why ∆θ > 1/ √ N is not the SQL for estimating θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' The SQL for estimating θ is ∆θ > 1/(ΩT √ N), where Ω is the interaction strength between the atom and the phase element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} +page_content=' 4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE3T4oBgHgl3EQfOwlv/content/2301.04396v1.pdf'} diff --git a/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf b/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0c1a4c3a42e14060bb51d6482f2dbbc4ab14661f --- /dev/null +++ b/xtE0T4oBgHgl3EQftQEc/content/2301.02588v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0a34f2c8f47cca0bc0785eb38094ddab570aa347df5b216f42c2793424cdeba6 +size 2621986 diff 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JORGENSEN +Abstract. The purpose of the present paper is to place a number of geometric (and hands- +on) configurations relating to spectrum and geometry inside a general framework for the Fuglede +conjecture. Note that in its general form, the Fuglede conjecture concerns general Borel sets Ω in +a fixed number of dimensions d such that Ω has finite positive Lebesgue measure. The conjecture +proposes a correspondence between two properties for Ω, one takes the form of spectrum, while +the other refers to a translation-tiling property. We focus here on the case of dimension one, and +the connections between the Fuglede conjecture and properties of the self-adjoint extensions of the +momentum operator +1 +2πi +d +dx, realized in L2 of a union of intervals. +Contents +1. +Introduction +1 +2. +Notations and preliminaries +2 +3. +Symmetric and self-adjoint extensions +7 +4. +Spectral decomposition +11 +5. +The unitary group +13 +6. +Spectral sets +14 +References +19 +1. Introduction +For bounded open domains Ω in Rd, the Fuglede problem deals with two properties that Ω may +or may not have, one (called spectral) is relative to the Hilbert space L2(Ω), the question of whether +L2(Ω) has an orthogonal d-variable Fourier basis, and the other is geometric (tiling), whether Ω +tiles Rd by some set of translation vectors. The original problem asked whether the two properties +are equivalent. +In this paper we show how tools from operator theory (especially choices of spectral representa- +tions for unbounded operators), serve to link the two sides of the problem, spectrum vs geometry. +Since the inception, stated this way, the Fuglede conjecture is now known to be negative, more +precisely that the two properties are not equivalent in dimension 3 and higher [Tao04, Mat05, +KM06b, KM06a, FMM06]. Nonetheless, the Fuglede problem is even open for d = 1. +2010 Mathematics Subject Classification. 47E05,42A16. +Key words and phrases. momentum operator, self-adjoint extension, Fourier bases, Fuglede conjecture . +∗Corresponding author. +1 + +2 +DORIN ERVIN DUTKAY∗ AND PALLE E.T. JORGENSEN +Parallel to this we note that there are many closely related new research directions, including +analysis on fractals, which deal with various notions of interplay between spectral theoretic prop- +erties on the one hand, and geometry on the other, e.g., direct problems and inverse problems. We +further stress that the original formulation was stated in terms of properties for the set of d partial +derivative operators for the coordinate directions in Ω, specifically the possible extensions of partial +derivative operators in the form of commuting generators for unitary one-parameter groups acting +in L2(Ω). Such extensions are known to necessarily be local translation generators. Moreover, +following quantum theory, such generators may be viewed as momentum operators, a viewpoint +motivated by the canonical duality from quantum mechanics for momentum and position observ- +ables. This formulation in turn makes a direct connection to scattering theoretic properties, again +related to the Hilbert space L2(Ω). And in this form, the problem is of interest even for d = 1; and +so the case when Ω is a union of intervals. +Continuing earlier work (e.g., [DJ15]) we aim here at presenting new results for the d = 1 Fuglede +problem, and making the presentation as self-contained as possible, for the readers who might not +be experts in this field. +Starting with its classical roots, the Fuglede problem addresses two related properties for bounded +domains Ω in Rd . More precisely, Fuglede’s question asks for a specific linking between multivari- +able spectra on one side, and geometry of Ω on the other (spectral vs tiling). But by now, the +Fuglede problem/conjecture has become distinctly interdisciplinary. +It has come to encompass +a diverse variety of neighboring fields of mathematics each of which in turn lies at the cross- +roads of at least the following six separate disciplines: (i) harmonic analysis, (ii) spectral and +scattering theory for operators in Hilbert space, (iii) metric/convex geometry, (iv) fractals, (v) +operator algebras, and (vi) representation theory. +For the benefit of readers, we include cita- +tions to the following list of papers, each dealing with one or the other of the above six areas, +[Jor18, Bir22, LM22, CNo22, KS21, Mat20, GL20, FS20, GL17, IMP17, Hak10]. +The paper is organized as follows: in Section 1, we introduce some definitions related to un- +bounded symmetric operators and their extensions, the associated one-parameter unitary group, +and we recall Fuglede’s result and conjecture, which serve as the main motivation for our paper. +In the next sections we focus on the case when the set Ω is a finite union of intervals in dimension +d = 1. In Section 2, we study the symmetric and the self-adjoint extensions A of the momentum +operator D = +1 +2πi +d +dx on the space C∞ +c (Ω) of infinitely differentiable functions with compact support +in Ω. In Section 3, we describe the spectral decomposition of such self-adjoint extensions. Section +4 is devoted to the one-parameter unitary group U(t) = exp(2πitA), t ∈ R, which acts as transla- +tions inside the intervals of Ω and has a different behavior at the end-points. In Section 5, we make +the connections between the existence of orthogonal Fourier bases on Ω and the properties of the +self-adjoint extensions A or of the unitary group U(t). +2. Notations and preliminaries +When d = 1, with a choice of an open set Ω, the corresponding connected components will then +be intervals, and so Ω takes the form of a finite union of intervals as follows. +Definition 2.1. Let +Ω = +n +� +i=1 +(αi, βi), where − ∞ < α1 < β1 < α2 < β2 < · · · < αn < βn < ∞. + +THE MOMENTUM OPERATOR ON A UNION OF INTERVALS +3 +So +Ω = +n� +i=1 +Ji, where Ji = (αi, βi) for all i ∈ {1, . . . , n}. +On Ω we consider the Lebesgue measure dx. We denote by ∂Ω the boundary of Ω, +∂Ω = {αi, βi : i ∈ {1, . . . , n}}. +For a function f on ∂Ω we use the notation +� +∂Ω +f = +n +� +i=1 +(f(βi) − f(αi)). +Consider the subspace of infinitely differentiable compactly supported functions on Ω, C∞ +c (Ω). We +define the differential/momentum operator D on C∞ +c (Ω): +Df = +1 +2πif ′, +(f ∈ C∞ +c (Ω)). +Define also the subspaces +(2.1) +D0(Ω) = {f : Ω → C : f is absolutely continuous on each Ji, +f(αi+) = f(βi−) = 0 for all i and f ′ ∈ L2(Ω) +� +, +(2.2) +Dmax = +� +f : Ω → C : f is absolutely continuous on each interval Ji and f ′ ∈ L2(Ω) +� +, +Remark 2.2. If a function f is absolutely continuous on each interval Ji and f ′ ∈ L2(Ω), then the +values of the function f at the endpoints αi and βi are well defined. Indeed, fix i ∈ {1, . . . , n} and +a point x0 ∈ (αi, βi). Then, since f is absolutely continuous and with f ′ ∈ L2(Ω) ⊂ L1(Ω), one has +f(x) = f(x0) + +� x +x0 +f ′(t) dt, for all x ∈ (αi, βi), +and therefore +f(αi+) = f(x0) − +� x0 +αi +f ′(t) dt and, f(βi−) = f(x0) + +� βi +x0 +f ′(t) dt. +This means that we can define f(αi) := f(αi+), and similarly f(βi) by continuity. +For f ∈ Dmax, we denote by f(⃗α) = (f(α1), . . . , f(αn)) and similarly for f(⃗β). +We are looking for closed symmetric and for self-adjoint extensions of the operator D on C∞ +c (Ω). +Definition 2.3. We recall some notions about unbounded linear operators, see for example [Con90, +Chapter X]. Let H be a Hilbert space. +• We denote by B(H) the set of bounded linear operators on H. +• We denote the domain of an unbounded operator T by D(T). +• An operator T2 is an extension of T1 if D(T2) contains D(T1) and T2f = T1f, for all +f ∈ D(T1). We write T1 ⊆ T2. +• An operator T is closed if its graph is closed, i.e., if {fn} in D(T) converges to f and {Tfn} +converges to g then f ∈ D(T) and Tf = g. + +4 +DORIN ERVIN DUTKAY∗ AND PALLE E.T. JORGENSEN +• For a densely defined unbounded operator T : H1 → H2, the adjoint operator T ∗ : H2 → +H1 is defined on the set of vectors g ∈ H2 with the property that the linear functional +D(T) ∋ f �→ ⟨Tf , g⟩ is bounded. In this case, by Riesz’s lemma, there exists a unique +element T ∗g ∈ H1 such that ⟨Tf , g⟩ = ⟨f , T ∗g⟩ for all f ∈ D(T). +• An densely defined operator is called symmetric if +⟨Tf , g⟩ = ⟨f , Tg⟩ , for all f, g ∈ D(T). +• An operator is called self-adjoint if D(T ∗) = D(T) and T ∗f = Tf for all f ∈ D(T). +• An operator N is called normal if it is closed, densely defined and N ∗N = NN ∗. +Definition 2.4. Recall that a (possibly unbounded) linear operator T : H → K is called boundedly +invertible if there is a bounded operator S : K → H such that TS = I and ST ⊆ I. The resolvent +set ρ(T) for the operator T is defined by +ρ(T) = {λ ∈ C : λI − T is boundedly invertible}. +The spectrum of T is defined as σ(T) = C \ ρ(T). +Definition 2.5. If X is a set, B is a σ-algebra of subsets of X, and H is a Hilbert space, a spectral +measure/spectral resolution for (X, B, H) is a function E : B → B(H) such that: +(a) for each ∆ in B, E(∆) is a projection; +(b) E(∅) = 0 and E(X) = IH; +(c) E(∆1 ∩ ∆2) = E(∆1)E(∆2) for ∆1 and ∆2 in B; +(d) if {∆n}∞ +n=1 are pairwise disjoint sets from B, then +E +� ∞ +� +n=1 +∆n +� += +∞ +� +n=1 +E(∆n). +Theorem 2.6. Spectral theorem for unbounded self-adjoint operators. If A is a self- +adjoint operator on H, then there exists a unique spectral measure E defined on the Borel subsets +of R such that +(a) A = +� +x dE(x); +(b) E(∆) = 0 if ∆ ∩ σ(A) = ∅; +(c) if U is an open subset of R and U ∩ σ(A) ̸= ∅, then E(U) ̸= 0; +(d) if B ∈ B(H) such that BA ⊆ AB, then B( +� +φ dE) ⊆ ( +� +φ dE)B for every Borel function +φ on R. +Remark 2.7. Note that, starting with a fixed selfadjoint operator A in a Hilbert space H, the +corresponding projection valued measure E then induces a spectral representation for A via a +system of scalar measures indexed by the vectors in H. These are defined for h ∈ H by +(2.3) +Eh,h(∆) = ⟨E(∆)h , h⟩ , for all Borel sets ∆. +Let A be a self-adjoint operator in a Hilbert space H with spectral resolution E; see Definition +2.5. Then the following three conclusions follow immediately : +(i) If ϕ : R → C is measurable, then the operator +(2.4) +ϕ(A) := +� +R +ϕ(x) dE(x) +is well defined in H and normal. + +THE MOMENTUM OPERATOR ON A UNION OF INTERVALS +5 +(ii) The dense domain of ϕ(A), denoted D(ϕ(A)), consists of all vectors h ∈ H such that +(2.5) +� +R +|ϕ(x)|2 dEh,h < ∞, +(iii) For all h ∈ D(ϕ(A)), +(2.6) +∥ϕ(A)h∥2 = +� +R +|ϕ(x)|2 dEh,h. +Definition 2.8. A strongly continuous one parameter unitary group is a function U : R → B(H) +such that for all s and t in R: +(a) U(t) is a unitary operator; +(b) U(s + t) = U(s)U(t); +(c) if h ∈ H and t0 ∈ R then U(t)h → U(t0)h as t → t0. +Theorem 2.9. Let A be a self-adjoint operator on H and let E on (X, B, H) be its spectral measure. +Define +U(t) = exp(2πitA) = +� +e2πitx dE(x), +(t ∈ R). +Then +(a) (U(t))t∈R is a strongly continuous one parameter group; +(b) if h ∈ D(A), then +lim +t→0 +1 +t (U(t)h − h) = 2πiAh; +(c) if h ∈ H and limt→0 1 +t (U(t)h − h) exists, then h ∈ D(A). Consequently, D(A) is invariant +under U(t). +Theorem 2.10. Stone’s Theorem. If U is a strongly continuous one parameter unitary group, +then there exists a self-adjoint operator A such that U(t) = exp(2πitA), t ∈ R, and conversely. The +self-adjoint operator A is called the infinitesimal generator for U. +We will also need the following well-known lemma about multiplication operators. +Lemma 2.11. Let (X, µ) be a σ-finite measure space and let φ : X → C be a measurable function. +Let D = {f ∈ L2(µ) : φf ∈ L2(µ)} and define Af = φf for all f ∈ D. Then A is a closed operator, +D(A∗) = D, and A∗f = φf for f ∈ D. In particular, if f is real-valued, then A is self-adjoint. +Proof. First, the domain D is dense, because one can consider functions of the form +χ{x∈M:−n≤Re(φ(x))≤n,−m≤Im(φ(x))≤m}, +m, n ∈ N, M ⊆ X measurable, +and they are in the domain D and span L2(Ω). Thus the operator A is densely defined. +Consider now g ∈ D(A∗). +Then the map D ∋ f �→ ⟨Af , g⟩ =: ϕg(f) is a bounded linear +functional, i.e., +���� +� +φfg dµ +���� ≤ C∥f∥, +(f ∈ D). +This implies that ϕg can be extended continuously to the whole space L2(Ω) and therefore there +exists an element A∗g ∈ L2(Ω) such that ϕg(f) = ⟨f , A∗g⟩ for all f ∈ D. Then +� +fφg dµ = +� +f · A∗g dµ, +(f ∈ D). + +6 +DORIN ERVIN DUTKAY∗ AND PALLE E.T. JORGENSEN +But this means that A∗g = φg a.e. +Conversely, if g ∈ D we want to see that g ∈ D(A∗). But, if φg ∈ L2(Ω), then φg ∈ L2(Ω), so +g ∈ D(A∗). +□ +Corollary 2.12. Let Λ be some nonempty index set. The multiplication operator +MI(aλ)λ∈Λ = (λaλ)λ∈Λ, +D(MI) = +� +(aλ)λ∈Λ ∈ l2(Λ) : +� +λ∈Λ +|λ|2|aλ|2 < ∞ +� +, +is self-adjoint. +Proof. Using Lemma 2.11 for the space Λ with the discrete measure and φ(λ) = λ ∈ R, we get that +the operator MI is self-adjoint. +□ +Next, we recall some of the main ideas in Fuglede’s paper [Fug74]. +Definition 2.13. For an open set Ω in Rd, consider the partial differential operators +1 +2πi +∂ +∂xj , j = +1, . . . , d, defined on the space of infinitely differentiable functions with compact support contained +in Ω, C∞ +c (Ω). These operators are symmetric (by integration by parts). +• We say that Ω has the extension property if there are commuting self-adjoint extension +operators Hj, i.e., +1 +2πi +∂ +∂xj +⊆ Hj, +j = 1, . . . , d. +• Commutativity for the extension operators Hj is in the strong sense of spectral resolutions. +More precisely, all projections associated to the spectral resolutions of the operators Hj +must commute. +• A set Ω of finite Lebesgue measure is called spectral if there exists a set of frequencies +Λ ⊂ Rd, such that the family of exponential functions {eλ(x) = e2πiλ·x : λ ∈ Λ} forms an +orthogonal basis for L2(Ω). The set Λ is called a spectrum for the set Ω. +• A set Ω of finite Lebesgue measure is said to tile Rd by translations if there exists a set of +vectors Γ ⊂ Rd such that the translates {Ω + γ : γ ∈ Γ} cover Rd up to measure zero, and +if the intersections (Ω + γ) ∩ (Ω + γ′) have measure zero for γ ̸= γ′ in Γ. +Remark 2.14. In general, when Ω is given, the individual symmetric operators will have self- +adjoint extensions, but the added condition that there is a choice of d mutually commuting self- +adjoint extensions is a strong restriction. For example, if d = 2, and if Ω is a triangle or a disk, +then there will not be commuting self-adjoint extensions (see [Fug74]). This point is clarified in +the next theorem: +Theorem 2.15. [Fug74, Jr82, Ped87, JP92, JP00] Let Ω ⊂ Rd be open and connected, with finite +and positive Lebesgue measure. Then Ω has the extension property if and only if it is a spectral set. +Moreover, with Ω given, there is a one-to-one correspondence between the two sets of subsets: +(2.7) +{Λ ⊂ Rd : Λ is a spectrum for Ω} +and +(2.8) +� +Λ ⊂ Rd : Λ is the joint spectrum of some commutative family +(H1, . . . , Hd) of self-adjoint extensions} . + +THE MOMENTUM OPERATOR ON A UNION OF INTERVALS +7 +This correspondence is determined as follows: +(a) If the extensions (H1, . . . , Hd) are given, then λ ∈ Λ if and only if +(2.9) +eλ ∈ +� +j +D(Hj). +(b) If, conversely, Λ is a spectrum for Ω at the outset, then the ansatz (2.9) and +(2.10) +Hjeλ = λjeλ, +λ ∈ Λ +determine uniquely a set of commuting extensions. +If Ω is only assumed open, the the spectral-set property implies the extension property, but not +conversely. +Conjecture 2.16. The Fuglede Conjecture.[Fug74] A set Ω of finite Lebesgue measure is spec- +tral if and only if it tiles Rd by translations. +Remark 2.17. We note the following conclusions from Theorem 2.15: The link between the +theorem and Conjecture 2.16 is as follows: when commuting self-adjoint extensions exist, then +automatically the joint spectrum is purely discrete, and the corresponding eigenspaces will be one- +dimensional. And they are necessarily orthogonal in L2(Ω) and spanned by Fourier frequencies. +The link to geometry is on account of the fact that, when commuting self-adjoint extensions exist +for a given Ω, then they generate a unitary representation U of Rd, with U acting on L2(Ω). But +locally (i.e., in the interior of Ω), U will then act by translations. +3. Symmetric and self-adjoint extensions +In this section we investigate the symmetric and the self-adjoint extensions of the momentum +operator D on C∞ +c (Ω), where Ω is a union of intervals as in Definition 2.1. +Theorem 3.1. The operator D is symmetric. The adjoint D∗ has domain Dmax as in (2.2) and +(3.1) +D∗f = +1 +2πif ′, for f ∈ D(D∗) = Dmax. +The operator D on C∞ +c (Ω) has a closed extension to D0(Ω) (see (2.1)), and, for f in D0(Ω), we +also have Df = +1 +2πif ′. The adjoint of the operator D|D0(Ω) is the same as the one described above. +The adjoint of D∗ is +(3.2) +� +D|C∞ +c (Ω) +�∗∗ = D|D0(Ω) = D∗|D0(Ω). +Proof. Let g ∈ D(D∗). By definition, this means that, for all f ∈ C∞ +c (Ω), ⟨Df , g⟩ = ⟨f , D∗g⟩ +which means that +1 +2πi +� +Ω +f ′(x)g(x) dx = +� +Ω +f(x)D∗g(x) dx. +Define the function ϕ(x) := +� x +αi D∗g(t) dt, for all x ∈ Ji, i ∈ {1, . . . , n}. Then ϕ is absolutely +continuous and ϕ′(x) = D∗g(x) for almost every x ∈ Ω. Then, using integration by parts, and the +fact that f|∂Ω = 0, we have: +1 +2πi +� +Ω +f ′(x)g(x) dx = +� +Ω +f(x)ϕ′(x) dx = +� +∂Ω +fϕ − +� +Ω +f ′(x)ϕ(x) dx = − +� +Ω +f ′(x)ϕ(x) dx. + +8 +DORIN ERVIN DUTKAY∗ AND PALLE E.T. JORGENSEN +Then +� +Ω +f ′(x) +� 1 +2πig(x) − ϕ(x) +� +dx = 0, for all f ∈ C∞ +c (Ω). +This means that the function ϕ − +1 +2πig is orthogonal to the range of the operator D. +Next, we compute the orthogonal complement of the range of the operator D. Note that, if +f ∈ C∞ +c (Ω), then f|∂Ω = 0, and f(x) = +� x +αi f ′(t) dt for all x ∈ Ji, so +� βi +αi f ′(t) dt = f(βi) = 0. +This implies that, for every function h ∈ L2(Ω) which is constant on each interval Ji, we have +� +Ω Df(x)h(x) dx = 0, so h is orthogonal to the range of D. +Conversely, let h ∈ L2(Ω) be orthogonal to the range of D. Fix i ∈ {1, . . . , n}. We will show +that h has to be constant on Ji. +First, we show that the range of D contains the functions f ∈ C∞ +c (Ji) with +� βi +αi f(t) dt = 0. Let +f be such a function and let ψ(x) = 2πi +� x +αi f(t) dt, for x ∈ Ji, and ψ(x) = 0 otherwise. Then, +ψ ∈ C∞ +c (Ω) and Dϕ = f, thus f is in the range of D. +Take now a function f ∈ L2(Ω), which is zero outside Ji and with +� βi +αi f(t) dt = 0. One can +approximate f in L2 by a sequence of functions fn in C∞ +c (Ji) with +� βi +αi f(t) dt = 0, therefore the +functions fn are in the range of the operator D. It follows that h is orthogonal to the functions fn, +hence to f. +Now take a function ˜f ∈ L2(Ω) which is zero outside Ji. Define f(x) = ˜f(x) − +1 +βi−αi +� βi +αi ˜f(t) dt, +for x ∈ Ji, and f(x) = 0 outside Ji. Then +� βi +αi f(t) dt = 0. Then +� +Ω h(x)f(x) dx = 0, which means +that +� +Ω +� +h(x) − +1 +βi − αi +� βi +αi +h(t) dt +� +˜f(x) dx = +� βi +αi +h(x) ˜f(x) dx − +1 +βi − αi +� βi +αi +h(x) dx · +� βi +αi +˜f(x) dx += +� +Ω +h(x) · +� +˜f(x) − +1 +βi − αi +� βi +αi +˜f(t) dt +� +dx = +� +Ω +h(x)f(x) dx = 0. +Since ˜f is arbitrary, it follows that the function h(x) − +1 +βi−αi +� βi +αi h(t) dt is zero a.e., on each +interval Ji, which means that h is constant a.e., on each interval Ji. +Returning to the computation of the domain of D∗, we obtain that ϕ − +1 +2πig is constant on each +interval Ji. But then since D∗g is in L2(Ω) ⊂ L1(Ω), it follows that ϕ(x) = +� x +αi D∗g dx is absolutely +continuous, on each interval Ji, so g is as well, and +1 +2πig′ = ϕ′ = D∗g a.e. on Ω. +Conversely, if g is absolutely continuous on each interval Ji, and g′ ∈ L2(Ω), then using integra- +tion by parts as above, we have ⟨Df , g⟩ = +� +f , +1 +2πig′� +, and therefore g ∈ D(D∗) and D∗g = +1 +2πig′. +Next, we prove that the operator D on D0(Ω) is closed. Take a sequence {fn} in D0(Ω) which +converges in L2(Ω) to some function f, and such that the sequence {Dfn} converges in L2(Ω) to +some other function +1 +2πig. We want to prove that f is in D0(Ω) and f ′ = g. +Define ϕ(x) = +� x +αi g(t) dt = +� +g , χ(αi,x) +� +, for all x ∈ Ji. Then, for x ∈ Ω, ϕ(x) is the limit of +� +f ′ +n , χ(αi,x) +� += +� x +αi f ′ +n(t) dt = fn(x)−fn(αi) = fn(x). Since {fn} converges to f in L2(Ω), we obtain +that f = ϕ a.e. This implies that f is absolutely continuous, and f ′ = ϕ′ = g a.e. Since ϕ(αi) = 0, +it follows that f(αi) = 0. Also, +f(βi) = ϕ(βi) = +� βi +αi +f ′(t) dt = lim +n +� βi +αi +f ′ +n(t) dt = lim +n (fn(βi) − fn(αi)) = 0. + +THE MOMENTUM OPERATOR ON A UNION OF INTERVALS +9 +To prove that the adjoint, of D|D0(Ω) is as before, the same arguments can be used. +Since +A = D|D0(Ω) is closed, A∗∗ = A, see [Con90, Corollary 1.8, page 305]. +□ +Theorem 3.2. If T is closed symmetric extension of D|C∞ +c (Ω), then there exists a partial isometry +B between subspaces Bl and Br of Cn such that +D(T) = +� +f ∈ Dmax : f(⃗α) ∈ Bl, Bf(⃗α) = f(⃗β) +� +, +and Tf = Df for f ∈ D(T). Conversely, if the unbounded operator T is defined as such, then it is +a closed symmetric extension of D|C∞ +c (Ω). +The adjoint T ∗ has domain +D(T ∗) = +� +f ∈ Dmax : (B∗ ⃗f(β) − ⃗f(α)) ⊥ Bl +� +, +and T ∗f = Df, for f ∈ D(T ∗). +The operator T is a self-adjoint extension if and only if B is unitary, i.e., Bl = Br = Cn. +Proof. Let T be a closed symmetric extension of D. Then, for all f ∈ C∞ +c (Ω), and g ∈ D(T), we +have +⟨f , Tg⟩ = ⟨Tf , g⟩ = ⟨Df , g⟩ . +By Theorem 3.1, this implies that g ∈ Dmax and Tg = D∗g = Dg. Thus D(T) ⊆ Dmax and Tg = Dg, +for all g ∈ D(T), in other words, D on Dmax is an extension of T. +Using integration by parts we have, for all f, g ∈ Dmax: +⟨f , Dg⟩ = +� +Ω +f 1 +2πig′ = − 1 +2πi +� +Ω +fg′ = − 1 +2πi +� +− +� +Ω +f ′g + +n +� +i=1 +(f(βi)g(βi) − f(αi)g(αi)) +� += +1 +2πi +� +Ω +f ′g − +1 +2πi +�� +f(⃗β) , g(⃗β) +� +− ⟨f(⃗α) , g(⃗α)⟩ +� +, +(see the notation in Remark 2.2). Thus, we have +(3.3) +⟨f , Dg⟩ = ⟨Df , g⟩ − +1 +2πi +�� +f(⃗β) , g(⃗β) +� +− ⟨f(⃗α) , g(⃗α)⟩ +� +, for all f, g ∈ Dmax. +Now, if T is a closed symmetric extension of D on C∞ +c (Ω), then, for f, g ∈ D(T), we have, +f, g ∈ Dmax and +0 = ⟨f , Tg⟩ − ⟨Tf , g⟩ = ⟨f , Dg⟩ − ⟨Df , g⟩ = − 1 +2πi +�� +f(⃗β) , g(⃗β) +� +− ⟨f(⃗α) , g(⃗α)⟩ +� +, +therefore +(3.4) +⟨f(⃗α) , g(⃗α)⟩ = +� +f(⃗β) , g(⃗β) +� +, for all f, g ∈ D(T). +Taking f = g in (3.4), we get that ∥f(⃗α)∥2 = ∥f(⃗β)∥2 for all f ∈ D(T); in particular, if f(⃗α) = 0, +then f(⃗β) = 0. This implies that, if f1, f2 ∈ D(T) and f1(⃗α) = f2(⃗α) then (f1 − f2)(⃗α) = 0 so +(f1 − f2)(⃗β) = 0, and f1(⃗β) = f2(⃗β). This means that there exists a well defined function B, +B(f(⃗α)) = f(⃗β), for all f ∈ D(T), +B : Bl → Br, +where +Bl := {f(⃗α) : f ∈ D(T)}, +Br := {f(⃗β) : f ∈ D(T)}. + +10 +DORIN ERVIN DUTKAY∗ AND PALLE E.T. JORGENSEN +In addition B is a linear isometry between the subspaces Bl and Br of Cn, and, by definition, +f(⃗β) = Bf(⃗α), for f ∈ D(T). We can define B to be zero on the orthogonal complement of Bl. +Because of this, we obtain that D(T ) is contained in +DB := +� +f ∈ Dmax : f(⃗α) ∈ Bl, f(⃗β) ∈ Br, Bf(⃗α) = f(⃗β) +� +. +We prove that the reverse inclusion also holds. Let f ∈ DB. Then f(⃗α) ∈ Bl. Then, there +exists f0 ∈ D(T) such that f0(⃗α) = f(⃗α). Then also f0(⃗β) = Bf0(⃗α) = Bf(⃗α) = f(⃗β). Hence +(f − f0)(⃗α) = (f − f0)(⃗β) = 0, and so f − f0 ∈ D0(Ω) ⊆ D(T). Then f = (f − f0) + f0 ∈ D(T). +Assume now, conversely, that we are given an partial isometry B from Bl to Br, and we prove +that D on DB is symmetric and closed. +For symmetry, we use (3.3); for f, g ∈ D(T), we have +⟨f , Dg⟩ = ⟨Df , g⟩ − +1 +2πi +�� +f(⃗β) , g(⃗β) +� +− ⟨f(⃗α) , g(⃗α)⟩ +� += ⟨Df , g⟩ − +1 +2πi (⟨Bf(⃗α) , Bg(⃗α)⟩ − ⟨f(⃗α) , g(⃗α)⟩) = ⟨Df , g⟩ . +To see that the operator is closed, take {fn} in D(T) convergent to f in L2(Ω), and {f ′ +n} +convergent to g in L2(Ω). For i ∈ {1, . . . , n} and x ∈ Ji, we have +fn(x) = fn(αi) + +� x +αi +f ′ +n(t) dt. +On the left hand side {fn} converges to f in L2(Ω); on the right hand side +� x +αi f ′ +n(t) dt converges +to +� x +αi g(t) dt for all x ∈ Ji. Therefore we obtain that {fn(αi)} converges to ci = f(x) − +� x +αi g(t) dt, +for a.e. x. But then f(x) = ci + +� x +αi g(t) dt for a.e. x and therefore f ′ = g a.e., so the operator is +closed. +Consider now the extension T = D on DB. We will compute its adjoint. For g ∈ D(T ∗), we +have that DB ∋ f �→ ⟨Df , g⟩ is bounded. In particular, it is bounded on D0(Ω) so g ∈ Dmax and +T ∗g = Dg, ⟨Df , g⟩ = ⟨f , T ∗g⟩ = ⟨f , Dg⟩. Then, with integration by parts (3.3), we have that +� +f(⃗α) , B∗g(⃗β) +� += +� +Bf(⃗α) , g(⃗β) +� += +� +f(⃗β) , g(⃗β) +� += ⟨f(⃗α) , g(⃗α)⟩ , for all f ∈ DB. +This implies that +� +B∗g(⃗β) − g(⃗α) +� +is orthogonal to the subspace Bl. +Conversely, if g ∈ Dmax and +� +B∗g(⃗β) − g(⃗α) +� +⊥ Bl, then from the previous computation, and +using integration by parts (3.3), we get that, for f ∈ DB, +| ⟨Df , g⟩ | = | ⟨f , Dg⟩ | ≤ ∥f∥∥Dg∥, +and so the linear map DB ∋ f �→ ⟨Df , g⟩ is bounded and g ∈ D(T ∗). +Now, let’s consider the case when the extension T is self-adjoint. In this case D(T ∗) is contained +in D(T), and therefore, if g ∈ Dmax with B∗g(⃗β) − g(⃗α) orthogonal to Bl, then we have that +Bg(⃗α) = g(⃗β). We will prove that Bl must be the entire space Cn. Suppose there exists a non-zero +vector v orthogonal to Bl. Then, since B is a partial isometry, there exists a non-zero vector w +orthogonal to Br. Let g ∈ Dmax such that g(⃗α) = 0 and g(⃗β) = w (for example, make g piecewise +linear on the intervals Ji). Then B∗g(⃗β) − ⃗g(α) = 0 − 0 = 0 ⊥ Bl. Thus g ∈ D(T ∗) = D(T), and + +THE MOMENTUM OPERATOR ON A UNION OF INTERVALS +11 +therefore Bg(⃗α) = g(⃗β), which implies that B(0) = w, a contradiction. Thus, when the extension +is self-adjoint, we get that Br = Cn and Bl = Cn and B is unitary. +For the converse, if B is unitary and Bl = Cn, Br = Cn, then, g ∈ D(T ∗), if and only if +B∗g(⃗β) − g(⃗α) is orthogonal to Bl so it must be zero, i.e., B∗g(⃗β) = g(⃗α), which is equivalent to +Bg(⃗α) = g(⃗β). This means that D(T ∗) = D(T) = DB. +□ +Definition 3.3. For a self-adjoint extension A of the operator D|C∞ +c (Ω) as in Theorem 3.2, we call +the unitary matrix B, the boundary matrix associated to A. +4. Spectral decomposition +Having a self-adjoint extension A of the momentum operator D, we can use the Spectral Theorem +2.6 to obtain a spectral resolution of the self-adjoint operator A. We present in this section an +explicit description of this spectral resolution. +Definition 4.1. For ⃗z = (z1, z2, . . . , zn) ∈ Cn denote by E(⃗z), the n × n diagonal matrix with +entries (e2πiz1, e2πiz2, . . . , e2πizn). +Theorem 4.2. Let A be a self-adjoint extension of the operator D|C∞ +c (Ω) and let B its unitary +boundary matrix. Let P be the spectral measure for the operator A, so +A = +� +R +t dP(t) +Then the spectral measure is atomic, supported on the spectrum +σ(A) = +� +λ ∈ C : det(I − E(λ⃗β)−1BE(λ⃗α)) = 0 +� +⊆ R +which is a discrete unbounded set. For λ ∈ σ(A), the eigenspace P({λ})L2(Ω) has dimension at +most n, and it consists of functions of the form +f(x) = e2πiλx +n +� +i=1 +ciχJi(x), where c = (ci)n +i=1 ∈ Cn, and BE(λ⃗α)c = E(λ⃗β)c. +Proof. We begin with a proposition. +Proposition 4.3. Let A be a self-adjoint extension as in Theorem 4.2. Let λ ∈ C. The following +statements are equivalent: +(i) λ is in the resolvent set of A. +(ii) The operator A − λI is onto. +(iii) The matrix E(λ⃗β)−1BE(λ⃗α) − I is onto. +(iv) The operator A − λI in one-to-one. +(v) The matrix E(λ⃗β)−1BE(λ⃗α) − I is one-to-one. +Proof. We prove that (ii) and (iii) are equivalent. +The operator A − λI is onto, means that for every g ∈ L2(Ω), there exists f ∈ DB = D(A), such +that +1 +2πif ′ − λf = g. + +12 +DORIN ERVIN DUTKAY∗ AND PALLE E.T. JORGENSEN +We solve this first order linear differential equation on each interval Ji of Ω. We have f ′ − 2πiλf = +2πig. Multiplying by the integrating factor e−2πiλx, we get +� +e−2πiλtf(t) +�′ += 2πig(t)e−2πiλt. +Integrating, we get the general solution +f(x) = e2πiλx +� +2πi +� x +αi +g(t)e−2πiλt dt + ci +� +, +for some constant ci, for all x ∈ Ji, and all i ∈ {1, . . . , n}. +Since g is in L2(Ω), we see that f is absolutely continuous and f ′ = 2πi(λf + g) is in L2(Ω), +which means that f is in the domain Dmax, and the only thing that we have to insure is that +B ⃗f(α) = ⃗f(β), by picking the right constants (ci). +Let’s see what the condition B ⃗f(α) = ⃗f(β) means. We have +f(αi) = cie2πiλαi, +f(βi) = e2πiλβi(Ai + ci), +where Ai = 2πi +� βi +αi g(t)e−2πiλt dt. +Let ⃗A = (Ai)n +i=1, ⃗c = (ci)n +i=1. Note that, by varying g, any vector in Cn can be obtained as ⃗A. +Then, the condition f(⃗β) = Bf(⃗α) is equivalent to +E(λ⃗β)( ⃗A + ⃗c) = BE(λ⃗α)⃗c, +or, equivalently, +⃗A = (E(λ⃗β)−1BE(λ⃗α) − I)⃗c. +This shows that, the operator A − λI is onto if and only if the matrix E(λ⃗β)−1BE(λ⃗α) − I is onto. +Next, we prove that (iv) and (v) are equivalent. The operator A−λI is not one-to-one means that +there exists a non-zero f ∈ DB with ⃗f(β) = B ⃗f(α), such that +1 +2πif ′ = λf. Solving this differential +equation on each interval Ji, we obtain that f(x) = cie2πiλx, for x ∈ Ji, for some constant ci. Then, +the relation ⃗f(β) = B ⃗f(α) implies that, (cie2πiλβi)n +i=1 = B(cie2πiλαi)n +i=1; with ⃗c := (ci)n +i=1, this can +be rewritten as (E(λ⃗β)−1BE(λ⃗α) − I)⃗c = 0. Thus the operator A − λI if and only if the matrix +E(λ⃗β)−1BE(λ⃗α) − I is not one-to-one. +Finally, the statements (iii) and (v) are equivalent because they refer to a square matrix in a +finite dimensional space. +Now, since A is self-adjoint, so also closed, λ ∈ ρ(A) if and only if A − λI is both one-to-one and +onto; but these two properties are equivalent, so (i) is equivalent to all the other statements. +□ +Returning to the proof of Theorem 4.2, we see that λ is in the spectrum of A if and only if +det(I − E(λ⃗β)−1BE(λ⃗α)) = 0. This is an analytic function of λ, therefore the zero set is discrete +and at most countable. +From the proof of Proposition 4.3, we see that the eigenspace P({λ})L2(Ω) is as in the statement +of the theorem, and hence has dimension at most n. Since, by the Spectral Theorem the orthogonal +sum of the eigenspaces spans the entire Hilbert space L2(Ω), it follows that σ(A) cannot be finite, +and since it is discrete, it has to be unbounded. +□ + +THE MOMENTUM OPERATOR ON A UNION OF INTERVALS +13 +Theorem 4.4. Let A be as in Theorem 4.2 and let {λn : n ∈ Z} a list of the eigenvalues of A +repeated according to multiplicity. Let {ǫn : n ∈ Z} be an orthonormal basis of eigenvectors for A, +Aǫn = λnǫn, for all n ∈ Z. Then +(4.1) +D(A) = +� +f = +� +n∈Z +fnǫn ∈ L2(Ω) : +� +n∈Z +|λn|2|fn|2 < ∞ +� +. +Proof. Let g = � +n∈Z anǫn ∈ L2(Ω) with � +n∈Z |λn|2|an|2 < ∞. Then, for f ∈ D(A), we have +⟨Df , g⟩ = +� +n +an ⟨Df , ǫn⟩ = +� +n +an ⟨f , D∗ǫn⟩ = +� +n +anλn ⟨f , ǫn⟩ = +� +f , +� +n +λnanǫn +� +. +This implies that g ∈ D(A∗) = D(A) and Ag = A∗g = � +n λnanǫn. +Thus DI := {� +n anǫn : +� +n |λn|2|an|2 < ∞} is contained in D(A). +But, by Corollary 2.12, the diagonal operator MI(� +n anǫn) = � +n λnanǫn defined on DI is +self-adjoint. Since self-adjoints operators are maximally symmetric, it follows that DI = D(A). +□ +5. The unitary group +If we have a self-adjoint extension A of the momentum operator D, we can associate to it a +one-parameter unitary group U(t) = exp(2πitA), t ∈ R as in Theorem 2.9. In this section we +present some basic properties of this unitary group and show that it acts as translations inside the +intervals and it splits points at the endpoints, with probabilities given by the boundary matrix B. +Theorem 5.1. Let A = D on DB a self-adjoint extension with boundary matrix B. Let +U(t) = exp (2πitA), +(t ∈ R), +be the associated one-parameter unitary group. +(i) The domain DB is invariant for U(t) for all t ∈ R, i.e., if f ∈ Dmax with Bf(⃗α) = f(⃗β), +then U(t)f ∈ Dmax with B(U(t)f)(⃗α) = (U(t)f)(⃗β). +(ii) Fix i ∈ {1, . . . , n} and let t ∈ R such that Ji ∩ (Ji − t) ̸= ∅. Then, for f ∈ L2(Ω), +(5.1) +(U(t)f)(x) = f(x + t), for a.e. x ∈ Ji ∩ (Ji − t). +In particular, +(5.2) +(U(βi − x)f)(x) = f(βi), (U(αi − x)f)(x) = f(αi), for f ∈ Dmax, x ∈ Ji. +(iii) For f ∈ Dmax, if x ∈ Ji and t > βi − x, then +(5.3) +[U(t)f] (x) = πi (B [U(t − (βi − x))f] (⃗α)) . +Here πi : Cn → C denotes the projection onto the i-th component π(x1, x2 . . . , xn) = xi. +Proof. (i) This follows from more general rules, see Theorem 2.9(c), but we include a more direct +proof. With the notation as in Theorem 4.4, f = � +n fnǫn is in the domain of A, if and only +if � +n |λn|2|fn|2 < ∞. +Then, for t ∈ R, U(t)f = � +n fne2πiλntǫn, and � +n |λn|2|fne2πiλnt|2 = +� +n |λn|2|fn|2 < ∞, which means that U(t)f is also in the domain of A. + +14 +DORIN ERVIN DUTKAY∗ AND PALLE E.T. JORGENSEN +(ii) Let vλ be an eigenvector for A with eigenvalue λ. Then, by Theorem 4.2, we have +vλ(x) = +n +� +k=1 +ckχJk(x)e2πiλx, +(x ∈ Ω), +for some constants ck ∈ C. Then, for x ∈ Ji ∩ (Ji − t), we have x, x + t ∈ Ji, and +(5.4) +(U(t)vλ)(x) = e2πiλtvλ(x) = +n +� +k=1 +ckχJk(x)e2πiλ(x+t) = vλ(x + t). +Now let f ∈ L2(Ω). One can find a sequence {fn} of finite linear combinations of eigenvectors, +such that lim fn = f in L2(Ω). Then lim U(t)fn = U(t)f, passing to subsequences, we can assume +in addition that {fn} converges to f pointwise a.e. Ω, and U(t)fn converges to U(t)f pointwise +a.e. in Ω. Then, for a.e. x ∈ Ji ∩ (Ji − t), we have +(U(t)f)(x) = lim(U(t)fn)(x) = lim fn(x + t) = f(x + t). +(Note that we used also the fact that translation by t preserves measure zero sets.) +The first relation in (5.2) follows from (5.4), by taking t = βi −x−ǫ and letting ǫ → 0. Similarly +for the second relation. +(iii) Indeed, we have, with (i) and (ii), +[U(t)f] (x) = [U(βi − x)U(t − (βi − x))f] (x) = [U(t − (βi − x))f] (βi) += πi +� +[U(t − (βi − x))f] (⃗β) +� += πi (B [U(t − (βi − x))f] (⃗α)) . +□ +6. Spectral sets +In this section we consider the case when Ω is a spectral set. We present various characterizations +of this property in terms of the self-adjoint extensions of the momentum operator D and in terms +of the associated unitary groups. +Definition 6.1. Assume that Ω is a spectral set with spectrum Λ. Recall that eλ denotes the +exponential function eλ(x) = e2πiλx. In order to make the vectors eλ in L2(Ω) of norm one, we +renormalize the Lebesgue measure on Ω by +1 +|Ω| dx (or we can simply assume that Ω has measure +1). +The Fourier transform (associated to the spectrum Λ) is the unitary operator +(6.1) +FΛ : L2(Ω) → l2(Λ), +FΛf = (⟨f , eλ⟩)λ∈Λ . +Define also the unbounded operator of multiplication by the identity function on l2(Ω): +(6.2) +MI(aλ)λ∈Λ = (λaλ)λ∈Λ, +D(MI) = +� +(aλ)λ∈Λ ∈ l2(Λ) : +� +λ∈Λ +|λ|2|aλ|2 < ∞ +� +. +Define the unitary group associated to Λ on L2(Ω), by +(6.3) +UΛ(t) +�� +λ∈Λ +aλeλ +� += +� +λ∈Λ +e2πiλtaλeλ, for +� +λ∈Λ +aλeλ ∈ L2(Ω), t ∈ R. + +THE MOMENTUM OPERATOR ON A UNION OF INTERVALS +15 +Definition 6.2. A unitary group of local translations on Ω is a strongly continuous one parameter +unitary group U(t) on L2(Ω) with the property that, for any f ∈ L2(Ω) and any t ∈ R, +(6.4) +(U(t)f)(x) = f(x + t) for a.e. x ∈ Ω ∩ (Ω − t). +Remark 6.3. Note the difference between the Definition 6.2, and the property (ii) in Theorem +5.1. The Definition 6.2 is a stronger condition, because it allows jumps between different intervals +Ji of Ω. +Definition 6.4. A unitary boundary matrix B is called spectral if, for every λ ∈ R, the equation +BEλ(⃗α)c = Eλ(⃗β)c, c ∈ Cn has either only the trivial solution c = 0, or only constant solutions of +the form c = α(1, 1, . . . , 1), α ∈ C. +Theorem 6.5. Assume the Ω is a spectral set with spectrum Λ. Define the unbounded operator A +on L2(Ω) by +A +�� +λ∈Λ +fλeλ +� += +� +λ∈Λ +λfλeλ, +D(A) = +� +f = +� +λ∈Λ +fλeλ : +� +λ∈Λ +|λ|2|fλ|2 < ∞ +� +. +Then, the operator A is conjugate to the multiplication operator MI, by the Fourier transform, i.e., +(6.5) +A = F−1 +Λ MIFΛ. +The domain D(A) contains D0(Ω) and all functions eλ, λ ∈ Λ, and A is a self-adjoint extension of +D|C∞ +c (Ω) with the property that all eigenvectors are constant multiples of exponential functions ceλ, +c ∈ C, λ ∈ Λ. +Conversely, if there exists a self-adjoint extension A of D|C∞ +c (Ω) with the property that all eigen- +vectors are constant multiples of exponential functions ceλ, c ∈ C, λ ∈ R, then Ω is spectral, with +spectrum +(6.6) +Λ := {λ ∈ R : eλ ∈ D(A)} = σ(A). +Proof. Equation (6.5) follows from a direct computation. To see that A is self-adjoint, it is enough +to check that MI is self-adjoint, and this is a consequence of Corollary 2.12. +Clearly, the exponential functions eλ are in the domain D(A). Let’s check that D0(Ω) is also +contained in D(A), and Af = Df for f ∈ D0(Ω). Let f = � +λ fλeλ ∈ D0(Ω) and λ ∈ Λ(⊆ R). +Then, +cλ := ⟨Df , eλ⟩ = ⟨f , D∗eλ⟩ = +� +f , +1 +2πie′ +λ +� += ⟨f , λeλ⟩ = λ ⟨f , eλ⟩ = λfλ. +Therefore, � +λ |λ|2|fλ|2 = � +λ |cλ|2 = ∥Df∥2 < ∞, and +Df = +� +λ +λfλeλ = Af. +This shows that A is indeed a self-adjoint extension of D|C∞ +c (Ω). +Now we check that all eigenvectors of A are constant multiples of eλ. This follows from the next +Lemma, and the fact that self-adjoint operators are closed. +Lemma 6.6. Let A be a closed unbounded operator on a Hilbert space H. Assume that there exists +an orthonormal basis of H, {eλ}λ∈Λ, where Λ ⊂ C and Aeλ = λeλ, for all λ ∈ Λ. Then every + +16 +DORIN ERVIN DUTKAY∗ AND PALLE E.T. JORGENSEN +eigenvector f = � +λ fλeλ with the property that � +λ |λ|2|fλ|2 < ∞ is of the form ceλ, for some +c ∈ C, λ ∈ Λ. +Proof. Suppose f = � +λ fλeλ is an eigenvector for A with eigenvalue λ0. Then +Af = λ0f = +� +λ +λ0fλeλ. +On the other hand , we have that � +finite fλeλ converges in L2(Ω) to � +λ fλeλ; also +A + + � +finite +fλeλ + + = +� +finite +λfλeλ → +� +λ +λfλeλ, +because � +λ |λ|2|fλ|2 < ∞. Since A is closed we get +Af = A +�� +λ +fλeλ +� += +� +λ +fλeλ. +This means that λ0fλ = λfλ for all λ ∈ Λ, which implies that either λ0 = λ or fλ = 0. Thus all +the coefficients fλ are zero except for fλ0, so f = fλ0eλ0. +□ +For the converse in the Theorem 6.5, assume A is a self-adjoint extension of D|C∞ +c (Ω) with the +property that all eigenvectors are constant multiples of exponential functions. Then, with Theorem +4.2, for λ in the spectrum σ(A) =: Λ, the subspace P({λ})L2(Ω) is one-dimensional, spanned by eλ. +Then the set of all eigenvectors for A, {eλ : λ ∈ Λ} is an orthonormal basis for L2(Ω). Moreover, +if eλ ∈ D(A) for some λ ∈ R, then Aeλ = +1 +2πie′ +λ = λeλ, so (6.6) holds. +□ +Theorem 6.7. Assume that Ω is spectral with spectrum Λ. Let A = AΛ be the self-adjoint extension +of D|C∞ +c (Ω) defined in Theorem 6.5, and let B = BΛ be the unitary boundary matrix associated to +this extension as in Theorem 3.2. Then B is a spectral boundary matrix and it is uniquely and +well-defined by the conditions +(6.7) +Beλ(⃗α) = eλ(⃗β), for all λ ∈ Λ. +Moreover +(6.8) +span{eλ(⃗α) : λ ∈ Λ} = span{eλ(⃗β) : λ ∈ Λ} = Cn. +Conversely, if there exists a spectral boundary matrix B, then Ω is spectral with spectrum +(6.9) +Λ = {λ ∈ R : Beλ(⃗α) = eλ(⃗β)}. +Proof. By Theorem 6.5, the only eigenvectors of the operator A are multiples of eλ, λ ∈ Λ. On the +other hand, by Theorem 4.2, the eigenvectors are functions of the form +f = +n +� +i=1 +(ciχJi) eλ, +where c ∈ Cn, with BEλ(⃗α)c = Eλ(⃗β)c. Thus, if BEλ(⃗α)c = Eλ(⃗β)c for some non-zero c ∈ Cn, +then f is an eigenvector, but then it must be a constant multiple of eλ so all the components of c +are the same. This means that B is a spectral boundary matrix. + +THE MOMENTUM OPERATOR ON A UNION OF INTERVALS +17 +We check now that the relation (6.7) completely determines B as a well-defined unitary matrix. +Since Λ is a spectrum, for λ ̸= λ′ in Λ we have +0 = ⟨eλ , eλ′⟩ = +n +� +k=1 +� βk +αk +e2πi(λ−λ′)x dx = +n +� +k=1 +1 +2πi(λ − λ′) +� +e2πi(λ−λ′)βk − e2πi(λ−λ′)αk +� +, +which means that +(6.10) +⟨eλ(⃗α) , eλ′(⃗α)⟩ = +� +eλ(⃗β) , eλ′(⃗β) +� +, for all λ, λ′ ∈ Λ. +Define the linear operator B from span{eλ(⃗α) : λ ∈ Λ} to span{eλ(⃗β) : λ ∈ Λ}, by +B +�� +λ∈Λ +aλeλ(⃗α) +� += +� +λ∈Λ +aλeλ(⃗β), +where only finitely many coefficients are non-zero. +The operator is well-defined, because, if � +λ aλeλ(⃗α) = 0, then +0 = +�� +λ +aλeλ(⃗α) , +� +λ +aλeλ(⃗α) +� += +� +λ,λ′ +aλaλ′ ⟨eλ(⃗α) , eλ′(⃗α)⟩ += +� +λ,λ′ +aλaλ′ +� +eλ(⃗β) , eλ′(⃗β) +� += +�� +λ +aλeλ(⃗β) , +� +λ +aλeλ(⃗β) +� +A similar argument shows that B is unitary. +Next we prove that Bl := span{eλ(⃗α) : λ ∈ Λ} = Cn. Since B is unitary, the same will be true for +⃗β. We proceed by contradiction, if Bl is not the entire space Cn then it has a non-trivial orthogonal +complement, and therefore we can extend the partial isometry B, in two different ways to unitaries +˜B and ˜B′. Both of them give rise, by Theorem 3.2, to self-adjoint extensions of D|C∞ +c (Ω), and, since +Beλ(⃗α) = eλ(⃗β), we have eλ ∈ D ˜B ∩ D ˜B′. Then, with Lemma 6.6, all eigenvectors are multiples of +eλ, λ ∈ Λ, and with Theorem 4.4, we get that +D ˜B = +�� +λ +fλeλ : +� +λ +|λ|2|fλ|2 < ∞ +� += D ˜B′. +But this means that ˜B = ˜B′, a contradiction, and (6.8) follows. +For the converse, if a spectral unitary boundary matrix is given, then the self-adjoint extension +associated to it has all eigenvectors of the form ceλ, with c ∈ C and λ ∈ Λ as in (6.9), and they +form an orthonormal basis. Thus Ω is spectral. +□ +Theorem 6.8. Assume that Ω is spectral with spectrum Λ. +Then the unitary group U = UΛ +associated to Λ is a unitary group of local translations. In addition, if A is the self-adjoint extension +associated to Λ as in Theorem 6.5, then +(6.11) +U(t) = exp(2πitA), +(t ∈ R). +Conversely, if there exists a unitary group of local translations (U(t))t∈R for Ω, then Ω is spectral. + +18 +DORIN ERVIN DUTKAY∗ AND PALLE E.T. JORGENSEN +Proof. Note first that U(t)eλ = e2πiλteλ, for all λ ∈ Λ and t ∈ R. Then, for t ∈ R and x ∈ Ω∩(Ω−t), +we have +(U(t)eλ)(x) = e2πiλte2πiλx = e2πiλ(x+t) = eλ(x + t). +Now fix t ∈ R and let f ∈ L2(Ω). We want to check (6.4). Since {eλ : λ ∈ Λ} form an orthonormal +basis for L2(Ω), we can find a sequence of functions {fn} which are finite linear combinations of +function eλ, such that {fn} converges to f in L2(Ω). Passing to a subsequence, we can assume +that {fn} converges to f almost everywhere. Since U(t) is unitary, {U(t)fn} converges to U(t)f in +L2(Ω), and again passing to a subsequence we can assume in addition that {U(t)fn} converges to +U(t)f almost everywhere. +We have (U(t)fn)(x) = fn(x+t) for a.e. x ∈ Ω∩(Ω−t), for all n. Taking the limit (U(t)f)(x) = +f(x + t) for a.e x ∈ Ω ∩ (Ω − t). This proves (6.4) so U is a unitary group of local translations. +The relation (6.11) follows immediately, because we have the operators A and U(t) in diagonal +form. +Assume now that U(t) is a unitary group of local translations. By Stone’s Theorem 2.10, there +exists a self-adjoint operator A such that U(t) = exp(2πitA). We claim that A is a self-adjoint +extension of D|C∞ +c (Ω). We will prove that, for f ∈ C∞ +c (Ω), +(6.12) +1 +2πit(U(t)f − f) converges in L2(Ω) to Df as t → 0, +which, by Theorem 2.9(b) and (c), implies that f is in the domain of A and Af = Df. This is to +be expected, since, especially for small values of t, the operator U(t) acts as a translation. +We need a Lemma. +Lemma 6.9. Assume that the function f ∈ L2(Ω) is supported on the set Ωǫ = ∪n +i=1[αi + ǫ, βi − ǫ] +for some ǫ > 0, and that |t| < ǫ. Then +(6.13) +(U(t)f)(x) = f(x + t) for a.e. x ∈ Ω, +where f(x) := 0 for x not in Ω. +Proof. If |t| < ǫ, then Ωǫ−t ⊂ (Ω∩(Ω−t)), and therefore (U(t)f)(x) = f(x+t), for a.e. x ∈ Ωǫ −t. +We prove that g(x) := (U(t)f)(x) = 0 for x ∈ Ω \ (Ωǫ − t). We have +∥f∥2 +L2(Ω) = ∥U(t)f∥2 +L2(Ω) = +� +Ωǫ−t +|f(x + t)|2 dx + +� +Ω\(Ωǫ−t) +|g(x)|2 dx += +� +Ωǫ +|f(x)|2 dx + +� +Ω\(Ωǫ−t) +|g(x)|2 dx = ∥f∥2 +L2(Ω) + ∥g∥2 +L2(Ω). +This implies that g is 0 and we obtain the Lemma. +□ +Take now f ∈ C∞ +c (Ω). This means that f is supported on a set Ωǫ for some ǫ > 0. Using Lemma +6.9, for |t| < ǫ and for a.e. x ∈ Ω, we have +1 +2πit((U(t)f)(x) − f(x)) = +1 +2πit(f(x + t) − f(x)), +which converges uniformly to +1 +2πif ′(x) (since f ∈ C∞ +c (Ω)) . Then (6.12) follows and therefore A is +a self-adjoint extension of D|C∞ +c (Ω). +With Theorem 4.2, the eigenvectors for A are of the form f = (�n +i=1 ciχJi) eλ, Af = λf. Then +U(t)f = e2πitλf. + +THE MOMENTUM OPERATOR ON A UNION OF INTERVALS +19 +Since U(t) is a unitary group of local translations, for a.e. x ∈ Ω ∩ (Ω − t), we have +e2πitλ +� n +� +i=1 +ciχJi(x) +� +eλ(x) = (U(t)f)(x) = f(x + t) = +� n +� +i=1 +ciχJi(x + t) +� +eλ(x + t). +Fix i ̸= k in {1, . . . , n} and choose t such that Ji ∩ (Jk − t) ̸= ∅ and then pick x ∈ Ji ∩ (Jk − t) +such that the previous relation holds. Then, we get +e2πitλcie2πiλx = cke2πiλ(x+t). +This means that ci = ck, and thus f = c1eλ. Therefore, all the eigenvectors of A are multiples of +eλ, and, by Theorem 6.5, it follows that Ω is spectral. +□ +Concluding remarks. 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Spectral theory of commuting selfadjoint partial differential operators. J. Funct. Anal., +73(1):122–134, 1987. +[PL22] +Alberto Debernardi Pinos and Nir Lev. Gabor orthonormal bases, tiling and periodicity. Math. Ann., +384(3-4):1461–1467, 2022. +[PPTW15] Steen Pedersen, Jason D. Phillips, Feng Tian, and Cody E. Watson. On the spectra of momentum +operators. Complex Anal. Oper. Theory, 9(7):1557–1587, 2015. +[Tao04] +Terence Tao. Fuglede’s conjecture is false in 5 and higher dimensions. Math. Res. Lett., 11(2-3):251–258, +2004. +[Dorin Ervin Dutkay] University of Central Florida, Department of Mathematics, 4000 Central +Florida Blvd., P.O. Box 161364, Orlando, FL 32816-1364, U.S.A., +Email address: Dorin.Dutkay@ucf.edu +[Palle E.T. Jorgensen]University of Iowa, Department of Mathematics, 14 MacLean Hall, Iowa +City, IA 52242-1419, +Email address: palle-jorgensen@uiowa.edu + diff --git a/z9FIT4oBgHgl3EQf2isN/content/tmp_files/load_file.txt b/z9FIT4oBgHgl3EQf2isN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f29c6b2c41ba837ea146793fe3e5ae1e734de596 --- /dev/null +++ b/z9FIT4oBgHgl3EQf2isN/content/tmp_files/load_file.txt @@ -0,0 +1,928 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf,len=927 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='11377v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='SP] 26 Jan 2023 THE MOMENTUM OPERATOR ON A UNION OF INTERVALS AND THE FUGLEDE CONJECTURE DORIN ERVIN DUTKAY∗ AND PALLE E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' JORGENSEN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The purpose of the present paper is to place a number of geometric (and hands- on) configurations relating to spectrum and geometry inside a general framework for the Fuglede conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Note that in its general form, the Fuglede conjecture concerns general Borel sets Ω in a fixed number of dimensions d such that Ω has finite positive Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The conjecture proposes a correspondence between two properties for Ω, one takes the form of spectrum, while the other refers to a translation-tiling property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We focus here on the case of dimension one, and the connections between the Fuglede conjecture and properties of the self-adjoint extensions of the momentum operator 1 2πi d dx, realized in L2 of a union of intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Notations and preliminaries 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Symmetric and self-adjoint extensions 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Spectral decomposition 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The unitary group 13 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Spectral sets 14 References 19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Introduction For bounded open domains Ω in Rd, the Fuglede problem deals with two properties that Ω may or may not have, one (called spectral) is relative to the Hilbert space L2(Ω), the question of whether L2(Ω) has an orthogonal d-variable Fourier basis, and the other is geometric (tiling), whether Ω tiles Rd by some set of translation vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The original problem asked whether the two properties are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' In this paper we show how tools from operator theory (especially choices of spectral representa- tions for unbounded operators), serve to link the two sides of the problem, spectrum vs geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Since the inception, stated this way, the Fuglede conjecture is now known to be negative, more precisely that the two properties are not equivalent in dimension 3 and higher [Tao04, Mat05, KM06b, KM06a, FMM06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Nonetheless, the Fuglede problem is even open for d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' 47E05,42A16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' momentum operator, self-adjoint extension, Fourier bases, Fuglede conjecture .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' 1 2 DORIN ERVIN DUTKAY∗ AND PALLE E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' JORGENSEN Parallel to this we note that there are many closely related new research directions, including analysis on fractals, which deal with various notions of interplay between spectral theoretic prop- erties on the one hand, and geometry on the other, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', direct problems and inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We further stress that the original formulation was stated in terms of properties for the set of d partial derivative operators for the coordinate directions in Ω, specifically the possible extensions of partial derivative operators in the form of commuting generators for unitary one-parameter groups acting in L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Such extensions are known to necessarily be local translation generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Moreover, following quantum theory, such generators may be viewed as momentum operators, a viewpoint motivated by the canonical duality from quantum mechanics for momentum and position observ- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This formulation in turn makes a direct connection to scattering theoretic properties, again related to the Hilbert space L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' And in this form, the problem is of interest even for d = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' and so the case when Ω is a union of intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Continuing earlier work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', [DJ15]) we aim here at presenting new results for the d = 1 Fuglede problem, and making the presentation as self-contained as possible, for the readers who might not be experts in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Starting with its classical roots, the Fuglede problem addresses two related properties for bounded domains Ω in Rd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' More precisely, Fuglede’s question asks for a specific linking between multivari- able spectra on one side, and geometry of Ω on the other (spectral vs tiling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' But by now, the Fuglede problem/conjecture has become distinctly interdisciplinary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' It has come to encompass a diverse variety of neighboring fields of mathematics each of which in turn lies at the cross- roads of at least the following six separate disciplines: (i) harmonic analysis, (ii) spectral and scattering theory for operators in Hilbert space, (iii) metric/convex geometry, (iv) fractals, (v) operator algebras, and (vi) representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' For the benefit of readers, we include cita- tions to the following list of papers, each dealing with one or the other of the above six areas, [Jor18, Bir22, LM22, CNo22, KS21, Mat20, GL20, FS20, GL17, IMP17, Hak10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The paper is organized as follows: in Section 1, we introduce some definitions related to un- bounded symmetric operators and their extensions, the associated one-parameter unitary group, and we recall Fuglede’s result and conjecture, which serve as the main motivation for our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' In the next sections we focus on the case when the set Ω is a finite union of intervals in dimension d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' In Section 2, we study the symmetric and the self-adjoint extensions A of the momentum operator D = 1 2πi d dx on the space C∞ c (Ω) of infinitely differentiable functions with compact support in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' In Section 3, we describe the spectral decomposition of such self-adjoint extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Section 4 is devoted to the one-parameter unitary group U(t) = exp(2πitA), t ∈ R, which acts as transla- tions inside the intervals of Ω and has a different behavior at the end-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' In Section 5, we make the connections between the existence of orthogonal Fourier bases on Ω and the properties of the self-adjoint extensions A or of the unitary group U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Notations and preliminaries When d = 1, with a choice of an open set Ω, the corresponding connected components will then be intervals, and so Ω takes the form of a finite union of intervals as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let Ω = n � i=1 (αi, βi), where − ∞ < α1 < β1 < α2 < β2 < · · · < αn < βn < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' THE MOMENTUM OPERATOR ON A UNION OF INTERVALS 3 So Ω = n� i=1 Ji, where Ji = (αi, βi) for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' On Ω we consider the Lebesgue measure dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We denote by ∂Ω the boundary of Ω, ∂Ω = {αi, βi : i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , n}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' For a function f on ∂Ω we use the notation � ∂Ω f = n � i=1 (f(βi) − f(αi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Consider the subspace of infinitely differentiable compactly supported functions on Ω, C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We define the differential/momentum operator D on C∞ c (Ω): Df = 1 2πif ′, (f ∈ C∞ c (Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Define also the subspaces (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='1) D0(Ω) = {f : Ω → C : f is absolutely continuous on each Ji, f(αi+) = f(βi−) = 0 for all i and f ′ ∈ L2(Ω) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2) Dmax = � f : Ω → C : f is absolutely continuous on each interval Ji and f ′ ∈ L2(Ω) � , Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' If a function f is absolutely continuous on each interval Ji and f ′ ∈ L2(Ω), then the values of the function f at the endpoints αi and βi are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Indeed, fix i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , n} and a point x0 ∈ (αi, βi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, since f is absolutely continuous and with f ′ ∈ L2(Ω) ⊂ L1(Ω), one has f(x) = f(x0) + � x x0 f ′(t) dt, for all x ∈ (αi, βi), and therefore f(αi+) = f(x0) − � x0 αi f ′(t) dt and, f(βi−) = f(x0) + � βi x0 f ′(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This means that we can define f(αi) := f(αi+), and similarly f(βi) by continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' For f ∈ Dmax, we denote by f(⃗α) = (f(α1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , f(αn)) and similarly for f(⃗β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We are looking for closed symmetric and for self-adjoint extensions of the operator D on C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We recall some notions about unbounded linear operators, see for example [Con90, Chapter X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let H be a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We denote by B(H) the set of bounded linear operators on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We denote the domain of an unbounded operator T by D(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' An operator T2 is an extension of T1 if D(T2) contains D(T1) and T2f = T1f, for all f ∈ D(T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We write T1 ⊆ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' An operator T is closed if its graph is closed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', if {fn} in D(T) converges to f and {Tfn} converges to g then f ∈ D(T) and Tf = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' 4 DORIN ERVIN DUTKAY∗ AND PALLE E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' JORGENSEN For a densely defined unbounded operator T : H1 → H2, the adjoint operator T ∗ : H2 → H1 is defined on the set of vectors g ∈ H2 with the property that the linear functional D(T) ∋ f �→ ⟨Tf , g⟩ is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' In this case, by Riesz’s lemma, there exists a unique element T ∗g ∈ H1 such that ⟨Tf , g⟩ = ⟨f , T ∗g⟩ for all f ∈ D(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' An densely defined operator is called symmetric if ⟨Tf , g⟩ = ⟨f , Tg⟩ , for all f, g ∈ D(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' An operator is called self-adjoint if D(T ∗) = D(T) and T ∗f = Tf for all f ∈ D(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' An operator N is called normal if it is closed, densely defined and N ∗N = NN ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Recall that a (possibly unbounded) linear operator T : H → K is called boundedly invertible if there is a bounded operator S : K → H such that TS = I and ST ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The resolvent set ρ(T) for the operator T is defined by ρ(T) = {λ ∈ C : λI − T is boundedly invertible}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The spectrum of T is defined as σ(T) = C \\ ρ(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' If X is a set, B is a σ-algebra of subsets of X, and H is a Hilbert space, a spectral measure/spectral resolution for (X, B, H) is a function E : B → B(H) such that: (a) for each ∆ in B, E(∆) is a projection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (b) E(∅) = 0 and E(X) = IH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (c) E(∆1 ∩ ∆2) = E(∆1)E(∆2) for ∆1 and ∆2 in B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (d) if {∆n}∞ n=1 are pairwise disjoint sets from B, then E � ∞ � n=1 ∆n � = ∞ � n=1 E(∆n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Spectral theorem for unbounded self-adjoint operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' If A is a self- adjoint operator on H, then there exists a unique spectral measure E defined on the Borel subsets of R such that (a) A = � x dE(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (b) E(∆) = 0 if ∆ ∩ σ(A) = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (c) if U is an open subset of R and U ∩ σ(A) ̸= ∅, then E(U) ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (d) if B ∈ B(H) such that BA ⊆ AB, then B( � φ dE) ⊆ ( � φ dE)B for every Borel function φ on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Note that, starting with a fixed selfadjoint operator A in a Hilbert space H, the corresponding projection valued measure E then induces a spectral representation for A via a system of scalar measures indexed by the vectors in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' These are defined for h ∈ H by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='3) Eh,h(∆) = ⟨E(∆)h , h⟩ , for all Borel sets ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let A be a self-adjoint operator in a Hilbert space H with spectral resolution E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then the following three conclusions follow immediately : (i) If ϕ : R → C is measurable, then the operator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='4) ϕ(A) := � R ϕ(x) dE(x) is well defined in H and normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' THE MOMENTUM OPERATOR ON A UNION OF INTERVALS 5 (ii) The dense domain of ϕ(A), denoted D(ϕ(A)), consists of all vectors h ∈ H such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='5) � R |ϕ(x)|2 dEh,h < ∞, (iii) For all h ∈ D(ϕ(A)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='6) ∥ϕ(A)h∥2 = � R |ϕ(x)|2 dEh,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' A strongly continuous one parameter unitary group is a function U : R → B(H) such that for all s and t in R: (a) U(t) is a unitary operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (b) U(s + t) = U(s)U(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (c) if h ∈ H and t0 ∈ R then U(t)h → U(t0)h as t → t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let A be a self-adjoint operator on H and let E on (X, B, H) be its spectral measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Define U(t) = exp(2πitA) = � e2πitx dE(x), (t ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then (a) (U(t))t∈R is a strongly continuous one parameter group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (b) if h ∈ D(A), then lim t→0 1 t (U(t)h − h) = 2πiAh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (c) if h ∈ H and limt→0 1 t (U(t)h − h) exists, then h ∈ D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Consequently, D(A) is invariant under U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Stone’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' If U is a strongly continuous one parameter unitary group, then there exists a self-adjoint operator A such that U(t) = exp(2πitA), t ∈ R, and conversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The self-adjoint operator A is called the infinitesimal generator for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We will also need the following well-known lemma about multiplication operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let (X, µ) be a σ-finite measure space and let φ : X → C be a measurable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let D = {f ∈ L2(µ) : φf ∈ L2(µ)} and define Af = φf for all f ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then A is a closed operator, D(A∗) = D, and A∗f = φf for f ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' In particular, if f is real-valued, then A is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' First, the domain D is dense, because one can consider functions of the form χ{x∈M:−n≤Re(φ(x))≤n,−m≤Im(φ(x))≤m}, m, n ∈ N, M ⊆ X measurable, and they are in the domain D and span L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Thus the operator A is densely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Consider now g ∈ D(A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then the map D ∋ f �→ ⟨Af , g⟩ =: ϕg(f) is a bounded linear functional, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', ���� � φfg dµ ���� ≤ C∥f∥, (f ∈ D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This implies that ϕg can be extended continuously to the whole space L2(Ω) and therefore there exists an element A∗g ∈ L2(Ω) such that ϕg(f) = ⟨f , A∗g⟩ for all f ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then � fφg dµ = � f · A∗g dµ, (f ∈ D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' 6 DORIN ERVIN DUTKAY∗ AND PALLE E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' JORGENSEN But this means that A∗g = φg a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Conversely, if g ∈ D we want to see that g ∈ D(A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' But, if φg ∈ L2(Ω), then φg ∈ L2(Ω), so g ∈ D(A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let Λ be some nonempty index set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The multiplication operator MI(aλ)λ∈Λ = (λaλ)λ∈Λ, D(MI) = � (aλ)λ∈Λ ∈ l2(Λ) : � λ∈Λ |λ|2|aλ|2 < ∞ � , is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='11 for the space Λ with the discrete measure and φ(λ) = λ ∈ R, we get that the operator MI is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' □ Next, we recall some of the main ideas in Fuglede’s paper [Fug74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' For an open set Ω in Rd, consider the partial differential operators 1 2πi ∂ ∂xj , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , d, defined on the space of infinitely differentiable functions with compact support contained in Ω, C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' These operators are symmetric (by integration by parts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We say that Ω has the extension property if there are commuting self-adjoint extension operators Hj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', 1 2πi ∂ ∂xj ⊆ Hj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Commutativity for the extension operators Hj is in the strong sense of spectral resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' More precisely, all projections associated to the spectral resolutions of the operators Hj must commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' A set Ω of finite Lebesgue measure is called spectral if there exists a set of frequencies Λ ⊂ Rd, such that the family of exponential functions {eλ(x) = e2πiλ·x : λ ∈ Λ} forms an orthogonal basis for L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The set Λ is called a spectrum for the set Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' A set Ω of finite Lebesgue measure is said to tile Rd by translations if there exists a set of vectors Γ ⊂ Rd such that the translates {Ω + γ : γ ∈ Γ} cover Rd up to measure zero, and if the intersections (Ω + γ) ∩ (Ω + γ′) have measure zero for γ ̸= γ′ in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' In general, when Ω is given, the individual symmetric operators will have self- adjoint extensions, but the added condition that there is a choice of d mutually commuting self- adjoint extensions is a strong restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' For example, if d = 2, and if Ω is a triangle or a disk, then there will not be commuting self-adjoint extensions (see [Fug74]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This point is clarified in the next theorem: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' [Fug74, Jr82, Ped87, JP92, JP00] Let Ω ⊂ Rd be open and connected, with finite and positive Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then Ω has the extension property if and only if it is a spectral set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Moreover, with Ω given, there is a one-to-one correspondence between the two sets of subsets: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='7) {Λ ⊂ Rd : Λ is a spectrum for Ω} and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='8) � Λ ⊂ Rd : Λ is the joint spectrum of some commutative family (H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , Hd) of self-adjoint extensions} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' THE MOMENTUM OPERATOR ON A UNION OF INTERVALS 7 This correspondence is determined as follows: (a) If the extensions (H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , Hd) are given, then λ ∈ Λ if and only if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='9) eλ ∈ � j D(Hj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (b) If, conversely, Λ is a spectrum for Ω at the outset, then the ansatz (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='10) Hjeλ = λjeλ, λ ∈ Λ determine uniquely a set of commuting extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' If Ω is only assumed open, the the spectral-set property implies the extension property, but not conversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The Fuglede Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' [Fug74] A set Ω of finite Lebesgue measure is spec- tral if and only if it tiles Rd by translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We note the following conclusions from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='15: The link between the theorem and Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='16 is as follows: when commuting self-adjoint extensions exist, then automatically the joint spectrum is purely discrete, and the corresponding eigenspaces will be one- dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' And they are necessarily orthogonal in L2(Ω) and spanned by Fourier frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The link to geometry is on account of the fact that, when commuting self-adjoint extensions exist for a given Ω, then they generate a unitary representation U of Rd, with U acting on L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' But locally (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', in the interior of Ω), U will then act by translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Symmetric and self-adjoint extensions In this section we investigate the symmetric and the self-adjoint extensions of the momentum operator D on C∞ c (Ω), where Ω is a union of intervals as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The operator D is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The adjoint D∗ has domain Dmax as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='1) D∗f = 1 2πif ′, for f ∈ D(D∗) = Dmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The operator D on C∞ c (Ω) has a closed extension to D0(Ω) (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='1)), and, for f in D0(Ω), we also have Df = 1 2πif ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The adjoint of the operator D|D0(Ω) is the same as the one described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The adjoint of D∗ is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2) � D|C∞ c (Ω) �∗∗ = D|D0(Ω) = D∗|D0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let g ∈ D(D∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' By definition, this means that, for all f ∈ C∞ c (Ω), ⟨Df , g⟩ = ⟨f , D∗g⟩ which means that 1 2πi � Ω f ′(x)g(x) dx = � Ω f(x)D∗g(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Define the function ϕ(x) := � x αi D∗g(t) dt, for all x ∈ Ji, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then ϕ is absolutely continuous and ϕ′(x) = D∗g(x) for almost every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, using integration by parts, and the fact that f|∂Ω = 0, we have: 1 2πi � Ω f ′(x)g(x) dx = � Ω f(x)ϕ′(x) dx = � ∂Ω fϕ − � Ω f ′(x)ϕ(x) dx = − � Ω f ′(x)ϕ(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' 8 DORIN ERVIN DUTKAY∗ AND PALLE E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' JORGENSEN Then � Ω f ′(x) � 1 2πig(x) − ϕ(x) � dx = 0, for all f ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This means that the function ϕ − 1 2πig is orthogonal to the range of the operator D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Next, we compute the orthogonal complement of the range of the operator D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Note that, if f ∈ C∞ c (Ω), then f|∂Ω = 0, and f(x) = � x αi f ′(t) dt for all x ∈ Ji, so � βi αi f ′(t) dt = f(βi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This implies that, for every function h ∈ L2(Ω) which is constant on each interval Ji, we have � Ω Df(x)h(x) dx = 0, so h is orthogonal to the range of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Conversely, let h ∈ L2(Ω) be orthogonal to the range of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Fix i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We will show that h has to be constant on Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' First, we show that the range of D contains the functions f ∈ C∞ c (Ji) with � βi αi f(t) dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let f be such a function and let ψ(x) = 2πi � x αi f(t) dt, for x ∈ Ji, and ψ(x) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, ψ ∈ C∞ c (Ω) and Dϕ = f, thus f is in the range of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Take now a function f ∈ L2(Ω), which is zero outside Ji and with � βi αi f(t) dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' One can approximate f in L2 by a sequence of functions fn in C∞ c (Ji) with � βi αi f(t) dt = 0, therefore the functions fn are in the range of the operator D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' It follows that h is orthogonal to the functions fn, hence to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Now take a function ˜f ∈ L2(Ω) which is zero outside Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Define f(x) = ˜f(x) − 1 βi−αi � βi αi ˜f(t) dt, for x ∈ Ji, and f(x) = 0 outside Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then � βi αi f(t) dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then � Ω h(x)f(x) dx = 0, which means that � Ω � h(x) − 1 βi − αi � βi αi h(t) dt � ˜f(x) dx = � βi αi h(x) ˜f(x) dx − 1 βi − αi � βi αi h(x) dx · � βi αi ˜f(x) dx = � Ω h(x) · � ˜f(x) − 1 βi − αi � βi αi ˜f(t) dt � dx = � Ω h(x)f(x) dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Since ˜f is arbitrary, it follows that the function h(x) − 1 βi−αi � βi αi h(t) dt is zero a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', on each interval Ji, which means that h is constant a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', on each interval Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Returning to the computation of the domain of D∗, we obtain that ϕ − 1 2πig is constant on each interval Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' But then since D∗g is in L2(Ω) ⊂ L1(Ω), it follows that ϕ(x) = � x αi D∗g dx is absolutely continuous, on each interval Ji, so g is as well, and 1 2πig′ = ϕ′ = D∗g a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Conversely, if g is absolutely continuous on each interval Ji, and g′ ∈ L2(Ω), then using integra- tion by parts as above, we have ⟨Df , g⟩ = � f , 1 2πig′� , and therefore g ∈ D(D∗) and D∗g = 1 2πig′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Next, we prove that the operator D on D0(Ω) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Take a sequence {fn} in D0(Ω) which converges in L2(Ω) to some function f, and such that the sequence {Dfn} converges in L2(Ω) to some other function 1 2πig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We want to prove that f is in D0(Ω) and f ′ = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Define ϕ(x) = � x αi g(t) dt = � g , χ(αi,x) � , for all x ∈ Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, for x ∈ Ω, ϕ(x) is the limit of � f ′ n , χ(αi,x) � = � x αi f ′ n(t) dt = fn(x)−fn(αi) = fn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Since {fn} converges to f in L2(Ω), we obtain that f = ϕ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This implies that f is absolutely continuous, and f ′ = ϕ′ = g a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Since ϕ(αi) = 0, it follows that f(αi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Also, f(βi) = ϕ(βi) = � βi αi f ′(t) dt = lim n � βi αi f ′ n(t) dt = lim n (fn(βi) − fn(αi)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' THE MOMENTUM OPERATOR ON A UNION OF INTERVALS 9 To prove that the adjoint, of D|D0(Ω) is as before, the same arguments can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Since A = D|D0(Ω) is closed, A∗∗ = A, see [Con90, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='8, page 305].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' If T is closed symmetric extension of D|C∞ c (Ω), then there exists a partial isometry B between subspaces Bl and Br of Cn such that D(T) = � f ∈ Dmax : f(⃗α) ∈ Bl, Bf(⃗α) = f(⃗β) � , and Tf = Df for f ∈ D(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Conversely, if the unbounded operator T is defined as such, then it is a closed symmetric extension of D|C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The adjoint T ∗ has domain D(T ∗) = � f ∈ Dmax : (B∗ ⃗f(β) − ⃗f(α)) ⊥ Bl � , and T ∗f = Df, for f ∈ D(T ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The operator T is a self-adjoint extension if and only if B is unitary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', Bl = Br = Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let T be a closed symmetric extension of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, for all f ∈ C∞ c (Ω), and g ∈ D(T), we have ⟨f , Tg⟩ = ⟨Tf , g⟩ = ⟨Df , g⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='1, this implies that g ∈ Dmax and Tg = D∗g = Dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Thus D(T) ⊆ Dmax and Tg = Dg, for all g ∈ D(T), in other words, D on Dmax is an extension of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Using integration by parts we have, for all f, g ∈ Dmax: ⟨f , Dg⟩ = � Ω f 1 2πig′ = − 1 2πi � Ω fg′ = − 1 2πi � − � Ω f ′g + n � i=1 (f(βi)g(βi) − f(αi)g(αi)) � = 1 2πi � Ω f ′g − 1 2πi �� f(⃗β) , g(⃗β) � − ⟨f(⃗α) , g(⃗α)⟩ � , (see the notation in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Thus, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='3) ⟨f , Dg⟩ = ⟨Df , g⟩ − 1 2πi �� f(⃗β) , g(⃗β) � − ⟨f(⃗α) , g(⃗α)⟩ � , for all f, g ∈ Dmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Now, if T is a closed symmetric extension of D on C∞ c (Ω), then, for f, g ∈ D(T), we have, f, g ∈ Dmax and 0 = ⟨f , Tg⟩ − ⟨Tf , g⟩ = ⟨f , Dg⟩ − ⟨Df , g⟩ = − 1 2πi �� f(⃗β) , g(⃗β) � − ⟨f(⃗α) , g(⃗α)⟩ � , therefore (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='4) ⟨f(⃗α) , g(⃗α)⟩ = � f(⃗β) , g(⃗β) � , for all f, g ∈ D(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Taking f = g in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='4), we get that ∥f(⃗α)∥2 = ∥f(⃗β)∥2 for all f ∈ D(T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' in particular, if f(⃗α) = 0, then f(⃗β) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This implies that, if f1, f2 ∈ D(T) and f1(⃗α) = f2(⃗α) then (f1 − f2)(⃗α) = 0 so (f1 − f2)(⃗β) = 0, and f1(⃗β) = f2(⃗β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This means that there exists a well defined function B, B(f(⃗α)) = f(⃗β), for all f ∈ D(T), B : Bl → Br, where Bl := {f(⃗α) : f ∈ D(T)}, Br := {f(⃗β) : f ∈ D(T)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' 10 DORIN ERVIN DUTKAY∗ AND PALLE E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' JORGENSEN In addition B is a linear isometry between the subspaces Bl and Br of Cn, and, by definition, f(⃗β) = Bf(⃗α), for f ∈ D(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We can define B to be zero on the orthogonal complement of Bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Because of this, we obtain that D(T ) is contained in DB := � f ∈ Dmax : f(⃗α) ∈ Bl, f(⃗β) ∈ Br, Bf(⃗α) = f(⃗β) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We prove that the reverse inclusion also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let f ∈ DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then f(⃗α) ∈ Bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, there exists f0 ∈ D(T) such that f0(⃗α) = f(⃗α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then also f0(⃗β) = Bf0(⃗α) = Bf(⃗α) = f(⃗β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Hence (f − f0)(⃗α) = (f − f0)(⃗β) = 0, and so f − f0 ∈ D0(Ω) ⊆ D(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then f = (f − f0) + f0 ∈ D(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Assume now, conversely, that we are given an partial isometry B from Bl to Br, and we prove that D on DB is symmetric and closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' For symmetry, we use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' for f, g ∈ D(T), we have ⟨f , Dg⟩ = ⟨Df , g⟩ − 1 2πi �� f(⃗β) , g(⃗β) � − ⟨f(⃗α) , g(⃗α)⟩ � = ⟨Df , g⟩ − 1 2πi (⟨Bf(⃗α) , Bg(⃗α)⟩ − ⟨f(⃗α) , g(⃗α)⟩) = ⟨Df , g⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' To see that the operator is closed, take {fn} in D(T) convergent to f in L2(Ω), and {f ′ n} convergent to g in L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , n} and x ∈ Ji, we have fn(x) = fn(αi) + � x αi f ′ n(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' On the left hand side {fn} converges to f in L2(Ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' on the right hand side � x αi f ′ n(t) dt converges to � x αi g(t) dt for all x ∈ Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Therefore we obtain that {fn(αi)} converges to ci = f(x) − � x αi g(t) dt, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' But then f(x) = ci + � x αi g(t) dt for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' x and therefore f ′ = g a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', so the operator is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Consider now the extension T = D on DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We will compute its adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' For g ∈ D(T ∗), we have that DB ∋ f �→ ⟨Df , g⟩ is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' In particular, it is bounded on D0(Ω) so g ∈ Dmax and T ∗g = Dg, ⟨Df , g⟩ = ⟨f , T ∗g⟩ = ⟨f , Dg⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, with integration by parts (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='3), we have that � f(⃗α) , B∗g(⃗β) � = � Bf(⃗α) , g(⃗β) � = � f(⃗β) , g(⃗β) � = ⟨f(⃗α) , g(⃗α)⟩ , for all f ∈ DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This implies that � B∗g(⃗β) − g(⃗α) � is orthogonal to the subspace Bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Conversely, if g ∈ Dmax and � B∗g(⃗β) − g(⃗α) � ⊥ Bl, then from the previous computation, and using integration by parts (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='3), we get that, for f ∈ DB, | ⟨Df , g⟩ | = | ⟨f , Dg⟩ | ≤ ∥f∥∥Dg∥, and so the linear map DB ∋ f �→ ⟨Df , g⟩ is bounded and g ∈ D(T ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Now, let’s consider the case when the extension T is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' In this case D(T ∗) is contained in D(T), and therefore, if g ∈ Dmax with B∗g(⃗β) − g(⃗α) orthogonal to Bl, then we have that Bg(⃗α) = g(⃗β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We will prove that Bl must be the entire space Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Suppose there exists a non-zero vector v orthogonal to Bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, since B is a partial isometry, there exists a non-zero vector w orthogonal to Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let g ∈ Dmax such that g(⃗α) = 0 and g(⃗β) = w (for example, make g piecewise linear on the intervals Ji).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then B∗g(⃗β) − ⃗g(α) = 0 − 0 = 0 ⊥ Bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Thus g ∈ D(T ∗) = D(T), and THE MOMENTUM OPERATOR ON A UNION OF INTERVALS 11 therefore Bg(⃗α) = g(⃗β), which implies that B(0) = w, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Thus, when the extension is self-adjoint, we get that Br = Cn and Bl = Cn and B is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' For the converse, if B is unitary and Bl = Cn, Br = Cn, then, g ∈ D(T ∗), if and only if B∗g(⃗β) − g(⃗α) is orthogonal to Bl so it must be zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', B∗g(⃗β) = g(⃗α), which is equivalent to Bg(⃗α) = g(⃗β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This means that D(T ∗) = D(T) = DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' For a self-adjoint extension A of the operator D|C∞ c (Ω) as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2, we call the unitary matrix B, the boundary matrix associated to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Spectral decomposition Having a self-adjoint extension A of the momentum operator D, we can use the Spectral Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='6 to obtain a spectral resolution of the self-adjoint operator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We present in this section an explicit description of this spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' For ⃗z = (z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , zn) ∈ Cn denote by E(⃗z), the n × n diagonal matrix with entries (e2πiz1, e2πiz2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , e2πizn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let A be a self-adjoint extension of the operator D|C∞ c (Ω) and let B its unitary boundary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let P be the spectral measure for the operator A, so A = � R t dP(t) Then the spectral measure is atomic, supported on the spectrum σ(A) = � λ ∈ C : det(I − E(λ⃗β)−1BE(λ⃗α)) = 0 � ⊆ R which is a discrete unbounded set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' For λ ∈ σ(A), the eigenspace P({λ})L2(Ω) has dimension at most n, and it consists of functions of the form f(x) = e2πiλx n � i=1 ciχJi(x), where c = (ci)n i=1 ∈ Cn, and BE(λ⃗α)c = E(λ⃗β)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We begin with a proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let A be a self-adjoint extension as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let λ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The following statements are equivalent: (i) λ is in the resolvent set of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (ii) The operator A − λI is onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (iii) The matrix E(λ⃗β)−1BE(λ⃗α) − I is onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (iv) The operator A − λI in one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (v) The matrix E(λ⃗β)−1BE(λ⃗α) − I is one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We prove that (ii) and (iii) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The operator A − λI is onto, means that for every g ∈ L2(Ω), there exists f ∈ DB = D(A), such that 1 2πif ′ − λf = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' 12 DORIN ERVIN DUTKAY∗ AND PALLE E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' JORGENSEN We solve this first order linear differential equation on each interval Ji of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We have f ′ − 2πiλf = 2πig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Multiplying by the integrating factor e−2πiλx, we get � e−2πiλtf(t) �′ = 2πig(t)e−2πiλt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Integrating, we get the general solution f(x) = e2πiλx � 2πi � x αi g(t)e−2πiλt dt + ci � , for some constant ci, for all x ∈ Ji, and all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Since g is in L2(Ω), we see that f is absolutely continuous and f ′ = 2πi(λf + g) is in L2(Ω), which means that f is in the domain Dmax, and the only thing that we have to insure is that B ⃗f(α) = ⃗f(β), by picking the right constants (ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let’s see what the condition B ⃗f(α) = ⃗f(β) means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We have f(αi) = cie2πiλαi, f(βi) = e2πiλβi(Ai + ci), where Ai = 2πi � βi αi g(t)e−2πiλt dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let ⃗A = (Ai)n i=1, ⃗c = (ci)n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Note that, by varying g, any vector in Cn can be obtained as ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, the condition f(⃗β) = Bf(⃗α) is equivalent to E(λ⃗β)( ⃗A + ⃗c) = BE(λ⃗α)⃗c, or, equivalently, ⃗A = (E(λ⃗β)−1BE(λ⃗α) − I)⃗c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This shows that, the operator A − λI is onto if and only if the matrix E(λ⃗β)−1BE(λ⃗α) − I is onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Next, we prove that (iv) and (v) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The operator A−λI is not one-to-one means that there exists a non-zero f ∈ DB with ⃗f(β) = B ⃗f(α), such that 1 2πif ′ = λf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Solving this differential equation on each interval Ji, we obtain that f(x) = cie2πiλx, for x ∈ Ji, for some constant ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, the relation ⃗f(β) = B ⃗f(α) implies that, (cie2πiλβi)n i=1 = B(cie2πiλαi)n i=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' with ⃗c := (ci)n i=1, this can be rewritten as (E(λ⃗β)−1BE(λ⃗α) − I)⃗c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Thus the operator A − λI if and only if the matrix E(λ⃗β)−1BE(λ⃗α) − I is not one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Finally, the statements (iii) and (v) are equivalent because they refer to a square matrix in a finite dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Now, since A is self-adjoint, so also closed, λ ∈ ρ(A) if and only if A − λI is both one-to-one and onto;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' but these two properties are equivalent, so (i) is equivalent to all the other statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' □ Returning to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2, we see that λ is in the spectrum of A if and only if det(I − E(λ⃗β)−1BE(λ⃗α)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This is an analytic function of λ, therefore the zero set is discrete and at most countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' From the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='3, we see that the eigenspace P({λ})L2(Ω) is as in the statement of the theorem, and hence has dimension at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Since, by the Spectral Theorem the orthogonal sum of the eigenspaces spans the entire Hilbert space L2(Ω), it follows that σ(A) cannot be finite, and since it is discrete, it has to be unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' □ THE MOMENTUM OPERATOR ON A UNION OF INTERVALS 13 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let A be as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2 and let {λn : n ∈ Z} a list of the eigenvalues of A repeated according to multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let {ǫn : n ∈ Z} be an orthonormal basis of eigenvectors for A, Aǫn = λnǫn, for all n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='1) D(A) = � f = � n∈Z fnǫn ∈ L2(Ω) : � n∈Z |λn|2|fn|2 < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let g = � n∈Z anǫn ∈ L2(Ω) with � n∈Z |λn|2|an|2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, for f ∈ D(A), we have ⟨Df , g⟩ = � n an ⟨Df , ǫn⟩ = � n an ⟨f , D∗ǫn⟩ = � n anλn ⟨f , ǫn⟩ = � f , � n λnanǫn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This implies that g ∈ D(A∗) = D(A) and Ag = A∗g = � n λnanǫn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Thus DI := {� n anǫn : � n |λn|2|an|2 < ∞} is contained in D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' But, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='12, the diagonal operator MI(� n anǫn) = � n λnanǫn defined on DI is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Since self-adjoints operators are maximally symmetric, it follows that DI = D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The unitary group If we have a self-adjoint extension A of the momentum operator D, we can associate to it a one-parameter unitary group U(t) = exp(2πitA), t ∈ R as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' In this section we present some basic properties of this unitary group and show that it acts as translations inside the intervals and it splits points at the endpoints, with probabilities given by the boundary matrix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let A = D on DB a self-adjoint extension with boundary matrix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let U(t) = exp (2πitA), (t ∈ R), be the associated one-parameter unitary group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (i) The domain DB is invariant for U(t) for all t ∈ R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', if f ∈ Dmax with Bf(⃗α) = f(⃗β), then U(t)f ∈ Dmax with B(U(t)f)(⃗α) = (U(t)f)(⃗β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (ii) Fix i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , n} and let t ∈ R such that Ji ∩ (Ji − t) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, for f ∈ L2(Ω), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='1) (U(t)f)(x) = f(x + t), for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' x ∈ Ji ∩ (Ji − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' In particular, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2) (U(βi − x)f)(x) = f(βi), (U(αi − x)f)(x) = f(αi), for f ∈ Dmax, x ∈ Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (iii) For f ∈ Dmax, if x ∈ Ji and t > βi − x, then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='3) [U(t)f] (x) = πi (B [U(t − (βi − x))f] (⃗α)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Here πi : Cn → C denotes the projection onto the i-th component π(x1, x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , xn) = xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (i) This follows from more general rules, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='9(c), but we include a more direct proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' With the notation as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='4, f = � n fnǫn is in the domain of A, if and only if � n |λn|2|fn|2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, for t ∈ R, U(t)f = � n fne2πiλntǫn, and � n |λn|2|fne2πiλnt|2 = � n |λn|2|fn|2 < ∞, which means that U(t)f is also in the domain of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' 14 DORIN ERVIN DUTKAY∗ AND PALLE E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' JORGENSEN (ii) Let vλ be an eigenvector for A with eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2, we have vλ(x) = n � k=1 ckχJk(x)e2πiλx, (x ∈ Ω), for some constants ck ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, for x ∈ Ji ∩ (Ji − t), we have x, x + t ∈ Ji, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='4) (U(t)vλ)(x) = e2πiλtvλ(x) = n � k=1 ckχJk(x)e2πiλ(x+t) = vλ(x + t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Now let f ∈ L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' One can find a sequence {fn} of finite linear combinations of eigenvectors, such that lim fn = f in L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then lim U(t)fn = U(t)f, passing to subsequences, we can assume in addition that {fn} converges to f pointwise a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Ω, and U(t)fn converges to U(t)f pointwise a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' x ∈ Ji ∩ (Ji − t), we have (U(t)f)(x) = lim(U(t)fn)(x) = lim fn(x + t) = f(x + t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (Note that we used also the fact that translation by t preserves measure zero sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=') The first relation in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2) follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='4), by taking t = βi −x−ǫ and letting ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Similarly for the second relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' (iii) Indeed, we have, with (i) and (ii), [U(t)f] (x) = [U(βi − x)U(t − (βi − x))f] (x) = [U(t − (βi − x))f] (βi) = πi � [U(t − (βi − x))f] (⃗β) � = πi (B [U(t − (βi − x))f] (⃗α)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Spectral sets In this section we consider the case when Ω is a spectral set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We present various characterizations of this property in terms of the self-adjoint extensions of the momentum operator D and in terms of the associated unitary groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Assume that Ω is a spectral set with spectrum Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Recall that eλ denotes the exponential function eλ(x) = e2πiλx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' In order to make the vectors eλ in L2(Ω) of norm one, we renormalize the Lebesgue measure on Ω by 1 |Ω| dx (or we can simply assume that Ω has measure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The Fourier transform (associated to the spectrum Λ) is the unitary operator (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='1) FΛ : L2(Ω) → l2(Λ), FΛf = (⟨f , eλ⟩)λ∈Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Define also the unbounded operator of multiplication by the identity function on l2(Ω): (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2) MI(aλ)λ∈Λ = (λaλ)λ∈Λ, D(MI) = � (aλ)λ∈Λ ∈ l2(Λ) : � λ∈Λ |λ|2|aλ|2 < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Define the unitary group associated to Λ on L2(Ω), by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='3) UΛ(t) �� λ∈Λ aλeλ � = � λ∈Λ e2πiλtaλeλ, for � λ∈Λ aλeλ ∈ L2(Ω), t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' THE MOMENTUM OPERATOR ON A UNION OF INTERVALS 15 Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' A unitary group of local translations on Ω is a strongly continuous one parameter unitary group U(t) on L2(Ω) with the property that, for any f ∈ L2(Ω) and any t ∈ R, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='4) (U(t)f)(x) = f(x + t) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' x ∈ Ω ∩ (Ω − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Note the difference between the Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2, and the property (ii) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2 is a stronger condition, because it allows jumps between different intervals Ji of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' A unitary boundary matrix B is called spectral if, for every λ ∈ R, the equation BEλ(⃗α)c = Eλ(⃗β)c, c ∈ Cn has either only the trivial solution c = 0, or only constant solutions of the form c = α(1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , 1), α ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Assume the Ω is a spectral set with spectrum Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Define the unbounded operator A on L2(Ω) by A �� λ∈Λ fλeλ � = � λ∈Λ λfλeλ, D(A) = � f = � λ∈Λ fλeλ : � λ∈Λ |λ|2|fλ|2 < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, the operator A is conjugate to the multiplication operator MI, by the Fourier transform, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='5) A = F−1 Λ MIFΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The domain D(A) contains D0(Ω) and all functions eλ, λ ∈ Λ, and A is a self-adjoint extension of D|C∞ c (Ω) with the property that all eigenvectors are constant multiples of exponential functions ceλ, c ∈ C, λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Conversely, if there exists a self-adjoint extension A of D|C∞ c (Ω) with the property that all eigen- vectors are constant multiples of exponential functions ceλ, c ∈ C, λ ∈ R, then Ω is spectral, with spectrum (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='6) Λ := {λ ∈ R : eλ ∈ D(A)} = σ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='5) follows from a direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' To see that A is self-adjoint, it is enough to check that MI is self-adjoint, and this is a consequence of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Clearly, the exponential functions eλ are in the domain D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let’s check that D0(Ω) is also contained in D(A), and Af = Df for f ∈ D0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let f = � λ fλeλ ∈ D0(Ω) and λ ∈ Λ(⊆ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, cλ := ⟨Df , eλ⟩ = ⟨f , D∗eλ⟩ = � f , 1 2πie′ λ � = ⟨f , λeλ⟩ = λ ⟨f , eλ⟩ = λfλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Therefore, � λ |λ|2|fλ|2 = � λ |cλ|2 = ∥Df∥2 < ∞, and Df = � λ λfλeλ = Af.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This shows that A is indeed a self-adjoint extension of D|C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Now we check that all eigenvectors of A are constant multiples of eλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This follows from the next Lemma, and the fact that self-adjoint operators are closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let A be a closed unbounded operator on a Hilbert space H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Assume that there exists an orthonormal basis of H, {eλ}λ∈Λ, where Λ ⊂ C and Aeλ = λeλ, for all λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then every 16 DORIN ERVIN DUTKAY∗ AND PALLE E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' JORGENSEN eigenvector f = � λ fλeλ with the property that � λ |λ|2|fλ|2 < ∞ is of the form ceλ, for some c ∈ C, λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Suppose f = � λ fλeλ is an eigenvector for A with eigenvalue λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then Af = λ0f = � λ λ0fλeλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' On the other hand , we have that � finite fλeλ converges in L2(Ω) to � λ fλeλ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' also A \uf8eb \uf8ed � finite fλeλ \uf8f6 \uf8f8 = � finite λfλeλ → � λ λfλeλ, because � λ |λ|2|fλ|2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Since A is closed we get Af = A �� λ fλeλ � = � λ fλeλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This means that λ0fλ = λfλ for all λ ∈ Λ, which implies that either λ0 = λ or fλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Thus all the coefficients fλ are zero except for fλ0, so f = fλ0eλ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' □ For the converse in the Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='5, assume A is a self-adjoint extension of D|C∞ c (Ω) with the property that all eigenvectors are constant multiples of exponential functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2, for λ in the spectrum σ(A) =: Λ, the subspace P({λ})L2(Ω) is one-dimensional, spanned by eλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then the set of all eigenvectors for A, {eλ : λ ∈ Λ} is an orthonormal basis for L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Moreover, if eλ ∈ D(A) for some λ ∈ R, then Aeλ = 1 2πie′ λ = λeλ, so (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='6) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Assume that Ω is spectral with spectrum Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Let A = AΛ be the self-adjoint extension of D|C∞ c (Ω) defined in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='5, and let B = BΛ be the unitary boundary matrix associated to this extension as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then B is a spectral boundary matrix and it is uniquely and well-defined by the conditions (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='7) Beλ(⃗α) = eλ(⃗β), for all λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Moreover (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='8) span{eλ(⃗α) : λ ∈ Λ} = span{eλ(⃗β) : λ ∈ Λ} = Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Conversely, if there exists a spectral boundary matrix B, then Ω is spectral with spectrum (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='9) Λ = {λ ∈ R : Beλ(⃗α) = eλ(⃗β)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='5, the only eigenvectors of the operator A are multiples of eλ, λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' On the other hand, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2, the eigenvectors are functions of the form f = n � i=1 (ciχJi) eλ, where c ∈ Cn, with BEλ(⃗α)c = Eλ(⃗β)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Thus, if BEλ(⃗α)c = Eλ(⃗β)c for some non-zero c ∈ Cn, then f is an eigenvector, but then it must be a constant multiple of eλ so all the components of c are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This means that B is a spectral boundary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' THE MOMENTUM OPERATOR ON A UNION OF INTERVALS 17 We check now that the relation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='7) completely determines B as a well-defined unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Since Λ is a spectrum, for λ ̸= λ′ in Λ we have 0 = ⟨eλ , eλ′⟩ = n � k=1 � βk αk e2πi(λ−λ′)x dx = n � k=1 1 2πi(λ − λ′) � e2πi(λ−λ′)βk − e2πi(λ−λ′)αk � , which means that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='10) ⟨eλ(⃗α) , eλ′(⃗α)⟩ = � eλ(⃗β) , eλ′(⃗β) � , for all λ, λ′ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Define the linear operator B from span{eλ(⃗α) : λ ∈ Λ} to span{eλ(⃗β) : λ ∈ Λ}, by B �� λ∈Λ aλeλ(⃗α) � = � λ∈Λ aλeλ(⃗β), where only finitely many coefficients are non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The operator is well-defined, because, if � λ aλeλ(⃗α) = 0, then 0 = �� λ aλeλ(⃗α) , � λ aλeλ(⃗α) � = � λ,λ′ aλaλ′ ⟨eλ(⃗α) , eλ′(⃗α)⟩ = � λ,λ′ aλaλ′ � eλ(⃗β) , eλ′(⃗β) � = �� λ aλeλ(⃗β) , � λ aλeλ(⃗β) � A similar argument shows that B is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Next we prove that Bl := span{eλ(⃗α) : λ ∈ Λ} = Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Since B is unitary, the same will be true for ⃗β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We proceed by contradiction, if Bl is not the entire space Cn then it has a non-trivial orthogonal complement, and therefore we can extend the partial isometry B, in two different ways to unitaries ˜B and ˜B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Both of them give rise, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2, to self-adjoint extensions of D|C∞ c (Ω), and, since Beλ(⃗α) = eλ(⃗β), we have eλ ∈ D ˜B ∩ D ˜B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='6, all eigenvectors are multiples of eλ, λ ∈ Λ, and with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='4, we get that D ˜B = �� λ fλeλ : � λ |λ|2|fλ|2 < ∞ � = D ˜B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' But this means that ˜B = ˜B′, a contradiction, and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='8) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' For the converse, if a spectral unitary boundary matrix is given, then the self-adjoint extension associated to it has all eigenvectors of the form ceλ, with c ∈ C and λ ∈ Λ as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='9), and they form an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Thus Ω is spectral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Assume that Ω is spectral with spectrum Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then the unitary group U = UΛ associated to Λ is a unitary group of local translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' In addition, if A is the self-adjoint extension associated to Λ as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='5, then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='11) U(t) = exp(2πitA), (t ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Conversely, if there exists a unitary group of local translations (U(t))t∈R for Ω, then Ω is spectral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' 18 DORIN ERVIN DUTKAY∗ AND PALLE E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' JORGENSEN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Note first that U(t)eλ = e2πiλteλ, for all λ ∈ Λ and t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, for t ∈ R and x ∈ Ω∩(Ω−t), we have (U(t)eλ)(x) = e2πiλte2πiλx = e2πiλ(x+t) = eλ(x + t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Now fix t ∈ R and let f ∈ L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We want to check (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Since {eλ : λ ∈ Λ} form an orthonormal basis for L2(Ω), we can find a sequence of functions {fn} which are finite linear combinations of function eλ, such that {fn} converges to f in L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Passing to a subsequence, we can assume that {fn} converges to f almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Since U(t) is unitary, {U(t)fn} converges to U(t)f in L2(Ω), and again passing to a subsequence we can assume in addition that {U(t)fn} converges to U(t)f almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We have (U(t)fn)(x) = fn(x+t) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' x ∈ Ω∩(Ω−t), for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Taking the limit (U(t)f)(x) = f(x + t) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e x ∈ Ω ∩ (Ω − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This proves (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='4) so U is a unitary group of local translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The relation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='11) follows immediately, because we have the operators A and U(t) in diagonal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Assume now that U(t) is a unitary group of local translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' By Stone’s Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='10, there exists a self-adjoint operator A such that U(t) = exp(2πitA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We claim that A is a self-adjoint extension of D|C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We will prove that, for f ∈ C∞ c (Ω), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='12) 1 2πit(U(t)f − f) converges in L2(Ω) to Df as t → 0, which, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='9(b) and (c), implies that f is in the domain of A and Af = Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This is to be expected, since, especially for small values of t, the operator U(t) acts as a translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We need a Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Assume that the function f ∈ L2(Ω) is supported on the set Ωǫ = ∪n i=1[αi + ǫ, βi − ǫ] for some ǫ > 0, and that |t| < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='13) (U(t)f)(x) = f(x + t) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' x ∈ Ω, where f(x) := 0 for x not in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' If |t| < ǫ, then Ωǫ−t ⊂ (Ω∩(Ω−t)), and therefore (U(t)f)(x) = f(x+t), for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' x ∈ Ωǫ −t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We prove that g(x) := (U(t)f)(x) = 0 for x ∈ Ω \\ (Ωǫ − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' We have ∥f∥2 L2(Ω) = ∥U(t)f∥2 L2(Ω) = � Ωǫ−t |f(x + t)|2 dx + � Ω\\(Ωǫ−t) |g(x)|2 dx = � Ωǫ |f(x)|2 dx + � Ω\\(Ωǫ−t) |g(x)|2 dx = ∥f∥2 L2(Ω) + ∥g∥2 L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This implies that g is 0 and we obtain the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' □ Take now f ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This means that f is supported on a set Ωǫ for some ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='9, for |t| < ǫ and for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' x ∈ Ω, we have 1 2πit((U(t)f)(x) − f(x)) = 1 2πit(f(x + t) − f(x)), which converges uniformly to 1 2πif ′(x) (since f ∈ C∞ c (Ω)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='12) follows and therefore A is a self-adjoint extension of D|C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' With Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='2, the eigenvectors for A are of the form f = (�n i=1 ciχJi) eλ, Af = λf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then U(t)f = e2πitλf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' THE MOMENTUM OPERATOR ON A UNION OF INTERVALS 19 Since U(t) is a unitary group of local translations, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' x ∈ Ω ∩ (Ω − t), we have e2πitλ � n � i=1 ciχJi(x) � eλ(x) = (U(t)f)(x) = f(x + t) = � n � i=1 ciχJi(x + t) � eλ(x + t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Fix i ̸= k in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' , n} and choose t such that Ji ∩ (Jk − t) ̸= ∅ and then pick x ∈ Ji ∩ (Jk − t) such that the previous relation holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Then, we get e2πitλcie2πiλx = cke2πiλ(x+t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' This means that ci = ck, and thus f = c1eλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Therefore, all the eigenvectors of A are multiples of eλ, and, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='5, it follows that Ω is spectral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' □ Concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' As noted in the body of our paper, our present focus for the Fuglede conjecture is based on our particular choices of notions from geometry, harmonic analysis, and from spectral theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Indeed, we have made these definite choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Naturally, there are others, and readers will be able to review such alternative approaches in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' each one serving its purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' As a guide to the relevant papers, we conclude here with the following list of citations [Fug74, Fug67, FM21, IKL+21, JPT16, PPTW15, JPT15, Lev22, PL22, LM22, IP98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The paper does not generate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The authors do not have any conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' References [Bir22] Philipp Birklbauer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' The Fuglede conjecture holds in Z3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Exp.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' [Fug67] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Fuglede.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' On the relation PQ − QP = −iI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Scand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', 20:79–88, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', 228(2):385–420, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' [Mat05] M´at´e Matolcsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Fuglede’s conjecture fails in dimension 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Math.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Simon Stevin, 27(4):481–488, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' [Ped87] Steen Pedersen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Spectral theory of commuting selfadjoint partial differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', 73(1):122–134, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' [PL22] Alberto Debernardi Pinos and Nir Lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Gabor orthonormal bases, tiling and periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', 384(3-4):1461–1467, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' [PPTW15] Steen Pedersen, Jason D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Phillips, Feng Tian, and Cody E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Watson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' On the spectra of momentum operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Complex Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Theory, 9(7):1557–1587, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' [Tao04] Terence Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Fuglede’s conjecture is false in 5 and higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', 11(2-3):251–258, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=' [Dorin Ervin Dutkay] University of Central Florida, Department of Mathematics, 4000 Central Florida Blvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FIT4oBgHgl3EQf2isN/content/2301.11377v1.pdf'} +page_content='O.' metadata={'source': 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